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Deligne's theorem on tensor categories

Contents

Context

Representation theory

Monoidal categories

Super-Algebra and Super-Geometry

Contents

Idea

Deligne’s theorem on tensor categories (Deligne 02, recalled as theorem below) establishes Tannaka duality between linear tensor categories in characteristic zero subject to just a mild size constraint (subexponential growth, def. below), and supergroups (“supersymmetries”), realizing these tensor categories as categories of representations of these supergroups.

Relevance

For supersymmetry

Since the concept of linear tensor categories arises very naturally in mathematics, the theorem gives a purely mathematical “reason” for the relevance of superalgebra and supergeometry. It is reasonable to wonder why of all possible generalizations of commutative algebra, it is supercommutative superalgebras that are singled out (from alternatives such as plain /2\mathbb{Z}/2-graded algebras, or in fact /n\mathbb{Z}/n-graded algebras or general noncommutative algebras or the like), as they are notably in theoretical physics (“supersymmetry”), but also in mathematical fields such as spin geometry (e.g. via the relation between Majorana spinors and supersymmetry, here) and topological K-theory (for instance via its incarnation as Karoubi K-theory, or via the descriptioon of twisted K-theory by super line 2-bundles).

But with kk-linear tensor categories appearing on general abstract grounds as the canonical structure to consider in representation theory, Deligne’s theorem serves to base supercommutative superalgebra on this same general abstract foundation, showing that this is precisely the context in which full kk-linear tensor categories exhibit full Tannaka duality.

More concretely, in quantum field theory, under the Wigner classification, fundamental particles are identified with irreducible representations of the isometry group of the local model of spacetime (which are induced from finite dimensional representations of the “Wigner's little group” (Mackey 68) ). Forming the tensor product of two such representations corresponds to combining them as two subsystems of a joint system. Therefore it is natural to demand that physical particle species should form complex-linear tensor categories. Deligne’s theorem then gives that supersymmetry is the most general context in which this works out. (In physics the irreducible representation in this context here are called the supermultiplets.)

More exposition of this point is at:

For Tannaka duality of Hopf algebras

By Tannaka duality rigid monoidal category symmetric monoidal categories in general are categories of modules of triangular Hopf algebras. Hence Deligne’s theorem here implies that those triangular Hopf algebras over algebraically closed fields of characteristic zero whose category of representation has subexponential growth (def. below) are equivalent to supercommutative Hopf algebras. See (Etingof-Gelaki 02) for more.

Tannaka duality for categories of modules over monoids/associative algebras

monoid/associative algebracategory of modules
AAMod AMod_A
RR-algebraMod RMod_R-2-module
sesquialgebra2-ring = monoidal presentable category with colimit-preserving tensor product
bialgebrastrict 2-ring: monoidal category with fiber functor
Hopf algebrarigid monoidal category with fiber functor
hopfish algebra (correct version)rigid monoidal category (without fiber functor)
weak Hopf algebrafusion category with generalized fiber functor
quasitriangular bialgebrabraided monoidal category with fiber functor
triangular bialgebrasymmetric monoidal category with fiber functor
quasitriangular Hopf algebra (quantum group)rigid braided monoidal category with fiber functor
triangular Hopf algebrarigid symmetric monoidal category with fiber functor
supercommutative Hopf algebra (supergroup)rigid symmetric monoidal category with fiber functor and Schur smallness
form Drinfeld doubleform Drinfeld center
trialgebraHopf monoidal category

2-Tannaka duality for module categories over monoidal categories

monoidal category2-category of module categories
AAMod AMod_A
RR-2-algebraMod RMod_R-3-module
Hopf monoidal categorymonoidal 2-category (with some duality and strictness structure)

3-Tannaka duality for module 2-categories over monoidal 2-categories

monoidal 2-category3-category of module 2-categories
AAMod AMod_A
RR-3-algebraMod RMod_R-4-module

Background

This section provides exposition of the necessary background for the statement of Deligne’s theorem (theorem below).

We start by introducing the basic concepts of tensor categories along with the basic examples of vector spaces and super vector spaces:

This allows to speak of commutative algebra internal to tensor categories. Specializing this to the tensor category of super vector spaces yields supercommutative superalgebras. The formal duals of these are the affine super schemes. This we discuss in

Next we introduce the concept of commutative monoids equipped with the structure of a commutative Hopf algebras and explain how these are formal duals to groups. Then we use this to motivate and explain the concept of (affine algebraic) supergroups as formal duals to commutative Hopf algebras internal to the tensor category of super vector spaces, namely supercommutative Hopf algebras:

Finally we discuss how under this relation linear representations of groups correspond to comodules over their formally dual commutative Hopf algebras, and we introduce the key class of categories of interest here: tensor-categories of representations of groups and of super-representations of super-groups:

Tensor products and Super vector spaces

Definition

For kk a field, we write Vect kVect_k for the category whose

When the ground field kk is understood or when its precise nature is irrelevant, we will often notationally suppress it and speak of just the category Vect of vector spaces.

This is the category inside which linear algebra takes place.

Of course the category Vect has some special properties. Not only are its objects “linear spaces”, but the whole category inherits linear structure of sorts. This is traditionally captured by the following terminology for additive and abelian categories. Notice that there are several different but equivalent ways to state the following properties (discussed behind the relevant links).

Definition

Let 𝒞\mathcal{C} be a category.

  1. Say that 𝒞\mathcal{C} has direct sums if it has finite products and finite coproducts and if the canonical comparison morphism between these is an isomorphism. We write VWV \oplus W for the direct sum of two objects of 𝒞\mathcal{C}.

  2. Say that 𝒞\mathcal{C} is an additive category if it has direct sums and in addition it is enriched in abelian groups, meaning that every hom-set is equipped with the structure of an abelian group such that composition of morphisms is a bilinear map.

  3. Say that 𝒞\mathcal{C} is an abelian category if it is an additive category and has property that its monomorphisms are precisely the inclusions of kernels and its epimorphisms are precisely the projections onto cokernels.

We also make the following definition of kk-linear category, but notice that conventions differ as to which extra properties beyond Vect-enrichment to require on a linear category:

Definition

For kk a field (or more generally just a commutative ring), call a category 𝒞\mathcal{C} a kk-linear category if

  1. it is an abelian category (def. );

  2. its hom-sets have the structure of kk-vector spaces (generally kk-modules) such that composition of morphisms in 𝒞\mathcal{C} is a bilinear map

and the underlying additive abelian group structure of these hom-spaces is that of the underlying abelian category.

In other words, a kk-linear category is an abelian category with the additional structure of a Vect-enriched category (generally kkMod-enriched) such that the underlying Ab-enrichment according to def. is obtained from the VectVect-enrichment under the forgetful functor VectAbVect \to Ab.

A functor between kk-linear categories is called a kk-linear functor if its component functions on hom-sets are linear maps with respect to the given kk-linear structure, hence if it is a Vect-enriched functor.

Example

The category Vect k{}_k of vector spaces (def. ) is a kk-linear category according to def. .

Here the abstract direct sum is the usual direct sum of vector spaces, whence the name of the general concept.

For V,WV,W two kk-vector spaces, the vector space structure on the hom-set Hom Vect(V,W)Hom_{Vect}(V,W) of linear maps ϕ:VW\phi \colon V \to W is given by “pointwise” multiplication and addition of functions:

(c 1ϕ 1+c 2ϕ 2):vc 1ϕ 1(v)+c 2ϕ 2(v) (c_1 \phi_1 + c_2 \phi_2) \;\colon\, v \;\mapsto\; c_1 \phi_1(v) + c_2 \phi_2(v)

for all c 1,c 2kc_1, c_2 \in k and ϕ 1,ϕ 2Hom Vect(V,W)\phi_1, \phi_2 \in Hom_{Vect}(V,W).

Recall the basic construction of the tensor product of vector spaces:

Definition

Given two vector spaces over some field kk, V 1,V 2Vect kV_1, V_2 \in Vect_k, their tensor product of vector spaces is the vector space denoted

V 1 kV 2Vect V_1 \otimes_k V_2 \;\in\; Vect

whose elements are equivalence classes of tuples of elements (v 1,v 2)(v_1,v_2) with v iV iv_i \in V_i, for the equivalence relation given by

(cv 1,v 2)(v 1,cv 2) (c v_1 , v_2) \;\sim\; (v_1, c v_2)
(v 1+v 1,v 2)(v 1,v 2)+(v 1,v 2) (v_1 + v'_1 , v_2) \; \sim \; (v_1,v_2) + (v'_1, v_2)
(v 1,v 2+v 2)(v 1,v 2)+(v 1,v 2) (v_1 , v_2 + v'_2) \; \sim \; (v_1,v_2) + (v_1, v'_2)

More abstractly this means that the tensor product of vector spaces is the vector space characterized by the fact that

  1. it receives a bilinear map

    V 1×V 2V 1V 2 V_1 \times V_2 \longrightarrow V_1 \otimes V_2

    (out of the Cartesian product of the underlying sets)

  2. any other bilinear map of the form

    V 1×V 2V 3 V_1 \times V_2 \longrightarrow V_3

    factors through the above bilinear map via a unique linear map

    V 1×V 2 bilinear V 3 !linear V 1 kV 2 \array{ V_1 \times V_2 &\overset{bilinear}{\longrightarrow}& V_3 \\ \downarrow & \nearrow_{\mathrlap{\exists ! \, linear}} \\ V_1 \otimes_k V_2 }

The existence of the tensor product of vector spaces, def. , equips the category Vect of vector spaces with extra structure, which is a “categorification” of the familiar structure of a semi-group. One also says “monoid” for semi-group and therefore categories equipped with a tensor product operation are also called monoidal categories:

Definition

A monoidal category is a category 𝒞\mathcal{C} equipped with

  1. a functor

    :𝒞×𝒞𝒞 \otimes \;\colon\; \mathcal{C} \times \mathcal{C} \longrightarrow \mathcal{C}

    out of the product category of 𝒞\mathcal{C} with itself, called the tensor product,

  2. an object

    1𝒞 1 \in \mathcal{C}

    called the unit object or tensor unit,

  3. a natural isomorphism

    a:(()())()()(()()) a \;\colon\; ((-)\otimes (-)) \otimes (-) \overset{\simeq}{\longrightarrow} (-) \otimes ((-)\otimes(-))

    called the associator,

  4. a natural isomorphism

    :(1())() \ell \;\colon\; (1 \otimes (-)) \overset{\simeq}{\longrightarrow} (-)

    called the left unitor, and a natural isomorphism

    r:()1() r \;\colon\; (-) \otimes 1 \overset{\simeq}{\longrightarrow} (-)

    called the right unitor,

such that the following two kinds of diagrams commute, for all objects involved:

  1. triangle identity:

    (x1)y a x,1,y x(1y) ρ x1 y 1 xλ y xy \array{ & (x \otimes 1) \otimes y &\stackrel{a_{x,1,y}}{\longrightarrow} & x \otimes (1 \otimes y) \\ & {}_{\rho_x \otimes 1_y}\searrow && \swarrow_{1_x \otimes \lambda_y} & \\ && x \otimes y && }
  2. the pentagon identity:

    (wx)(yz) α wx,y,z α w,x,yz ((wx)y)z (w(x(yz))) α w,x,yid z id wα x,y,z (w(xy))z α w,xy,z w((xy)z) \array{ && (w \otimes x) \otimes (y \otimes z) \\ & {}^{\mathllap{\alpha_{w \otimes x, y, z}}}\nearrow && \searrow^{\mathrlap{\alpha_{w,x,y \otimes z}}} \\ ((w \otimes x ) \otimes y) \otimes z && && (w \otimes (x \otimes (y \otimes z))) \\ {}^{\mathllap{\alpha_{w,x,y}} \otimes id_z }\downarrow && && \uparrow^{\mathrlap{ id_w \otimes \alpha_{x,y,z} }} \\ (w \otimes (x \otimes y)) \otimes z && \underset{\alpha_{w,x \otimes y, z}}{\longrightarrow} && w \otimes ( (x \otimes y) \otimes z ) }

As expected, we have the following basic example:

Example

For kk a field, the category Vect k{}_k of kk-vector spaces becomes a monoidal category (def. ) as follows

  • the abstract tensor product is the tensor product of vector spaces k\otimes_k from def. ;

  • the tensor unit is the field kk itself, regarded as a 1-dimensional vector space over itself;

  • the associator is the map that on representing tuples acts as

    α V 1,V 2,V 3:((v 1,v 2),v 3)(v 1,(v 2,v 3)) \alpha_{V_{1}, V_2, V_3} \;\colon\; ((v_1, v_2), v_3) \mapsto (v_1, (v_2,v_3))
  • the left unitor is the map that on representing tuples is given by

    V:(k,v)kv \ell_{V} \colon (k,v) \mapsto k v

    and the right unitor is similarly given by

    r V:(v,k)kv. r_V \colon (v,k) \mapsto k v \,.

That this satisifes the pentagon identity (def. ) and the left and right unit identities is immediate on representing tuples.

But the point of the abstract definition of monoidal categories is that there are also more exotic examples. The followig one is just a minimal enrichment of example , and yet it will be important.

Example

Let GG be a group (or in fact just a monoid/semi-group). A GG-graded vector space VV is a direct sum of vector spaces labeled by the elements in GG:

V=gGV g. V = \underset{g \in G}{\oplus} V_g \,.

A homomorphism

ϕ:VW \phi \;\colon\; V \longrightarrow W

of GG-graded vector spaces is a linear map that respects this direct sum structure, hence equivalently a direct sum of linear maps

ϕ g:V gW g \phi_g \;\colon\; V_g \longrightarrow W_g

for all gGg \in G, such that

ϕ=gGϕ g. \phi = \underset{g \in G}{\oplus} \phi_g \,.

This defines a category, denoted Vect GVect^G. Equip this category with a tensor product which on the underlying vector spaces is just the tensor product of vector spaces from def. , equipped with the GG-grading which is obtained by multiplying degree labels in GG:

(VW) gg 1,g 2Gg 1g 2=gV g 1 kV g 2. (V \otimes W)_g \;\coloneqq\; \underset{{g_1, g_2 \in G} \atop {g_1 g_2 = g}}{\oplus} V_{g_1} \otimes_k V_{g_2} \,.

The tensor unit for the tensor product is the ground field kk, regarded as being in the degree of the neutral element eGe \in G

1 g={k |g=e 0 |otherwise. 1_g \;=\; \left\{ \array{ k & | g = e \\ 0 & | otherwise } \right. \,.

The associator and unitors are just those of the monoidal structure on plain vector spaces, from example .

One advantage of abstracting the concept of a monoidal category is that it allows to prove general statements uniformly for all kinds of tensor products, familar ones and more exotic ones. The following lemma and remark are two important such statements.

Lemma

(Kelly 64)

Let (𝒞,,1)(\mathcal{C}, \otimes, 1) be a monoidal category, def. . Then the left and right unitors \ell and rr satisfy the following conditions:

  1. 1=r 1:111\ell_1 = r_1 \;\colon\; 1 \otimes 1 \overset{\simeq}{\longrightarrow} 1;

  2. for all objects x,y𝒞x,y \in \mathcal{C} the following diagrams commutes:

    (1x)y α 1,x,y xid y 1(xy) xy xy; \array{ (1 \otimes x) \otimes y & & \\ {}^\mathllap{\alpha_{1, x, y}} \downarrow & \searrow^\mathrlap{\ell_x \otimes id_y} & \\ 1 \otimes (x \otimes y) & \underset{\ell_{x \otimes y}}{\longrightarrow} & x \otimes y } \,;

    and

    x(y1) α 1,x,y 1 id xr y (xy)1 r xy xy; \array{ x \otimes (y \otimes 1) & & \\ {}^\mathllap{\alpha^{-1}_{1, x, y}} \downarrow & \searrow^\mathrlap{id_x \otimes r_y} & \\ (x \otimes y) \otimes 1 & \underset{r_{x \otimes y}}{\longrightarrow} & x \otimes y } \,;

For proof see at monoidal category this lemma and this lemma.

Remark

Just as for an associative algebra it is sufficient to demand 1a=a1 a = a and a1=aa 1 = a and (ab)c=a(bc)(a b) c = a (b c) in order to have that expressions of arbitrary length may be re-bracketed at will, so there is a coherence theorem for monoidal categories which states that all ways of freely composing the unitors and associators in a monoidal category (def. ) to go from one expression to another will coincide. Accordingly, much as one may drop the notation for the bracketing in an associative algebra altogether, so one may, with due care, reason about monoidal categories without always making all unitors and associators explicit.

(Here the qualifier “freely” means informally that we must not use any non-formal identification between objects, and formally it means that the diagram in question must be in the image of a strong monoidal functor from a free monoidal category. For example if in a particular monoidal category it so happens that the object X(YZ)X \otimes (Y \otimes Z) is actually equal to (XY)Z(X \otimes Y)\otimes Z, then the various ways of going from one expression to another using only associators and this “accidental” equality no longer need to coincide.)

The above discussion makes it clear that a monoidal category is like a monoid/semi-group, but “categorified”. Accordingly we may consider additional properties of monoids/semi-groups and correspondingly lift them to monoidal categories. A key such property is commutativity. But while for a monoid commutativity is just an extra property, for a monoidal category it involves choices of commutativity-isomorphisms and hence is extra structure. We will see below that this is the very source of superalgebra.

The categorification of “commutativity” comes in two stages: braiding and symmetric braiding.

Definition

A braided monoidal category, is a monoidal category 𝒞\mathcal{C} (def. ) equipped with a natural isomorphism

τ x,y:xyyx \tau_{x,y} \;\colon\; x \otimes y \to y \otimes x

(for all objects x,yin𝒞x,y in \mathcal{C}) called the braiding, such that the following two kinds of diagrams commute for all objects involved (“hexagon identities”):

(xy)z a x,y,z x(yz) τ x,yz (yz)x τ x,yId a y,z,x (yx)z a y,x,z y(xz) Idτ x,z y(zx) \array{ (x \otimes y) \otimes z &\stackrel{a_{x,y,z}}{\to}& x \otimes (y \otimes z) &\stackrel{\tau_{x,y \otimes z}}{\to}& (y \otimes z) \otimes x \\ \downarrow^{\tau_{x,y}\otimes Id} &&&& \downarrow^{a_{y,z,x}} \\ (y \otimes x) \otimes z &\stackrel{a_{y,x,z}}{\to}& y \otimes (x \otimes z) &\stackrel{Id \otimes \tau_{x,z}}{\to}& y \otimes (z \otimes x) }

and

x(yz) a x,y,z 1 (xy)z τ xy,z z(xy) Idτ y,z a z,x,y 1 x(zy) a x,z,y 1 (xz)y τ x,zId (zx)y, \array{ x \otimes (y \otimes z) &\stackrel{a^{-1}_{x,y,z}}{\to}& (x \otimes y) \otimes z &\stackrel{\tau_{x \otimes y, z}}{\to}& z \otimes (x \otimes y) \\ \downarrow^{Id \otimes \tau_{y,z}} &&&& \downarrow^{a^{-1}_{z,x,y}} \\ x \otimes (z \otimes y) &\stackrel{a^{-1}_{x,z,y}}{\to}& (x \otimes z) \otimes y &\stackrel{\tau_{x,z} \otimes Id}{\to}& (z \otimes x) \otimes y } \,,

where a x,y,z:(xy)zx(yz)a_{x,y,z} \colon (x \otimes y) \otimes z \to x \otimes (y \otimes z) denotes the components of the associator of 𝒞 \mathcal{C}^\otimes.

Definition

A symmetric monoidal category is a braided monoidal category (def. ) for which the braiding

τ x,y:xyyx \tau_{x,y} \colon x \otimes y \to y \otimes x

satisfies the condition:

τ y,xτ x,y=1 xy \tau_{y,x} \circ \tau_{x,y} = 1_{x \otimes y}

for all objects x,yx, y

Remark

In analogy to the coherence theorem for monoidal categories (remark ) there is a coherence theorem for symmetric monoidal categories (def. ), saying that every diagram built freely (see remark ) from associators, unitors and braidings such that both sides of the diagram correspond to the same permutation of objects, coincide.

Consider the simplest non-trivial special case of GG-graded vector spaces from example , the case where G=/2G = \mathbb{Z}/2 is the cyclic group of order two.

Example

A /2\mathbb{Z}/2-graded vector space is a direct sum of two vector spaces

V=V evenV odd, V = V_{even} \oplus V_{odd} \,,

where we think of V evenV_{even} as the summand that is graded by the neutral element in /2\mathbb{Z}/2, and of V oddV_{odd} as being the summand that is graded by the single non-trivial element.

A homomorphism of /2\mathbb{Z}/2-graded vector spaces

f:V 1V 2 f \;\colon\; V_1 \longrightarrow V_2

is a linear map of the underlying vector spaces that respects the grading, hence equivalently a pair of linear maps

f even:(V 1) even(V 1) even f_{even} \;\colon\; (V_1)_{even} \longrightarrow (V_1)_{even}
f odd:(V 1) odd(V 1) odd f_{odd} \;\colon\; (V_1)_{odd} \longrightarrow (V_1)_{odd}

between then summands in even degree and in odd degree, respectively:

f=f evenf odd. f = f_{even} \oplus f_{odd} \,.

The tensor product of /2\mathbb{Z}/2-graded vector space is the tensor product of vector spaces of the underlying vector spaces, but with the grading obtained from multiplying the original gradings in /2\mathbb{Z}/2. Hence

(V 1V 2) even((V 1) even(V 2) even)((V 1) odd(V 2) odd) (V_1 \otimes V_2)_{even} \;\coloneqq\; \left((V_1)_{even} \otimes (V_2)_{even}\right) \oplus \left((V_1)_{odd} \otimes (V_2)_{odd}\right)

and

(V 1V 2) odd((V 1) even(V 2) odd)((V 1) odd(V 2) even). (V_1 \otimes V_2)_{odd} \;\coloneqq\; \left((V_1)_{even} \otimes (V_2)_{odd}\right) \oplus \left((V_1)_{odd} \otimes (V_2)_{even}\right) \,.

As in example , this definition makes /2\mathbb{Z}/2 a monoidal category def. .

Proposition

There are, up to braided monoidal equivalence of categories, precisely two choices for a symmetric braiding (def. )

V 1V 2τ V 1,V 2V 2V 1 V_1 \otimes V_2 \stackrel{\tau_{V_1,V_2}}{\longrightarrow} V_2 \otimes V_1

on the monoidal category (Vect k /2, k)(Vect_k^{\mathbb{Z}/2}, \otimes_k) of /2\mathbb{Z}/2-graded vector spaces from def. :

  1. the trivial braiding which is the natural linear map given on tuples (v 1,v 2)(v_1,v_2) representing an element in V 1V 2V_1 \otimes V_2 (according to def. ) by

    τ V 1,V 2 triv:(v 1,v 2)(v 2,v 1) \tau^{triv}_{V_1, V_2} \;\colon\; (v_1,v_2) \mapsto (v_2, v_1)
  2. the super-braiding which is the natural linear function given on tuples (v 1,v 2)(v_1,v_2) of homogeneous degree (i.e. v i(V i) σ iV iv_i \in (V_i)_{\sigma_i} \hookrightarrow V_i, for σ i/2\sigma_i \in \mathbb{Z}/2) by

    τ V 1,V 2 super:(v 1,v 2)(1) deg(v 1)deg(v 2)(v 2,v 1). \tau^{super}_{V_1, V_2} \;\colon\; (v_1, v_2) \mapsto (-1)^{deg(v_1) deg(v_2)} \, (v_2,v_1) \,.
Proof

For (𝒞,,1)(\mathcal{C}, \otimes, 1) a monoidal category, write

(Line(𝒞),,1)(𝒞,,1) (Line(\mathcal{C}), \otimes, 1) \hookrightarrow (\mathcal{C}, \otimes, 1)

for the full subcategory on those L𝒞L \in \mathcal{C} which are invertible objects under the tensor product, i.e. such that there is an object L 1𝒞L^{-1} \in \mathcal{C} with LL 11L \otimes L^{-1} \simeq 1 and L 1L1L^{-1} \otimes L \simeq 1. Since the tensor unit is clearly in Line(L)Line(L) (with 1 111^{-1} \simeq 1) and since with L 1,L 2Line(𝒞)𝒞L_1, L_2 \in Line(\mathcal{C}) \hookrightarrow \mathcal{C} also L 1L 2Line(𝒞)L_1 \otimes L_2 \in Line(\mathcal{C}) (with (L 1L 2) 1L 2 1L 1 1(L_1 \otimes L_2)^{-1} \simeq L_2^{-1} \otimes L_1^{-1}) the monoidal category structure on 𝒞\mathcal{C} restricts to Line(𝒞)Line(\mathcal{C}).

Accordingly any braiding on (𝒞,,1)(\mathcal{C}, \otimes,1) restricts to a braiding on (Line(𝒞),,1)(Line(\mathcal{C}), \otimes, 1). Hence it is sufficient to show that there is an essentially unique non-trivial symmetric braiding on (Line(𝒞),,1)(Line(\mathcal{C}), \otimes, 1), and that this is the restriction of a braiding on (𝒞,,1)(\mathcal{C}, \otimes, 1).

Now (Line(𝒞,,1))(Line(\mathcal{C}, \otimes , 1)) is necessarily a groupoid (the “Picard groupoid” of 𝒞\mathcal{C}) and in fact is what is called a 2-group. As such we may regard it equivalently as a homotopy 1-type with group structure, and as such it it is equivalent to its delooping

B Line(𝒞) B_\otimes Line(\mathcal{C})

regarded as a pointed homotopy type. (See at looping and delooping).

The Grothendieck group of (𝒞,,1)(\mathcal{C}, \otimes, 1) is

π 0(Line(𝒞))π 1(BLine(𝒞)) \pi_0(Line(\mathcal{C})) \simeq \pi_1(B Line(\mathcal{C}))

the fundamental group of the delooping space.

Now a symmetric braiding on Line(𝒞)Line(\mathcal{C}) is precisely the structure that makes it a symmetric 2-group which is equivalently the structure of a second delooping B 2Line(𝒞)B^2 Line(\mathcal{C}) (for the braiding) and then a third delooping B 3Line(𝒞)B^3 Line(\mathcal{C}) (for the symmetry), regarded as a pointed homotopy type.

This way we have rephrased the question equivalently as a question about the possible k-invariants of spaces of this form.

Now in the case at hand, Line(Vect /2)Line(Vect^{\mathbb{Z}/2}) has precisely two isomorphism classes of objects, namely the ground field kk itself, regarded as being in even degree and regarded as being in odd degree. We write k 1|0k^{1\vert 0} and k 0|1k^{0 \vert 1} for these, respectively. By the rules of the tensor product of graded vector spaces we have

k 1|0 kk 1|0k 1|0 k^{1\vert 0} \otimes_k k^{1\vert 0} \simeq k^{1\vert 0}
k 1|0 kk 0|1k 0|1 k^{1\vert 0} \otimes_k k^{0\vert 1} \simeq k^{0\vert 1}

and

k 0|1 kk 0|1k 1|0. k^{0 \vert 1} \otimes_k k^{0 \vert 1} \simeq k^{1 \vert 0} \,.

In other words

π 0(Line(Vect /2))/2. \pi_0(Line(Vect^{\mathbb{Z}/2})) \simeq \mathbb{Z}/2 \,.

Now under the above homotopical identification the non-trivial braiding is identified with the elements

1=k 1|0k 0|1 kk 0|1τ k 0|1,k 0|1 superk 0|1 kk 0|1k 1|0=1 1 = k^{1 \vert 0} \simeq k^{0\vert 1} \otimes_k k^{0 \vert 1} \stackrel{\tau^{super}_{k^{0\vert 1}, k^{0 \vert 1}}}{\longrightarrow} k^{0\vert 1} \otimes_k k^{0\vert 1} \simeq k^{1 \vert 0} = 1

Due to the symmetry condition (def. ) we have

(τ k 0|1,k 0|1 super) 2=id (\tau^{super}_{k^{0\vert 1}, k^{0 \vert 1}})^2 = id

which implies that

τ k 0|1,k 0|1 super{+id,id}. \tau^{super}_{k^{0\vert 1}, k^{0 \vert 1}} \in \{+ id, -id\} \,.

Therefore for classifying just the symmetric braidings, it is sufficient to restrict the hom-spaces in Line(Vect /2)Line(Vect^{\mathbb{Z}/2}) from being either kk or empty, to hom-sets being /2={+11}k\mathbb{Z}/2 = \{+1-1\} \hookrightarrow k or empty. Write Line˜(Vect /2)\widetilde{Line}(Vect^{\mathbb{Z}/2}) for the resulting 2-group.

In conclusion then the equivalence classes of possible k-invariants of B 3Line(Vect /2)B^3 Line(Vect^{\mathbb{Z}/2}), hence the possible symmetric braiding on Line(Vect /2)Line(Vect^{\mathbb{Z}/2}) are in the degree-4 ordinary cohomology of the Eilenberg-MacLane space K(/2,3)K(\mathbb{Z}/2,3) with coefficients in /2\mathbb{Z}/2. One finds (…)

H 4(K(/2,3),/2)/2. H^4(K(\mathbb{Z}/2, 3), \mathbb{Z}/2) \;\simeq\; \mathbb{Z}/2 \,.
Definition

The symmetric monoidal category (def. )

sVect k(Vect k /2,= k,1=k,τ=τ super). sVect_k \;\coloneqq\; (Vect_k^{\mathbb{Z}/2}, \otimes = \otimes_k, 1 = k, \tau = \tau^{super} ) \,.
Remark

The non-full symmetric monoidal subcategory

(Line˜(sVect), k,k,τ super) (\widetilde{Line}(sVect), \otimes_k, k, \tau^{super})

of

(Line(sVect), k,k,τ super)(sVect, k,k,τ super) (Line(sVect) , \otimes_k, k, \tau^{super}) \hookrightarrow (sVect, \otimes_k, k, \tau^{super})

(on the two objects k 1|0k^{1\vert 0} and k 0|1k^{0\vert 1} and with hom-sets restricted to {+1,1}k\{+1,-1\} \subset k, as in the proof of prop. ) happens to be the 1-truncation of the looping of the sphere spectrum 𝕊\mathbb{S}, regarded as a group-like E-infinity space (“abelian infinity-group”)

(Line˜(sVect),,k,τ super)trunc 1Ω𝕊. (\widetilde{Line}(sVect), \otimes, k, \tau^{super}) \;\simeq\; \trunc_1 \Omega \mathbb{S} \,.

It has been suggested (in Kapranov 15) that this and other phenomena are evidence that in the wider context of homotopy theory/stable homotopy theory super-grading (and hence superalgebra) is to be regarded as but a shadow of grading in higher algebra over the sphere spectrum. Notice that the sphere spectrum is just the analog of the group of integers in stable homotopy theory.

The following is evident but important

Proposition

The canonical inclusion

Vect ksVect k Vect_k \hookrightarrow sVect_k

of the category of vector spaces (def. ) into that of super vector spaces (def. ) given by regarding a vector space as a super-vector space concentrated in even degree, extends to a braided monoidal functor (def. ).

Definition

Given a symmetric monoidal category 𝒞\mathcal{C} with tensor product \otimes (def. ) it is called a closed monoidal category if for each Y𝒞Y \in \mathcal{C} the functor Y()()YY \otimes(-)\simeq (-)\otimes Y has a right adjoint, denoted hom(Y,)hom(Y,-)

𝒞hom(Y,)()Y𝒞, \mathcal{C} \underoverset {\underset{hom(Y,-)}{\longrightarrow}} {\overset{(-) \otimes Y}{\longleftarrow}} {\bot} \mathcal{C} \,,

hence if there are natural bijections

Hom 𝒞(XY,Z)Hom 𝒞C(X,hom(Y,Z)) Hom_{\mathcal{C}}(X \otimes Y, Z) \;\simeq\; Hom_{\mathcal{C}}{C}(X, hom(Y,Z))

for all objects X,Z𝒞X,Z \in \mathcal{C}.

Since for the case that X=1X = 1 is the tensor unit of 𝒞\mathcal{C} this means that

Hom 𝒞(1,hom(Y,Z))Hom 𝒞(Y,Z), Hom_{\mathcal{C}}(1,hom(Y,Z)) \simeq Hom_{\mathcal{C}}(Y,Z) \,,

the object hom(Y,Z)𝒞hom(Y,Z) \in \mathcal{C} is an enhancement of the ordinary hom-set Hom 𝒞(Y,Z)Hom_{\mathcal{C}}(Y,Z) to an object in 𝒞\mathcal{C}. Accordingly, it is also called the internal hom between YY and ZZ.

In a closed monoidal category, the adjunction isomorphism between tensor product and internal hom even holds internally:

Proposition

In a symmetric closed monoidal category (def. ) there are natural isomorphisms

hom(XY,Z)hom(X,hom(Y,Z)) hom(X \otimes Y, Z) \;\simeq\; hom(X, hom(Y,Z))

whose image under Hom 𝒞(1,)Hom_{\mathcal{C}}(1,-) are the defining natural bijections of def. .

Proof

Let A𝒞A \in \mathcal{C} be any object. By applying the defining natural bijections twice, there are composite natural bijections

Hom 𝒞(A,hom(XY,Z)) Hom 𝒞(A(XY),Z) Hom 𝒞((AX)Y,Z) Hom 𝒞(AX,hom(Y,Z)) Hom 𝒞(A,hom(X,hom(Y,Z))). \begin{aligned} Hom_{\mathcal{C}}(A , hom(X \otimes Y, Z)) & \simeq Hom_{\mathcal{C}}(A \otimes (X \otimes Y), Z) \\ & \simeq Hom_{\mathcal{C}}((A \otimes X)\otimes Y, Z) \\ & \simeq Hom_{\mathcal{C}}(A \otimes X, hom(Y,Z)) \\ & \simeq Hom_{\mathcal{C}}(A, hom(X,hom(Y,Z))) \end{aligned} \,.

Since this holds for all AA, the Yoneda lemma (the fully faithfulness of the Yoneda embedding) says that there is an isomorphism hom(XY,Z)hom(X,hom(Y,Z))hom(X\otimes Y, Z) \simeq hom(X,hom(Y,Z)). Moreover, by taking A=1A = 1 in the above and using the left unitor isomorphisms A(XY)XYA \otimes (X \otimes Y) \simeq X \otimes Y and AXXA\otimes X \simeq X we get a commuting diagram

Hom 𝒞(1,hom(XY,)) Hom 𝒞(1,hom(X,hom(Y,Z))) Hom 𝒞(XY,Z) Hom 𝒞(X,hom(Y,Z)). \array{ Hom_{\mathcal{C}}(1,hom(X\otimes Y, )) &\overset{\simeq}{\longrightarrow}& Hom_{\mathcal{C}}(1,hom(X,hom(Y,Z))) \\ {}^{\mathllap{\simeq}}\downarrow && \downarrow^{\mathrlap{\simeq}} \\ Hom_{\mathcal{C}}(X \otimes Y, Z) &\overset{\simeq}{\longrightarrow}& Hom_{\mathcal{C}}(X, hom(Y,Z)) } \,.
Definition

Let (𝒞, 𝒞,1 𝒞)(\mathcal{C},\otimes_{\mathcal{C}}, 1_{\mathcal{C}}) and (𝒟, 𝒟,1 𝒟)(\mathcal{D},\otimes_{\mathcal{D}}, 1_{\mathcal{D}} ) be two (pointed) topologically enriched monoidal categories (def. ). A lax monoidal functor between them is

  1. a functor

    F:𝒞𝒟, F \;\colon\; \mathcal{C} \longrightarrow \mathcal{D} \,,
  2. a morphism

    ϵ:1 𝒟F(1 𝒞) \epsilon \;\colon\; 1_{\mathcal{D}} \longrightarrow F(1_{\mathcal{C}})
  3. a natural transformation

    μ x,y:F(x) 𝒟F(y)F(x 𝒞y) \mu_{x,y} \;\colon\; F(x) \otimes_{\mathcal{D}} F(y) \longrightarrow F(x \otimes_{\mathcal{C}} y)

    for all x,y𝒞x,y \in \mathcal{C}

satisfying the following conditions:

  1. (associativity) For all objects x,y,z𝒞x,y,z \in \mathcal{C} the following diagram commutes

    (F(x) 𝒟F(y)) 𝒟F(z) a F(x),F(y),F(z) 𝒟 F(x) 𝒟(F(y) 𝒟F(z)) μ x,yid idμ y,z F(x 𝒞y) 𝒟F(z) F(x) 𝒟(F(x 𝒞y)) μ x 𝒞y,z μ x,y 𝒞z F((x 𝒞y) 𝒞z) F(a x,y,z 𝒞) F(x 𝒞(y 𝒞z)), \array{ (F(x) \otimes_{\mathcal{D}} F(y)) \otimes_{\mathcal{D}} F(z) &\underoverset{\simeq}{a^{\mathcal{D}}_{F(x),F(y),F(z)}}{\longrightarrow}& F(x) \otimes_{\mathcal{D}}( F(y)\otimes_{\mathcal{D}} F(z) ) \\ {}^{\mathllap{\mu_{x,y} \otimes id}}\downarrow && \downarrow^{\mathrlap{id\otimes \mu_{y,z}}} \\ F(x \otimes_{\mathcal{C}} y) \otimes_{\mathcal{D}} F(z) && F(x) \otimes_{\mathcal{D}} ( F(x \otimes_{\mathcal{C}} y) ) \\ {}^{\mathllap{\mu_{x \otimes_{\mathcal{C}} y , z} } }\downarrow && \downarrow^{\mathrlap{\mu_{ x, y \otimes_{\mathcal{C}} z }}} \\ F( ( x \otimes_{\mathcal{C}} y ) \otimes_{\mathcal{C}} z ) &\underset{F(a^{\mathcal{C}}_{x,y,z})}{\longrightarrow}& F( x \otimes_{\mathcal{C}} ( y \otimes_{\mathcal{C}} z ) ) } \,,

    where a 𝒞a^{\mathcal{C}} and a 𝒟a^{\mathcal{D}} denote the associators of the monoidal categories;

  2. (unitality) For all x𝒞x \in \mathcal{C} the following diagrams commutes

    1 𝒟 𝒟F(x) ϵid F(1 𝒞) 𝒟F(x) F(x) 𝒟 μ 1 𝒞,x F(x) F( x 𝒞) F(1 𝒞x) \array{ 1_{\mathcal{D}} \otimes_{\mathcal{D}} F(x) &\overset{\epsilon \otimes id}{\longrightarrow}& F(1_{\mathcal{C}}) \otimes_{\mathcal{D}} F(x) \\ {}^{\mathllap{\ell^{\mathcal{D}}_{F(x)}}}\downarrow && \downarrow^{\mathrlap{\mu_{1_{\mathcal{C}}, x }}} \\ F(x) &\overset{F(\ell^{\mathcal{C}}_x )}{\longleftarrow}& F(1 \otimes_{\mathcal{C}} x ) }

    and

    F(x) 𝒟1 𝒟 idϵ F(x) 𝒟F(1 𝒞) r F(x) 𝒟 μ x,1 𝒞 F(x) F(r x 𝒞) F(x 𝒞1), \array{ F(x) \otimes_{\mathcal{D}} 1_{\mathcal{D}} &\overset{id \otimes \epsilon }{\longrightarrow}& F(x) \otimes_{\mathcal{D}} F(1_{\mathcal{C}}) \\ {}^{\mathllap{r^{\mathcal{D}}_{F(x)}}}\downarrow && \downarrow^{\mathrlap{\mu_{x, 1_{\mathcal{C}} }}} \\ F(x) &\overset{F(r^{\mathcal{C}}_x )}{\longleftarrow}& F(x \otimes_{\mathcal{C}} 1 ) } \,,

    where 𝒞\ell^{\mathcal{C}}, 𝒟\ell^{\mathcal{D}}, r 𝒞r^{\mathcal{C}}, r 𝒟r^{\mathcal{D}} denote the left and right unitors of the two monoidal categories, respectively.

If ϵ\epsilon and all μ x,y\mu_{x,y} are isomorphisms, then FF is called a strong monoidal functor.

If moreover (𝒞, 𝒞,1 𝒞)(\mathcal{C},\otimes_{\mathcal{C}}, 1_{\mathcal{C}}) and (𝒟, 𝒟,1 𝒟)(\mathcal{D},\otimes_{\mathcal{D}}, 1_{\mathcal{D}} ) are equipped with the structure of braided monoidal categories (def. ) with braidings τ 𝒞\tau^{\mathcal{C}} and τ 𝒟\tau^{\mathcal{D}}, respectively, then the lax monoidal functor FF is called a braided monoidal functor if in addition the following diagram commutes for all objects x,y𝒞x,y \in \mathcal{C}

F(x) 𝒞F(y) τ F(x),F(y) 𝒟 F(y) 𝒟F(x) μ x,y μ y,x F(x 𝒞y) F(τ x,y 𝒞) F(y 𝒞x). \array{ F(x) \otimes_{\mathcal{C}} F(y) &\overset{\tau^{\mathcal{D}}_{F(x), F(y)}}{\longrightarrow}& F(y) \otimes_{\mathcal{D}} F(x) \\ {}^{\mathllap{\mu_{x,y}}}\downarrow && \downarrow^{\mathrlap{\mu_{y,x}}} \\ F(x \otimes_{\mathcal{C}} y ) &\underset{F(\tau^{\mathcal{C}}_{x,y} )}{\longrightarrow}& F( y \otimes_{\mathcal{C}} x ) } \,.

A homomorphism f:(F 1,μ 1,ϵ 1)(F 2,μ 2,ϵ 2)f\;\colon\; (F_1,\mu_1, \epsilon_1) \longrightarrow (F_2, \mu_2, \epsilon_2) between two (braided) lax monoidal functors is a monoidal natural transformation, in that it is a natural transformation f x:F 1(x)F 2(x)f_x \;\colon\; F_1(x) \longrightarrow F_2(x) of the underlying functors

compatible with the product and the unit in that the following diagrams commute for all objects x,y𝒞x,y \in \mathcal{C}:

F 1(x) 𝒟F 1(y) f(x) 𝒟f(y) F 2(x) 𝒟F 2(y) (μ 1) x,y (μ 2) x,y F 1(x 𝒞y) f(x 𝒞y) F 2(x 𝒞y) \array{ F_1(x) \otimes_{\mathcal{D}} F_1(y) &\overset{f(x)\otimes_{\mathcal{D}} f(y)}{\longrightarrow}& F_2(x) \otimes_{\mathcal{D}} F_2(y) \\ {}^{\mathllap{(\mu_1)_{x,y}}}\downarrow && \downarrow^{\mathrlap{(\mu_2)_{x,y}}} \\ F_1(x\otimes_{\mathcal{C}} y) &\underset{f(x \otimes_{\mathcal{C}} y ) }{\longrightarrow}& F_2(x \otimes_{\mathcal{C}} y) }

and

1 𝒟 ϵ 1 ϵ 2 F 1(1 𝒞) f(1 𝒞) F 2(1 𝒞). \array{ && 1_{\mathcal{D}} \\ & {}^{\mathllap{\epsilon_1}}\swarrow && \searrow^{\mathrlap{\epsilon_2}} \\ F_1(1_{\mathcal{C}}) &&\underset{f(1_{\mathcal{C}})}{\longrightarrow}&& F_2(1_{\mathcal{C}}) } \,.

We write MonFun(𝒞,𝒟)MonFun(\mathcal{C},\mathcal{D}) for the resulting category of lax monoidal functors between monoidal categories 𝒞\mathcal{C} and 𝒟\mathcal{D}, similarly BraidMonFun(𝒞,𝒟)BraidMonFun(\mathcal{C},\mathcal{D}) for the category of braided monoidal functors between braided monoidal categories, and SymMonFun(𝒞,𝒟)SymMonFun(\mathcal{C},\mathcal{D}) for the category of braided monoidal functors between symmetric monoidal categories.

Remark

In the literature the term “monoidal functor” often refers by default to what in def. is called a strong monoidal functor.

If (𝒞, 𝒞,1 𝒞)(\mathcal{C},\otimes_{\mathcal{C}}, 1_{\mathcal{C}}) and (𝒟, 𝒟,1 𝒟)(\mathcal{D},\otimes_{\mathcal{D}}, 1_{\mathcal{D}} ) are symmetric monoidal categories (def. ) then a braided monoidal functor (def. ) between them is often called a symmetric monoidal functor.

Example

Let 𝒜\mathcal{A} be the symmetric monoidal category of /2\mathbb{Z}/2-graded vector spaces Vect /2Vect^{\mathbb{Z}/2} (example ) or of super vector spaces sVectsVect (example ). Then there is an evident forgetful functor

𝒜Vect \mathcal{A} \longrightarrow Vect

to the category of plain vector spaces, which forgets the grading.

In both cases this is a strong monoidal functor (def. ) For 𝒜=Vect /2\mathcal{A} = Vect^{\mathbb{Z}/2} it is also a braided monoidal functor, but for 𝒜=sVect\mathcal{A} = sVect it is not.

Proposition

For 𝒞F𝒟G\mathcal{C} \overset{F}{\longrightarrow} \mathcal{D} \overset{G}{\longrightarrow} \mathcal{E} two composable lax monoidal functors (def. ) between monoidal categories, then their composite FGF \circ G becomes a lax monoidal functor with structure morphisms

ϵ GF:1 ϵ GG(1 𝒟)G(ϵ F)G(F(1 𝒞)) \epsilon^{G\circ F} \;\colon\; 1_{\mathcal{E}} \overset{\epsilon^G}{\longrightarrow} G(1_{\mathcal{D}}) \overset{G(\epsilon^F)}{\longrightarrow} G(F(1_{\mathcal{C}}))

and

μ c 1,c 2 GF:G(F(c 1)) G(F(c 2))μ F(c 1),F(c 2) GG(F(c 1) 𝒟F(c 2))G(μ c 1,c 2 F)G(F(c 1 𝒞c 2)). \mu^{G \circ F}_{c_1,c_2} \;\colon\; G(F(c_1)) \otimes_{\mathcal{E}} G(F(c_2)) \overset{\mu^{G}_{F(c_1), F(c_2)}}{\longrightarrow} G( F(c_1) \otimes_{\mathcal{D}} F(c_2) ) \overset{G(\mu^F_{c_1,c_2})}{\longrightarrow} G(F( c_1 \otimes_{\mathcal{C}} c_2 )) \,.

We now discuss one more extra property on monoidal categories

Definition

Let (𝒞,,1)(\mathcal{C},\otimes, 1) be a monoidal category (def. )

Then right duality between objects A,A *(𝒞,,1)A, A^\ast \in (\mathcal{C}, \otimes, 1)

consists of

  1. a morphism of the form

    ev A:A *A1 ev_A\;\colon\;A^\ast \otimes A \longrightarrow 1

    called the counit of the duality, or the evaluation map;

  2. a morphism of the form

    i A:1AA * i_A \;\colon\; 1 \longrightarrow A \otimes A^\ast

    called the unit or coevaluation map

such that

  • (triangle identity) the following diagrams commute

    A *(AA *) id A *i A A *1 α A *,A,A * 1 A 1r A (A *A)A * ev Aid A * 1A * \array{ A^\ast \otimes (A \otimes A^\ast) &\overset{id_{A^\ast} \otimes i_A}{\longleftarrow}& A^\ast \otimes 1 \\ {}^{\mathllap{\alpha^{-1}_{A^\ast,A, A^\ast}}}_{\mathllap{\simeq}}\downarrow && \downarrow^{\mathrlap{\ell^{-1}_A \circ r_A}}_{\mathrlap{\simeq}} \\ (A^\ast \otimes A) \otimes A^\ast &\underset{ev_A \otimes id_{A^\ast}}{\longrightarrow}& 1 \otimes A^\ast }

    and

    (AA *)A i Aid A 1A α A,A *,A r A 1 A A(A *A) id Aev A A1 \array{ (A \otimes A^\ast) \otimes A &\overset{i_A \otimes id_A}{\longleftarrow}& 1 \otimes A \\ {}^{\mathllap{\alpha_{A,A^\ast, A}}}_{\mathllap{\simeq}}\downarrow && \downarrow^{\mathrlap{r_A^{-1}\circ \ell_A}}_{\mathrlap{\simeq}} \\ A \otimes (A^\ast \otimes A) &\underset{id_A \otimes ev_A}{\longrightarrow}& A \otimes 1 }

    where α\alpha denotes the associator of the monoidal category 𝒞\mathcal{C}, and \ell and rr denote the left and right unitors, respectively.

We say that A *A^\ast is the right dual object to AA. Similarly a left dual for AA is an object A *A^\ast and the structure of AA as a right dual of A *A^\ast. If (𝒞,,1)(\mathcal{C}, \otimes, 1) is equipped with the structure of a braided monoidal category, then every right dual is canonically also a left dual.

If in a monoidal category (𝒞,,1)(\mathcal{C}, \otimes, 1) every object has a left and right dual, then it is called a rigid monoidal category.

Example

Let

FinVect kVect k FinVect_k \hookrightarrow Vect_k

be the full subcategory of that of all vector spaces (over the given ground field kk) on those which are finite dimensional vector spaces.

Clearly the tensor product of vector spaces (def. ) restricts to those of finite dimension, and so there is the induced monoidal category structure from example

(FinVect k,= k,1=k)(Vect k, k,k). (FinVect_k, \otimes = \otimes_k, 1 = k ) \hookrightarrow (Vect_k, \otimes_k, k) \,.

This is a a rigid monoidal category (def. ) in that for VV any finite dimensional vector spaces, its ordinary linear dual vector space

V *hom(V,k) V^\ast \coloneqq hom(V,k)

is a dual object in the abstract sense of def. .

Here the evaluation map is literally the defining evaluation map of linear duals (whence the name of the abstract concept)

ev V:V * kVhom(V,k) kkk ev_{V} \;\colon\; V^\ast \otimes_k V \simeq hom(V,k) \otimes_k k \overset{}{\longrightarrow} k
ev:(Vϕk,v)ϕ(v). ev \;\colon\; (V \stackrel{\phi}{\to} k, v) \;\;\mapsto \;\; \phi(v) \,.

The co-evaluation map

i V:kVV * i_V \;\colon\; k \longrightarrow V \otimes V^\ast

is the linear map that sends 1k1 \in k to id VEnd(V)V kV *id_V \in End(V) \simeq V \otimes_k V^\ast under the canonical identification of V kV *V \otimes_k V^\ast with the linear space of linear endomorphisms of VV.

If we choose a linear basis {e i}\{e_i\} for VV and a corresponding dual bases {e i}\{e^i\} of V *V^\ast, then the evaluation map is given by

ev:(e i,e j)e i(e j)=δ j i ev \;\colon\; (e^i, e_j) \mapsto e^i(e_j) = \delta^i_j

(with the Kronecker delta on the right) and the co-evaluation map is given by

1i(e i,e i). 1 \mapsto \underset{i}{\sum} (e_i, e^i) \,.

In this perspective the triangle identities are the statements that

je i(e j)e j=e i \underset{j}{\sum} e^i(e_j) e^j = e^i

and

je je j(e i)=e i. \underset{j}{\sum} e_j e^j(e_i) = e_i \,.

Physicists will recognize this as just the basic rules for tensor calculus in index-notation.

Example

Similarly, the full subcategory

(sFinVect, k,k,τ super)(sVect, k,k,τ super) (sFinVect, \otimes_k, k, \tau^{super}) \hookrightarrow (sVect, \otimes_k, k, \tau^{super})

of the symmetric monoidal category of super vector spaces from example , on those of finite total dimension is a rigid monoidal category.

Here we say that a super vector space VV has dimension

(p|q)× (p\vert q) \in \mathbb{N} \times \mathbb{N}

if its even part has dimension pp and its odd part has dimension qq:

dim(V)=(p|q)(dim(V even)=panddim(V odd)=q). dim(V) = (p\vert q) \;\;\; \Leftrightarrow \;\;\; \left( dim(V_{even}) = p \;\;\;\;\text{and}\;\;\;\; dim(V_{odd}) = q \right) \,.

The dual object of such a finite dimensional super vector space is just the linear dual vector space as in example , equipped with the evident grading:

V=V evenV oddV *=(V even *)(V odd *). V = V_{even} \oplus V_{odd} \;\;\;\; \Rightarrow \;\;\;\; V^\ast = (V_{even}^\ast) \oplus (V_{odd}^\ast) \,.
Proposition

Every rigid symmetric monoidal category (def. ) is a closed monoidal category (def. ) with internal hom between two objects given by the tensor product of the codomain object with the dual object of the domain object

[A,B]A *B. [A,B] \simeq A^* \otimes B \,.

(The closed monoidal categories arising this way are called compact closed categories).

Proof

The natural isomorphism that characterizes the internal hom [A,][A,-] as being right adjoint to the tensor product A()A \otimes (-) is given by the adjunction natural isomorphism that characterizes dual objects:

𝒞(C,[A,B])𝒞(C,A *B)𝒞(CA,B). \mathcal{C}(C,[A,B]) \simeq \mathcal{C}(C, A^\ast \otimes B) \simeq \mathcal{C}(C \otimes A, B) \,.

There are many monoidal categories whose “tensor product” operation is quite unlike the tensor product of vector spaces. Hence one says tensor category for monoidal categories that are also kk-linear categories and such that the tensor product functor suitably reflects that linear structure. There are slight variants of what people mean by a “tensor category”. Here we mean precisely the following:

Definition

For kk a field, then a kk-tensor category 𝒜\mathcal{A} is an

  1. essentially small

  2. k-linear (def. )

  3. rigid (def. )

  4. symmetric (def. )

  5. monoidal category (def. )

such that

  1. the tensor product functor :𝒜×𝒜𝒜\otimes \colon \mathcal{A} \times \mathcal{A} \longrightarrow \mathcal{A} is in both arguments separately

    1. kk-linear (def. );

    2. exact.

  2. End(1)kEnd(1) \simeq k (the endomorphism ring of the tensor unit coincides with kk).

In this form this is considered in (Deligne 02, 0.1).

We consider now various types of size constraints on tensor categories. The Tannaka reconstruction theorem (theorem below) only assumes one of them (subexponential growth, def. ), but the others appear in the course of the proof of the theorem.

  1. finiteness (def. )

  2. finite \otimes-generation (def. )

  3. subexponential growth (def. )

Recall the concept of length of an object in an abelian category, a generalization of the concept of dimension of a free module/vector space.

Definition

Let 𝒞\mathcal{C} be an abelian category. Given an object X𝒞X \in \mathcal{C}, then a Jordan-Hölder sequence or composition series for XX is a finite filtration, i.e. a finite sequence of subobject unclusions into XX, starting with the zero objects

0=X 0X 1X n1X n=X 0 = X_0 \hookrightarrow X_1 \hookrightarrow \cdots \hookrightarrow X_{n-1} \hookrightarrow X_n = X

such that at each stage ii the quotient X i/X i1X_i/X_{i-1} (i.e. the coimage of the monomorphism X i1X iX_{i-1} \hookrightarrow X_i) is a simple object of 𝒞\mathcal{C}.

If a Jordan-Hölder sequence for XX exists at all, then XX is said to be of finite length.

(e.g. EGNO 15, def. 1.5.3)

Proposition

(Jordan-Hölder theorem)

If X𝒞X \in \mathcal{C} has finite length according to def. , then in fact all Jordan-Hölder sequences for XX have the same length nn \in \mathbb{N}.

(e.g. EGNO 15, theorem 1.5.4)

Definition

If an object X𝒞X \in \mathcal{C} has finite length according to def. , then the length nn \in \mathbb{N} of any of its Jordan-Hölder sequences, which is uniquely defined according to prop. , is called the length of the object XX.

(e.g. EGNO 15, def. 1.5.5)

Definition

A kk-tensor category (def. ) is called finite (over kk) if

  1. There are only finitely many simple objects in CC (hence it is a finite abelian category), and each of them admits a projective presentation.

  2. Each object aa is of finite length;

  3. For any two objects aa, bb of CC, the hom-object (kk-vector space) hom(a,b)\hom(a, b) has finite dimension;

Example

The category of finite dimensional vector spaces over kk is a finite tensor category according to def. . It has a single isomorphism class of simple objects, namely kk itself.

Also category of finite dimensional super vector spaces is a finite tensor category. This has two isomorphism classes of simple objects, k=k 1|0k = k^{1 \vert 0} regarded in even degree, and k 0|1k^{0\vert 1} regarded in odd degree.

The following finiteness condition is useful in the proof of the main theorem, but not necessary for its statement (according to Deligne 02, bottom of p. 3):

Definition

A kk-tensor category (def. ) is called finitely \otimes-generated if there exists an object E𝒜E \in \mathcal{A} such that every other object X𝒜X \in \mathcal{A} is a subquotient of a direct sum of tensor products E nE^{\otimes^n}, for some nn \in \mathbb{N}:

iE n i X (iE n i)/Q. \array{ && \underset{i}{\oplus} E^{\otimes^{n_i}} \\ && \downarrow \\ X &\hookrightarrow& (\underset{i}{\oplus} E^{\otimes^{n_i}})/Q } \,.

Such EE is called an \otimes-generator for 𝒜\mathcal{A}.

(Deligne 02, 0.1)

The following is the main size constraint needed in the theorem. Notice that it is a “mild” constraint at least in the intuitive sense that it states just a minimum assumption on the expected behaviour of dimension (length) under tensor powers.

Definition

A tensor category 𝒜\mathcal{A} (def. ) is said to have subexponential growth* if the length of tensor exponentials is no larger than the exponential of the length: for every object XX there exists a natural number N XN_X such that XX is of length at most N XN_X, and that also all tensor product powers of XX are of length bounded by the corresponding powers of N XN_X:

X𝒜N Xnlength(X n)(N X) n. \underset{X \in \mathcal{A}}{\forall} \, \underset{N_X \in \mathbb{N}}{\exists} \, \underset{n \in \mathbb{N}}{\forall} \;\; length(X^{\otimes^n}) \leq (N_X)^n \,.

(e.g. EGNO 15, def. 9.11.1)

The evident example is the following:

Example

The tensor category kk-FinVect of finite dimensional vector spaces from example has subexponential growth (def. ), for N X=dim(X)N_X = dim(X) the dimension of a vector space XX, we have

dim(X n)=(dim(X)) n. dim\left( X^{\otimes^n} \right) = \left(dim(X)\right)^n \,.

While many linear monoidal categories of interest do not satisfy finiteness or rigidity (def. ), often they are such that all their objects are (formal) inductive limits over “small” objects that do form a rigid monoidal category.

Proposition

Let 𝒜\mathcal{A} a tensor category (def. ), such that

  1. all objects have finite length;

  2. all hom spaces are of finite dimension over kk

then for its category of ind-objects Ind(𝒜)Ind(\mathcal{A}) the following holds

  1. Ind(𝒜)Ind(\mathcal{A}) is an abelian category

  2. 𝒜Ind(𝒜)\mathcal{A} \hookrightarrow Ind(\mathcal{A}) is a full subcategory

    1. which stable under forming subquotients

    2. such that that every object XInd(𝒜)X \in Ind(\mathcal{A}) is the filtered colimit of those of its subobjects that are in 𝒜\mathcal{A};

  3. Ind(𝒜)\Ind(\mathcal{A}) inherits a tensor product by

    XY (lim iX i)(lim iY i) lim i,j(X iX j) \begin{aligned} X \otimes Y & \simeq (\underset{\longrightarrow}{\lim}_i X_i) \otimes (\underset{\longrightarrow}{\lim}_i Y_i) \\ & \simeq \underset{\longrightarrow}{\lim}_{i,j} (X_i \otimes X_j) \end{aligned}

    where X i,X j𝒜X_i,X_j \in \mathcal{A}, by the above.

  4. Ind(𝒜)Ind(\mathcal{A}) satisfies all the axioms of def. except that it fails to be essentially small and rigid category. In detail

    • an object in Ind(𝒜)Ind(\mathcal{A}) is dualizable precisely if it is in 𝒜\mathcal{A}.
Example

The category of all vector spaces is the category of ind-objects of the tensor category of finite dimensional vector spaces (example ):

VectInd(FinVect). Vect \simeq Ind(FinVect) \,.

Similarly the category of all super vector spaces (def. ) is the category of ind-objects of that of finite-dimensional super vector spaces (example )

sVectInd(sFinVect). sVect \simeq Ind(sFinVect) \,.

Commutative algebra in tensor categories and Affine super-spaces

The key idea of supercommutative superalgebra is that it is nothing but plain commutative algebra but “internalized” not in ordinary vector spaces, but in super vector spaces. This is made precise by def. and ef. below.

The key idea then of supergeometry is to define super-spaces to be spaces whose algebras of functions are supercommutative superalgebras. This is not the case for any “ordinary” space such as a topological space or a smooth manifold. But these spaces may be characterized dually via their algebras of functions, and hence it makes sense to generalize the latter.

For smooth manifolds the duality statement is the following:

Proposition

(embedding of smooth manifolds into formal duals of R-algebras)

The functor

C ():SmoothMfdAlg op C^\infty(-) \;\colon\; SmoothMfd \longrightarrow Alg_{\mathbb{R}}^{op}

which sends a smooth manifold (finite dimensional, paracompact, second countable) to (the formal dual of) its \mathbb{R}-algebra of smooth functions is a full and faithful functor.

In other words, for two smooth manifolds X,YX,Y there is a natural bijection between the smooth functions XYX \to Y and the \mathbb{R}-algebra homomorphisms C (X)C (Y)C^\infty(X)\leftarrow C^\infty(Y).

A proof is for instance in (Kolar-Slovak-Michor 93, lemma 35.8, corollaries 35.9, 35.10).

This says that we may identify smooth manifolds as the “formal duals” of certain associative algebras, namely those in the image of the above full embedding. Accordingly then, any larger class of associative algebras than this may be thought of as the class of formal duals to a generalized kind of manifold, defined thereby. Given any associative algebra AA, then we may think of it as representing a space Spec(A)Spec(A) which is such that it has AA as its algebra of functions.

This duality between certain spaces and their algebras of functions is profound. In physics it has always been used implicitly, in fact it was so ingrained into theoretical physics that it took much effort to abstract away from coordinate functions to discover global Riemannian geometry in the guise of“general relativity”. As mathematics, an early prominent duality theorem is Gelfand duality (between topological spaces and C*-algebras) which served as motivation for the very definition of algebraic geometry, where affine schemes are nothing but the formal duals of commutative rings/commutative algebras. Passing to non-commutative algebras here yields non-commutative geometry, and so forth. In great generality this duality between spaces and their function algebras appears as “Isbell duality” between presheaves and copresheaves.

In supergeometry we are concerned with spaces that are formally dual to associative algebras which are “very mildly” non-commutative, namely supercommutative superalgebras. These are in fact commutative algebras when viewed internal to super vector spaces (def. below). The corresponding formal dual spaces are, depending on some technical details, super schemes or supermanifolds. In the physics literature, such spaces are usually just called superspaces.

We now make this precise.

Definition

Given a monoidal category (𝒞,,1)(\mathcal{C}, \otimes, 1) (def ), then a monoid internal to (𝒞,,1)(\mathcal{C}, \otimes, 1) is

  1. an object A𝒞A \in \mathcal{C};

  2. a morphism e:1Ae \;\colon\; 1 \longrightarrow A (called the unit)

  3. a morphism μ:AAA\mu \;\colon\; A \otimes A \longrightarrow A (called the product);

such that

  1. (associativity) the following diagram commutes

    (AA)A a A,A,A A(AA) Aμ AA μA μ AA μ A, \array{ (A\otimes A) \otimes A &\underoverset{\simeq}{a_{A,A,A}}{\longrightarrow}& A \otimes (A \otimes A) &\overset{A \otimes \mu}{\longrightarrow}& A \otimes A \\ {}^{\mathllap{\mu \otimes A}}\downarrow && && \downarrow^{\mathrlap{\mu}} \\ A \otimes A &\longrightarrow& &\overset{\mu}{\longrightarrow}& A } \,,

    where aa is the associator isomorphism of 𝒞\mathcal{C};

  2. (unitality) the following diagram commutes:

    1A eid AA ide A1 μ r A, \array{ 1 \otimes A &\overset{e \otimes id}{\longrightarrow}& A \otimes A &\overset{id \otimes e}{\longleftarrow}& A \otimes 1 \\ & {}_{\mathllap{\ell}}\searrow & \downarrow^{\mathrlap{\mu}} & & \swarrow_{\mathrlap{r}} \\ && A } \,,

    where \ell and rr are the left and right unitor isomorphisms of 𝒞\mathcal{C}.

Moreover, if (𝒞,,1)(\mathcal{C}, \otimes , 1) has the structure of a symmetric monoidal category (def. ) (𝒞,,1,τ)(\mathcal{C}, \otimes, 1, \tau) with symmetric braiding τ\tau, then a monoid (A,μ,e)(A,\mu, e) as above is called a commutative monoid in (𝒞,,1,B)(\mathcal{C}, \otimes, 1, B) if in addition

  • (commutativity) the following diagram commutes

    AA τ A,A AA μ μ A. \array{ A \otimes A && \underoverset{\simeq}{\tau_{A,A}}{\longrightarrow} && A \otimes A \\ & {}_{\mathllap{\mu}}\searrow && \swarrow_{\mathrlap{\mu}} \\ && A } \,.

A homomorphism of monoids (A 1,μ 1,e 1)(A 2,μ 2,f 2)(A_1, \mu_1, e_1)\longrightarrow (A_2, \mu_2, f_2) is a morphism

f:A 1A 2 f \;\colon\; A_1 \longrightarrow A_2

in 𝒞\mathcal{C}, such that the following two diagrams commute

A 1A 1 ff A 2A 2 μ 1 μ 2 A 1 f A 2 \array{ A_1 \otimes A_1 &\overset{f \otimes f}{\longrightarrow}& A_2 \otimes A_2 \\ {}^{\mathllap{\mu_1}}\downarrow && \downarrow^{\mathrlap{\mu_2}} \\ A_1 &\underset{f}{\longrightarrow}& A_2 }

and

1 𝒸 e 1 A 1 e 2 f A 2. \array{ 1_{\mathcal{c}} &\overset{e_1}{\longrightarrow}& A_1 \\ & {}_{\mathllap{e_2}}\searrow & \downarrow^{\mathrlap{f}} \\ && A_2 } \,.

Write Mon(𝒞,,1)Mon(\mathcal{C}, \otimes,1) for the category of monoids in 𝒞\mathcal{C}, and CMon(𝒞,,1)CMon(\mathcal{C}, \otimes, 1) for its subcategory of commutative monoids.

Example

A monoid object according to def. in the monoidal category of vector spaces from example is equivalently an ordinary associative algebra over the given ground field. Similarly a commutative monoid in VectVect is an ordinary commutative algebra. Moreover, in both cases the homomorphisms of monoids agree with usual algebra homomorphisms. Hence there are equivalences of categories.

Mon(Vect k)Alg k Mon(Vect_k) \simeq Alg_k
CMon(Vect k)CAlg k. CMon(Vect_k) \simeq CAlg_k \,.
Example

For GG a group, then a GG-graded associative algebra is a monoid object according to def. in the monoidal category of GG-graded vector spaces from example .

Alg k GMon(Vect k G). Alg_k^G \simeq Mon(Vect_k^G) \,.

This means that a GG-graded algebra is

  1. a GG-graded vector space A=gGA gA = \underset{g\in G}{\oplus} A_g

  2. an associative algebra structure on the underlying vector space AA

such that for two elements of homogeneous degree, i.e. a 1A g 1Aa_1 \in A_{g_1} \hookrightarrow A and a 2A g 2Aa_2 \in A_{g_2} \hookrightarrow A then their product is in degre g 1g 2g_1 g_2

a g 1a g 2A g 1g 2A. a_{g_1} a_{g_2} \in A_{g_1 g_2} \hookrightarrow A \,.

Example motivates the following definition:

Definition

A supercommutative superalgebra is a commutative monoid (def. ) in the symmetric monoidal category of super vector spaces (def. ). We write sCAlg ksCAlg_k for the category of supercommutative superalgebras with the induced homomorphisms between them:

sCAlg kCMon(sVect k). sCAlg_k \;\coloneqq\; CMon(sVect_k) \,.

Unwinding what this means, then a supercommutative superalgebra AA is

  1. a /2\mathbb{Z}/2-graded associative algebra according to example ;

  2. such that for any two elements a,ba, b of homogeneous degree, their product satisfies

    ab=(1) deg(a)deg(b)ba. a b \; = \; (-1)^{deg(a) deg(b)}\, b a \,.
Remark

In view of def. we might define a not-necessarily supercommutative superalgebra to be a monoid (not necessarily commutative) in sVect, and write

sAlg kMon(sVect). sAlg_k \coloneqq Mon(sVect) \,.

However, since the definition of not-necessarily commutative monoids (def. ) does not invoke the braiding of the ambient tensor category, and since super vector spaces differ from /2\mathbb{Z}/2-graded vector spaces only via their braiding (example ), this yields equivalently just the /2\mathbb{Z}/2-graded algebras froom example :

sAlg kAlg k /2. sAlg_k \simeq Alg_k^{\mathbb{Z}/2} \,.

Hence the heart of superalgebra is super-commutativity.

Example

The supercommutative superalgebra which is freely generated over kk from nn generators {θ i} i=1 n\{\theta_i\}_{i = 1}^n is the quotient of the tensor algebra T nT^\bullet \mathbb{R}^n, with the generators θ i\theta_i in odd degree, by the ideal generated by the relations

θ iθ j=θ jθ i \theta_i \theta_j = - \theta_j \theta_i

for all i,j{1,,n}i,j \in \{1, \cdots, n\}.

This is also called a Grassmann algebra, in honor of (Grassmann 1844), who introduced and studied the super-sign rule in def. a century ahead of his time.

We also denote this algebra by

( n)sCAlg . \wedge^\bullet_{\mathbb{R}}(\mathbb{R}^n) \;\in\; sCAlg_{\mathbb{R}} \,.
Example

Given a homotopy commutative ring spectrum EE (i.e., via the Brown representability theorem, a multiplicative generalized cohomology theory), then its stable homotopy groups π (E)\pi_\bullet(E) inherit the structure of a super-commutative ring.

See at Introduction to Stable homotopy theory in the section 1-2 Homotopy commutative ring spectra this proposition.

The following is an elementary but fundamental fact about the relation between commutative algbra and supercommutative superalgebra. It is implicit in much of the literature, but maybe the only place where it has been made explicit before is (Carchedi-Roytenberg 12, example 3.18).

Proposition

There is a full subcategory inclusion

CAlg k sCAlg k = = CMon(Vect k) CMon(sVect k) \array{ CAlg_k &\hookrightarrow& sCAlg_k \\ = && = \\ CMon(Vect_k) &\hookrightarrow& CMon(sVect_k) }

of commutative algebras (example ) into supercommutative superalgebras (def. ) induced via prop. from the full inclusion

i:VectksVect k i \;\colon\; Vectk \hookrightarrow sVect_k

of vector spaces (def. ) into super vector spaces (def. ), which is a braided monoidal functor by prop. . Hence this regards a commutative algebra as a superalgebra concentrated in even degree.

This inclusion functor has both a left adjoint functor and a right adjoint functor , (an adjoint triple exibiting a reflective subcategory and coreflective subcategory inclusion, an “adjoint cylinder”):

CAlg k() even()/() oddsCAlg k. CAlg_k \underoverset {\underset{(-)_{even}}{\longleftarrow}} {\overset{(-)/(-)_{odd}}{\longleftarrow}} {\hookrightarrow} sCAlg_k \,.

Here

  1. the right adjoint () even(-)_{even} sends a supercommutative superalgebra to its even part AA evenA \mapsto A_{even};

  2. the left adjoint ()/() even(-)/(-)_{even} sends a supercommutative superalgebra to the quotient by the ideal which is generated by its odd part AA/(A odd)A \mapsto A/(A_{odd}) (hence it sets all elements to zero which may be written as a product such that at least one factor is odd-graded).

Proof

The full inclusion ii is evident. To see the adjunctions observe their characteristic natural bijections between hom-sets: If A ordinaryA_{ordinary} is an ordinary commutative algebra regarded as a superalgeba i(A ordinary)i(A_{ordinary}) concentrated in even degree, and if BB is any superalgebra,

  1. then every super-algebra homomorphism of the form A ordinaryBA_{ordinary} \to B must factor through B evenB_{even}, simply because super-algebra homomorpism by definition respect the /2\mathbb{Z}/2-grading. This gives a natual bijection

    Hom sCAlg k(i(A ordinary),B)Hom CAlg k(A ordinary,B even), Hom_{sCAlg_k}(i(A_{ordinary}), B) \simeq Hom_{CAlg_k}(A_{ordinary,B_{even}}) \,,
  2. every super-algebra homomorphism of the form Bi(A ordinary)B \to i(A_{ordinary}) must send every odd element of BB to 0, again because homomorphism have to respect the /2\mathbb{Z}/2-grading, and since homomorphism of course also preserve products, this means that the entire ideal generated by B oddB_{odd} must be sent to zero, hence the homomorphism must facto through the projection BB/B oddB \to B/B_{odd}, which gives a natural bijection

    Hom sCalg k(B,i(A ordinary))Hom Alg k(B/B odd,A ordinary). Hom_{sCalg_k}(B, i(A_{ordinary})) \simeq Hom_{Alg_k}(B/B_{odd}, A_{ordinary}) \,.

It is useful to make explicit the following formally dual perspective on supercommutative superalgebras:

Definition

For 𝒞\mathcal{C} a symmetric monoidal category, then we write

Aff(𝒞)CMon(𝒞) op Aff(\mathcal{C}) \coloneqq CMon(\mathcal{C})^{op}

for the opposite category of the category of commutative monoids in 𝒞\mathcal{C}, according to def. .

For RCMon(𝒞)R \in CMon(\mathcal{C}) we write

Spec(A)Aff(𝒞) Spec(A) \in Aff(\mathcal{C})

for the same object, regarded in the opposite category. We also call this the affine scheme of AA. Conversely, for XAff(𝒞)X \in Aff(\mathcal{C}), we write

𝒪(X)CMon(𝒞) \mathcal{O}(X) \in CMon(\mathcal{C})

for the same object, regarded in the category of commutative monoids. We also call this the algebra of functions on XX.

Definition

For the special case that mathalC=\mathal{C} = sVect (def. ) in def. , then we say that the objects in

Aff(sVect k)=scAlg k op=CMon(sVect k) op Aff(sVect_k) = scAlg_k^{op} = CMon(sVect_k)^{op}

are affine super schemes over kk.

Example

For ACAlg A \in CAlg_{\mathbb{R}} an ordinary commutative algebra over \mathbb{R}, then of course this becomes a supercommutative superalgebra by regarding it as being concentrated in even degrees. Accordingly, via def. , ordinary affine schemes fully embed into affine super schemes (def. )

Aff(Vect k)Aff(sVect k). Aff(Vect_k) \hookrightarrow Aff(sVect_k) \,.

In particular for p\mathbb{R}^p an ordinary Cartesian space, this becomes an affine superscheme in even degree, under the above embedding. As such, it is usually written

p|0Aff(sVect k). \mathbb{R}^{p \vert 0} \in Aff(sVect_k) \,.
Example

The formal dual space, according to def. (example ) to a Grassmann algebra ( q)\wedge^\bullet_{\mathbb{R}}(\mathbb{R}^q) (example ) is to be thought of as a space which is “so tiny” that the coefficients of the Taylor expansion of any real-valued function on it become “so very small” as to be actually equal to zero, at least after the qq-th power.

For instance for q=2q = 2 then a general element of ( q)\wedge^\bullet_{\mathbb{R}}(\mathbb{R}^q) is of the form

f=a 0+a 1θ 1+a 2θ 2+a 12θ 1θ 2 ( q). f = a_0 + a_1 \theta_1 + a_2 \theta_2 + a_{12} \theta_1 \theta_2 \;\;\;\in \wedge^\bullet_{\mathbb{R}}(\mathbb{R}^q) \,.

for a 1,a 2,a 12a_1,a_2, a_{12} \in \mathbb{R}, to be compared with the Taylor expansion of a smooth function g: 2g \colon \mathbb{R}^2 \to \mathbb{R}, which is of the form

g(x 1,x 2)=g(0)+gx 1(0)x 1+gx 2(0)x 2+ 2gx 1x 2(0)x 1x 2+. g(x_1, x_2) = g(0) + \frac{\partial g}{\partial x_1}(0)\, x_1 + \frac{\partial g}{\partial x_2}(0)\, x_2 + \frac{\partial^2 g}{\partial x_1 \partial x_2}(0) \, x_1 x_2 + \cdots \,.

Therefore the formal dual space to a Grassmann algebra behaves like an infinitesimal neighbourhood of a point. Hence these are also called superpoints and one writes

0|qSpec( ( q)). \mathbb{R}^{0\vert q} \coloneqq Spec(\wedge^\bullet_{\mathbb{R}}(\mathbb{R}^q)) \,.
Example

Combining example with example , and using prop. , we obtain the affine super schemes

p|q p|0× 0|qSpec(C ( p) q). \mathbb{R}^{p \vert q} \coloneqq \mathbb{R}^{p\vert 0} \times \mathbb{R}^{0\vert q} \simeq Spec\left( \underbrace{C^\infty(\mathbb{R}^p)} \otimes_{\mathbb{R}} \wedge^\bullet_{\mathbb{R}} \mathbb{R}^q \right) \,.

These may be called the super Cartesian spaces. The play the same role in the theory of supermanifolds as the ordinary Cartesian spaces do for smooth manifolds. See at geometry of physics – supergeometry for more on this.

Definition

Given a supercommutative superalgebra AA (def. ), its parity involution is the algebra automorphism

par:AA par \;\colon\; A \overset{\simeq}{\longrightarrow} A

which on homogeneously graded elements aa of degree deg(a){even,odd}=/2deg(a) \in \{even,odd\} = \mathbb{Z}/2\mathbb{Z} is multiplication by the degree

a(1) deg(a)a. a \mapsto (-1)^{deg(a)}a \,.

(e.g. arXiv:1303.1916, 7.5)

Dually, via def. , this means that every affine super scheme has a canonical involution.

Here are more general and more abstract examples of commutative monoids, which will be useful to make explicit:

Example

Given a monoidal category (𝒞,,1)(\mathcal{C}, \otimes, 1) (def. ), then the tensor unit 11 is a monoid in 𝒞\mathcal{C} (def. ) with product given by either the left or right unitor

1=r 1:111. \ell_1 = r_1 \;\colon\; 1 \otimes 1 \overset{\simeq}{\longrightarrow} 1 \,.

By lemma , these two morphisms coincide and define an associative product with unit the identity id:11id \colon 1 \to 1.

If (𝒞,,1)(\mathcal{C}, \otimes , 1) is a symmetric monoidal category (def. ), then this monoid is a commutative monoid.

Example

Given a symmetric monoidal category (𝒞,,1)(\mathcal{C}, \otimes, 1) (def. ), and given two commutative monoids (E i,μ i,e i)(E_i, \mu_i, e_i), i{1,2}i \in \{1,2\} (def. ), then the tensor product E 1E 2E_1 \otimes E_2 becomes itself a commutative monoid with unit morphism

e:111e 1e 2E 1E 2 e \;\colon\; 1 \overset{\simeq}{\longrightarrow} 1 \otimes 1 \overset{e_1 \otimes e_2}{\longrightarrow} E_1 \otimes E_2

(where the first isomorphism is, 1 1=r 1 1\ell_1^{-1} = r_1^{-1} (lemma )) and with product morphism given by

E 1E 2E 1E 2idτ E 2,E 1idE 1E 1E 2E 2μ 1μ 2E 1E 2 E_1 \otimes E_2 \otimes E_1 \otimes E_2 \overset{id \otimes \tau_{E_2, E_1} \otimes id}{\longrightarrow} E_1 \otimes E_1 \otimes E_2 \otimes E_2 \overset{\mu_1 \otimes \mu_2}{\longrightarrow} E_1 \otimes E_2

(where we are notationally suppressing the associators and where τ\tau denotes the braiding of 𝒞\mathcal{C}).

That this definition indeed satisfies associativity and commutativity follows from the corresponding properties of (E i,μ i,e i)(E_i,\mu_i, e_i), and from the hexagon identities for the braiding (def. ) and from symmetry of the braiding.

Similarly one checks that for E 1=E 2=EE_1 = E_2 = E then the unit maps

EE1ideEE E \simeq E \otimes 1 \overset{id \otimes e}{\longrightarrow} E \otimes E
E1Ee1EE E \simeq 1 \otimes E \overset{e \otimes 1}{\longrightarrow} E \otimes E

and the product map

μ:EEE \mu \;\colon\; E \otimes E \longrightarrow E

and the braiding

τ E,E:EEEE \tau_{E,E} \;\colon\; E \otimes E \longrightarrow E \otimes E

are monoid homomorphisms, with EEE \otimes E equipped with the above monoid structure.

Monoids are preserved by lax monoidal functors:

Proposition

Let (𝒞, 𝒞,1 𝒞)(\mathcal{C},\otimes_{\mathcal{C}}, 1_{\mathcal{C}}) and (𝒟, 𝒟,1 𝒟)(\mathcal{D}, \otimes_{\mathcal{D}},1_{\mathcal{D}}) be two monoidal categories (def. ) and let F:𝒞𝒟F \;\colon\; \mathcal{C} \longrightarrow \mathcal{D} be a lax monoidal functor (def. ) between them.

Then for (A,μ A,e A)(A,\mu_A,e_A) a monoid in 𝒞\mathcal{C} (def. ), its image F(A)𝒟F(A) \in \mathcal{D} becomes a monoid (F(A),μ F(A),e F(A))(F(A), \mu_{F(A)}, e_{F(A)}) by setting

μ F(A):F(A) 𝒞F(A)F(A 𝒞A)F(μ A)F(A) \mu_{F(A)} \;\colon\; F(A) \otimes_{\mathcal{C}} F(A) \overset{}{\longrightarrow} F(A \otimes_{\mathcal{C}} A) \overset{F(\mu_A)}{\longrightarrow} F(A)

(where the first morphism is the structure morphism of FF) and setting

e F(A):1 𝒟F(1 𝒞)F(e A)F(A) e_{F(A)} \;\colon\; 1_{\mathcal{D}} \longrightarrow F(1_{\mathcal{C}}) \overset{F(e_A)}{\longrightarrow} F(A)

(where again the first morphism is the corresponding structure morphism of FF).

This construction extends to a functor

Mon(F):Mon(𝒞, 𝒞,1 𝒞)Mon(𝒟, 𝒟,1 𝒟) Mon(F) \;\colon\; Mon(\mathcal{C}, \otimes_{\mathcal{C}}, 1_{\mathcal{C}}) \longrightarrow Mon(\mathcal{D},\otimes_{\mathcal{D}}, 1_{\mathcal{D}})

from the category of monoids of 𝒞\mathcal{C} (def. ) to that of 𝒟\mathcal{D}.

Moreover, if 𝒞\mathcal{C} and 𝒟\mathcal{D} are symmetric monoidal categories (def. ) and FF is a braided monoidal functor (def. ) and AA is a commutative monoid (def. ) then so is F(A)F(A), and this construction extends to a functor

CMon(F):CMon(𝒞, 𝒞,1 𝒞)CMon(𝒟, 𝒟,1 𝒟). CMon(F) \;\colon\; CMon(\mathcal{C}, \otimes_{\mathcal{C}}, 1_{\mathcal{C}}) \longrightarrow CMon(\mathcal{D},\otimes_{\mathcal{D}}, 1_{\mathcal{D}}) \,.
Proof

This follows immediately from combining the associativity and unitality (and symmetry) constraints of FF with those of AA.

Modules in tensor categories and Super vector bundles

Above (in def. ) we considered spaces XX from a dual perspective, as determined by their algebras of functions 𝒪(X)\mathcal{O}(X). In the same spirit then we are to express various constructions on and with spaces in terms of dual algebraic constructions.

A key such construction is that of vector bundles over XX. Here we discuss the corresponding algebraic incarnation of these, namely as modules over algebras of functions.

Suppose that XX is a smooth manifold, and VpXV \stackrel{p}{\to} X is an ordinary smooth real vector bundle over XX. A section of this vector bundle is a smooth function σ:XV\sigma \colon X \to V such that pσ=idp \circ \sigma = id

V σ p X = X. \array{ && V \\ & {}^{\mathllap{\sigma}}\nearrow & \downarrow^{\mathrlap{p}} \\ X &=& X } \,.

Write Γ X(V)\Gamma_X(V) for the set of all such sections. Observe that this set inherits various extra structure.

First of all, since VXV \to X is a vector bundle, we have fiber-wise the vector space operations. This means that given two elements c 1,c 2c_1, c_2 \in \mathbb{R} in the real numbers, and given two sections σ 1\sigma_1 and σ 2\sigma_2, we may form in each fiber V xV_x the linear combination c 1σ 1(x)+c 2σ 2(x)c_1 \sigma_1(x) + c_2 \sigma_2(x). This hence yields a new section c 1σ 1+c 2σ 2c_1 \sigma_1 + c_2 \sigma_2. Hence the set of sections of a vector bundle naturally forms itself a vector space.

But there is more structure. We need not multiply with the same element cc \in \mathbb{R} in each fiber, but we may multiply the section in each fiber by a different element, as long as the choice of element varies smoothly with the fibers, so that the resulting section is still smooth.

In other words, every element fC (X)f \in C^\infty(X) in the \mathbb{R}-algebra of smooth functions on XX, takes a smooth section σ\sigma of VV to a new smooth section fσf \cdot \sigma. This operation enjoys some evident properties. It is bilinear in the real vector spaces C (X)C^\infty(X) and Γ X(V)\Gamma_X(V), and it satisfies the “action property

(fg)σ=f(gσ) (f g) \cdot \sigma = f\cdot (g \cdot \sigma)

for any two smooth functions f,gC (X)f,g \in C^\infty(X).

One says that a vector space such as Γ X(V)\Gamma_X(V) equipped with an action of an algebra RR this way is a module over RR.

In conclusion, any vector bundle VXV \to X gives rise to an C (X)C^\infty(X)-module Γ X(V)\Gamma_X(V) of sections.

The smooth Serre-Swan theorem states sufficient conditions on XX such that the converse holds. Together with the embedding of smooth manifolds into formal duals of R-algebras (prop ), this means that differential geometry is “more algebraic” than it might superficially seem, hence that its “algebraic deformation” to supergeometry is more natura than it might superficially seem:

Proposition

(smooth Serre-Swan theorem, Nestruev 03)

For XX a smooth manifold, then the construction which sends a smooth vector bundle VXV \to X to its C (X)C^\infty(X)-module Γ X(V)\Gamma_X(V) of sections is an equivalence of categories

VectBund X finC (X)Mod proj fingen VectBund_X^{fin} \stackrel{\simeq}{\longrightarrow} C^\infty(X) Mod_{proj}^{fin\,gen}

between that of smooth vector bundles of finite rank over XX and that of finitely generated projective modules over the \mathbb{R}-algebra C (X)C^\infty(X) of smooth functions on XX.

One may turn the Serre-Swan theorem around to regard for RR any commutative monoid in some symmetric monoidal category (def. ), the modules over RR as “generalized vector bundles” over the space Spec(R)Spec(R) (def. ). These “generalized vector bundles” are called “quasicoherent sheaves” over affines. Specified to the case that 𝒞=\mathcal{C} = sVect, this hence yields a concept of super vector bundles.

We now state the relevant definitions and constructions formally.

Definition

Given a monoidal category (𝒞,,1)(\mathcal{C}, \otimes, 1) (def. ), and given (A,μ,e)(A,\mu,e) a monoid in (𝒞,,1)(\mathcal{C}, \otimes, 1) (def. ), then a left module object in (𝒞,,1)(\mathcal{C}, \otimes, 1) over (A,μ,e)(A,\mu,e) is

  1. an object N𝒞N \in \mathcal{C};

  2. a morphism ρ:ANN\rho \;\colon\; A \otimes N \longrightarrow N (called the action);

such that

  1. (unitality) the following diagram commutes:

    1N eid AN ρ N, \array{ 1 \otimes N &\overset{e \otimes id}{\longrightarrow}& A \otimes N \\ & {}_{\mathllap{\ell}}\searrow & \downarrow^{\mathrlap{\rho}} \\ && N } \,,

    where \ell is the left unitor isomorphism of 𝒞\mathcal{C}.

  2. (action property) the following diagram commutes

    (AA)N a A,A,N A(AN) Aρ AN μN ρ AN ρ N, \array{ (A\otimes A) \otimes N &\underoverset{\simeq}{a_{A,A,N}}{\longrightarrow}& A \otimes (A \otimes N) &\overset{A \otimes \rho}{\longrightarrow}& A \otimes N \\ {}^{\mathllap{\mu \otimes N}}\downarrow && && \downarrow^{\mathrlap{\rho}} \\ A \otimes N &\longrightarrow& &\overset{\rho}{\longrightarrow}& N } \,,

A homomorphism of left AA-module objects

(N 1,ρ 1)(N 2,ρ 2) (N_1, \rho_1) \longrightarrow (N_2, \rho_2)

is a morphism

f:N 1N 2 f\;\colon\; N_1 \longrightarrow N_2

in 𝒞\mathcal{C}, such that the following diagram commutes:

AN 1 Af AN 2 ρ 1 ρ 2 N 1 f N 2. \array{ A\otimes N_1 &\overset{A \otimes f}{\longrightarrow}& A\otimes N_2 \\ {}^{\mathllap{\rho_1}}\downarrow && \downarrow^{\mathrlap{\rho_2}} \\ N_1 &\underset{f}{\longrightarrow}& N_2 } \,.

For the resulting category of modules of left AA-modules in 𝒞\mathcal{C} with AA-module homomorphisms between them, we write

AMod(𝒞). A Mod(\mathcal{C}) \,.

The following degenerate example turns out to be important for the general development of the theory below.

Example

Given a monoidal category (𝒞,,1)(\mathcal{C},\otimes, 1) (def. ) with the tensor unit 11 regarded as a monoid in a monoidal category via example , then the left unitor

C:1CC \ell_C \;\colon\; 1\otimes C \longrightarrow C

makes every object C𝒞C \in \mathcal{C} into a left module, according to def. , over CC. The action property holds due to lemma . This gives an equivalence of categories

𝒞1Mod(𝒞) \mathcal{C} \simeq 1 Mod(\mathcal{C})

of 𝒞\mathcal{C} with the category of modules over its tensor unit.

Example

The classical subject of algebra, not necessarily over ground fields, is the above general concepts of monoids and their modules specialized to the ambient symmetric monoidal category being the category Ab of abelian groups regarded as a symmetric monoidal category via the tensor product of abelian groups \otimes_{\mathbb{Z}} (whose tensor unit is the additive group of integers \mathbb{Z}):

  1. A monoid in (Ab, ,)(Ab, \otimes_{\mathbb{Z}}, \mathbb{Z}) (def. ) is equivalently a ring.

  2. A commutative monoid in in (Ab, ,)(Ab, \otimes_{\mathbb{Z}}, \mathbb{Z}) (def. ) is equivalently a commutative ring RR.

  3. An RR-module object in (Ab, ,)(Ab, \otimes_{\mathbb{Z}}, \mathbb{Z}) (def. ) is equivalently an RR-module;

  4. The tensor product of RR-module objects (def. ) is the standard tensor product of modules.

  5. The category of module objects RMod(Ab)R Mod(Ab) (def. ) is the standard category of modules RModR Mod.

Example

Let GG be a discrete group and write k[G]k[G] for its group algebra over the ground field kk. Then k[G]k[G]-modules in Vect are equivalently linear representations of GG.

Proposition

In the situation of def. , the monoid (A,μ,e)(A,\mu, e) canonically becomes a left module over itself by setting ρμ\rho \coloneqq \mu. More generally, for C𝒞C \in \mathcal{C} any object, then ACA \otimes C naturally becomes a left AA-module by setting:

ρ:A(AC)a A,A,C 1(AA)CμidAC. \rho \;\colon\; A \otimes (A \otimes C) \underoverset{\simeq}{a^{-1}_{A,A,C}}{\longrightarrow} (A \otimes A) \otimes C \overset{\mu \otimes id}{\longrightarrow} A \otimes C \,.

The AA-modules of this form are called free modules.

The free functor FF constructing free AA-modules is left adjoint to the forgetful functor UU which sends a module (N,ρ)(N,\rho) to the underlying object U(N,ρ)NU(N,\rho) \coloneqq N.

AMod(𝒞)UF𝒞. A Mod(\mathcal{C}) \underoverset {\underset{U}{\longrightarrow}} {\overset{F}{\longleftarrow}} {\bot} \mathcal{C} \,.
Proof

A homomorphism out of a free AA-module is a morphism in 𝒞\mathcal{C} of the form

f:ACN f \;\colon\; A\otimes C \longrightarrow N

fitting into the diagram (where we are notationally suppressing the associator)

AAC Af AN μid ρ AC f N. \array{ A \otimes A \otimes C &\overset{A \otimes f}{\longrightarrow}& A \otimes N \\ {}^{\mathllap{\mu \otimes id}}\downarrow && \downarrow^{\mathrlap{\rho}} \\ A \otimes C &\underset{f}{\longrightarrow}& N } \,.

Consider the composite

f˜:C C1CeidACfN, \tilde f \;\colon\; C \underoverset{\simeq}{\ell_C}{\longrightarrow} 1 \otimes C \overset{e\otimes id}{\longrightarrow} A \otimes C \overset{f}{\longrightarrow} N \,,

i.e. the restriction of ff to the unit “in” AA. By definition, this fits into a commuting square of the form (where we are now notationally suppressing the associator and the unitor)

AC idf˜ AN ideid = AAC idf AN. \array{ A \otimes C &\overset{id \otimes \tilde f}{\longrightarrow}& A \otimes N \\ {}^{\mathllap{id \otimes e \otimes id}}\downarrow && \downarrow^{\mathrlap{=}} \\ A \otimes A \otimes C &\underset{id \otimes f}{\longrightarrow}& A \otimes N } \,.

Pasting this square onto the top of the previous one yields

AC idf˜ AN ideid = AAC Af AN μid ρ AC f N, \array{ A \otimes C &\overset{id \otimes \tilde f}{\longrightarrow}& A \otimes N \\ {}^{\mathllap{id \otimes e \otimes id}}\downarrow && \downarrow^{\mathrlap{=}} \\ A \otimes A \otimes C &\overset{A \otimes f}{\longrightarrow}& A \otimes N \\ {}^{\mathllap{\mu \otimes id}}\downarrow && \downarrow^{\mathrlap{\rho}} \\ A \otimes C &\underset{f}{\longrightarrow}& N } \,,

where now the left vertical composite is the identity, by the unit law in AA. This shows that ff is uniquely determined by f˜\tilde f via the relation

f=ρ(id Af˜). f = \rho \circ (id_A \otimes \tilde f) \,.

This natural bijection between ff and f˜\tilde f establishes the adjunction.

Definition

Given a closed symmetric monoidal category (𝒞,,1)(\mathcal{C}, \otimes, 1) with braiding denoted τ\tau (def. , def. ), given (A,μ,e)(A,\mu,e) a commutative monoid in (𝒞,,1)(\mathcal{C}, \otimes, 1) (def. ), and given (N 1,ρ 1)(N_1, \rho_1) and (N 2,ρ 2)(N_2, \rho_2) two left AA-module objects (def.), then

  1. the tensor product of modules N 1 AN 2N_1 \otimes_A N_2 is, if it exists, the coequalizer

    N 1AN 2AAAAρ 1(τ N 1,AN 2)N 1ρ 2N 1N 1coeqN 1 AN 2 N_1 \otimes A \otimes N_2 \underoverset {\underset{\rho_{1}\circ (\tau_{N_1,A} \otimes N_2)}{\longrightarrow}} {\overset{N_1 \otimes \rho_2}{\longrightarrow}} {\phantom{AAAA}} N_1 \otimes N_1 \overset{coeq}{\longrightarrow} N_1 \otimes_A N_2

    and if A()A \otimes (-) preserves these coequalizers, then this is equipped with the left AA-action induced from the left AA-action on N 1N_1

  2. the function module hom A(N 1,N 2)hom_A(N_1,N_2) is, if it exists, the equalizer

    hom A(N 1,N 2)equhom(N 1,N 2)AAAAAAhom(AN 1,ρ 2)(A())hom(ρ 1,N 2)hom(AN 1,N 2). hom_A(N_1, N_2) \overset{equ}{\longrightarrow} hom(N_1, N_2) \underoverset {\underset{hom(A \otimes N_1, \rho_2)\circ (A \otimes(-))}{\longrightarrow}} {\overset{hom(\rho_1,N_2)}{\longrightarrow}} {\phantom{AAAAAA}} hom(A \otimes N_1, N_2) \,.

    equipped with the left AA-action that is induced by the left AA-action on N 2N_2 via

    Ahom(X,N 2)hom(X,N 2)Ahom(X,N 2)XidevAN 2ρ 2N 2. \frac{ A \otimes hom(X,N_2) \longrightarrow hom(X,N_2) }{ A \otimes hom(X,N_2) \otimes X \overset{id \otimes ev}{\longrightarrow} A \otimes N_2 \overset{\rho_2}{\longrightarrow} N_2 } \,.

(e.g. Hovey-Shipley-Smith 00, lemma 2.2.2 and lemma 2.2.8)

Proposition

Given a closed symmetric monoidal category (𝒞,,1)(\mathcal{C}, \otimes, 1) (def. , def. ), and given (A,μ,e)(A,\mu,e) a commutative monoid in (𝒞,,1)(\mathcal{C}, \otimes, 1) (def. ). If all coequalizers exist in 𝒞\mathcal{C}, then the tensor product of modules A\otimes_A from def. makes the category of modules AMod(𝒞)A Mod(\mathcal{C}) into a symmetric monoidal category, (AMod, A,A)(A Mod, \otimes_A, A) with tensor unit the object AA itself, regarded as an AA-module via prop. .

If moreover all equalizers exist, then this is a closed monoidal category (def. ) with internal hom given by the function modules hom Ahom_A of def. .

(e.g. Hovey-Shipley-Smith 00, lemma 2.2.2, lemma 2.2.8)

Proof sketch

The associators and braiding for A\otimes_{A} are induced directly from those of \otimes and the universal property of coequalizers. That AA is the tensor unit for A\otimes_{A} follows with the same kind of argument that we give in the proof of example below.

Example

For (A,μ,e)(A,\mu,e) a monoid (def. ) in a symmetric monoidal category (𝒞,,1)(\mathcal{C},\otimes, 1) (def. ), the tensor product of modules (def. ) of two free modules (def. ) AC 1A\otimes C_1 and AC 2A \otimes C_2 always exists and is the free module over the tensor product in 𝒞\mathcal{C} of the two generators:

(AC 1) A(AC 2)A(C 1C 2). (A \otimes C_1) \otimes_A (A \otimes C_2) \simeq A \otimes (C_1 \otimes C_2) \,.

Hence if 𝒞\mathcal{C} has all coequalizers, so that the category of modules is a monoidal category (AMod, A,A)(A Mod, \otimes_A, A) (prop. ) then the free module functor (def. ) is a strong monoidal functor (def. )

F:(𝒞,,1)(AMod, A,A). F \;\colon\; (\mathcal{C}, \otimes, 1) \longrightarrow (A Mod, \otimes_A, A) \,.
Proof

It is sufficient to show that the diagram

AAAAAAAidμμidAAμA A \otimes A \otimes A \underoverset {\underset{id \otimes \mu}{\longrightarrow}} {\overset{\mu \otimes id}{\longrightarrow}} {\phantom{AAAA}} A \otimes A \overset{\mu}{\longrightarrow} A

is a coequalizer diagram (we are notationally suppressing the associators), hence that A AAAA \otimes_A A \simeq A, hence that the claim holds for C 1=1C_1 = 1 and C 2=1C_2 = 1.

To that end, we check the universal property of the coequalizer:

First observe that μ\mu indeed coequalizes idμid \otimes \mu with μid\mu \otimes id, since this is just the associativity clause in def. . So for f:AAQf \colon A \otimes A \longrightarrow Q any other morphism with this property, we need to show that there is a unique morphism ϕ:AQ\phi \colon A \longrightarrow Q which makes this diagram commute:

AA μ A f ϕ Q. \array{ A \otimes A &\overset{\mu}{\longrightarrow}& A \\ {}^{\mathllap{f}}\downarrow & \swarrow_{\mathrlap{\phi}} \\ Q } \,.

We claim that

ϕ:Ar 1A1ideAAfQ, \phi \;\colon\; A \underoverset{\simeq}{r^{-1}}{\longrightarrow} A \otimes 1 \overset{id \otimes e}{\longrightarrow} A \otimes A \overset{f}{\longrightarrow} Q \,,

where the first morphism is the inverse of the right unitor of 𝒞\mathcal{C}.

First to see that this does make the required triangle commute, consider the following pasting composite of commuting diagrams

AA μ A idr 1 r 1 AA1 μid A1 ide ide AAA μid AA idμ f AA f Q. \array{ A \otimes A &\overset{\mu}{\longrightarrow}& A \\ {}^{\mathllap{id \otimes r^{-1}}}_{\mathllap{\simeq}}\downarrow && \downarrow^{\mathrlap{r^{-1}}}_{\simeq} \\ A \otimes A \otimes 1 &\overset{\mu \otimes id}{\longrightarrow}& A \otimes 1 \\ {}^{\mathllap{id \otimes e}}\downarrow && \downarrow^{\mathrlap{id \otimes e} } \\ A \otimes A \otimes A &\overset{\mu \otimes id}{\longrightarrow}& A \otimes A \\ {}^{\mathllap{id \otimes \mu}}\downarrow && \downarrow^{\mathrlap{f}} \\ A \otimes A &\underset{f}{\longrightarrow}& Q } \,.

Here the the top square is the naturality of the right unitor, the middle square commutes by the functoriality of the tensor product :𝒞×𝒞𝒞\otimes \;\colon\; \mathcal{C}\times \mathcal{C} \longrightarrow \mathcal{C} and the definition of the product category (def. ), while the commutativity of the bottom square is the assumption that ff coequalizes idμid \otimes \mu with μid\mu \otimes id.

Here the right vertical composite is ϕ\phi, while, by unitality of (A,μ,e)(A,\mu ,e), the left vertical composite is the identity on AA, Hence the diagram says that ϕμ=f\phi \circ \mu = f, which we needed to show.

It remains to see that ϕ\phi is the unique morphism with this property for given ff. For that let q:AQq \colon A \to Q be any other morphism with qμ=f q\circ \mu = f. Then consider the commuting diagram

A1 A ide = AA μ A f q Q, \array{ A \otimes 1 &\overset{\simeq}{\longleftarrow}& A \\ {}^{\mathllap{id\otimes e}}\downarrow & \searrow^{\simeq} & \downarrow^{\mathrlap{=}} \\ A \otimes A &\overset{\mu}{\longrightarrow}& A \\ {}^{\mathllap{f}}\downarrow & \swarrow_{\mathrlap{q}} \\ Q } \,,

where the top left triangle is the unitality condition and the two isomorphisms are the right unitor and its inverse. The commutativity of this diagram says that q=ϕq = \phi.

Definition

Given a monoidal category of modules (AMod, A,A)(A Mod , \otimes_A , A) as in prop. , then a monoid (E,μ,e)(E, \mu, e) in (AMod, A,A)(A Mod , \otimes_A , A) (def. ) is called an AA-algebra.

Propposition

Given a monoidal category of modules (AMod, A,A)(A Mod , \otimes_A , A) in a monoidal category (𝒞,,1)(\mathcal{C},\otimes, 1) as in prop. , and an AA-algebra (E,μ,e)(E,\mu,e) (def. ), then there is an equivalence of categories

AAlg comm(𝒞)CMon(AMod)CMon(𝒞) A/ A Alg_{comm}(\mathcal{C}) \coloneqq CMon(A Mod) \simeq CMon(\mathcal{C})^{A/}

between the category of commutative monoids in AModA Mod and the coslice category of commutative monoids in 𝒞\mathcal{C} under AA, hence between commutative AA-algebras in 𝒞\mathcal{C} and commutative monoids EE in 𝒞\mathcal{C} that are equipped with a homomorphism of monoids AEA \longrightarrow E.

(e.g. EKMM 97, VII lemma 1.3)

Proof

In one direction, consider a AA-algebra EE with unit e E:AEe_E \;\colon\; A \longrightarrow E and product μ E/A:E AEE\mu_{E/A} \colon E \otimes_A E \longrightarrow E. There is the underlying product μ E\mu_E

EAE AAA EE coeq E AE μ E μ E/A E. \array{ E \otimes A \otimes E & \underoverset {\underset{}{\longrightarrow}} {\overset{}{\longrightarrow}} {\phantom{AAA}} & E \otimes E &\overset{coeq}{\longrightarrow}& E \otimes_A E \\ && & {}_{\mathllap{\mu_E}}\searrow & \downarrow^{\mathrlap{\mu_{E/A}}} \\ && && E } \,.

By considering a diagram of such coequalizer diagrams with middle vertical morphism e Ee Ae_E\circ e_A, one find that this is a unit for μ E\mu_E and that (E,μ E,e Ee A)(E, \mu_E, e_E \circ e_A) is a commutative monoid in (𝒞,,1)(\mathcal{C}, \otimes, 1).

Then consider the two conditions on the unit e E:AEe_E \colon A \longrightarrow E. First of all this is an AA-module homomorphism, which means that

()AA ide E AE μ A ρ A e E E (\star) \;\;\;\;\; \;\;\;\;\; \array{ A \otimes A &\overset{id \otimes e_E}{\longrightarrow}& A \otimes E \\ {}^{\mathllap{\mu_A}}\downarrow && \downarrow^{\mathrlap{\rho}} \\ A &\underset{e_E}{\longrightarrow}& E }

commutes. Moreover it satisfies the unit property

A AE e Aid E AE μ E/A E. \array{ A \otimes_A E &\overset{e_A \otimes id}{\longrightarrow}& E \otimes_A E \\ & {}_{\mathllap{\simeq}}\searrow & \downarrow^{\mathrlap{\mu_{E/A}}} \\ && E } \,.

By forgetting the tensor product over AA, the latter gives

AE eid EE A AE e Eid E AE μ E/A E = EAE e Eid EE ρ μ E E id E, \array{ A \otimes E &\overset{e \otimes id}{\longrightarrow}& E \otimes E \\ \downarrow && \downarrow^{\mathrlap{}} \\ A \otimes_A E &\overset{e_E \otimes id}{\longrightarrow}& E \otimes_A E \\ {}^{\mathllap{\simeq}}\downarrow && \downarrow^{\mathrlap{\mu_{E/A}}} \\ E &=& E } \;\;\;\;\;\;\;\; \simeq \;\;\;\;\;\;\;\; \array{ A \otimes E &\overset{e_E \otimes id}{\longrightarrow}& E \otimes E \\ {}^{\mathllap{\rho}}\downarrow && \downarrow^{\mathrlap{\mu_{E}}} \\ E &\underset{id}{\longrightarrow}& E } \,,

where the top vertical morphisms on the left the canonical coequalizers, which identifies the vertical composites on the right as shown. Hence this may be pasted to the square ()(\star) above, to yield a commuting square

AA ide E AE e Eid EE μ A ρ μ E A e E E id E=AA e Ee E EE μ A μ E A e E E. \array{ A \otimes A &\overset{id\otimes e_E}{\longrightarrow}& A \otimes E &\overset{e_E \otimes id}{\longrightarrow}& E \otimes E \\ {}^{\mathllap{\mu_A}}\downarrow && {}^{\mathllap{\rho}}\downarrow && \downarrow^{\mathrlap{\mu_{E}}} \\ A &\underset{e_E}{\longrightarrow}& E &\underset{id}{\longrightarrow}& E } \;\;\;\;\;\;\;\;\;\; = \;\;\;\;\;\;\;\;\;\; \array{ A \otimes A &\overset{e_E \otimes e_E}{\longrightarrow}& E \otimes E \\ {}^{\mathllap{\mu_A}}\downarrow && \downarrow^{\mathrlap{\mu_E}} \\ A &\underset{e_E}{\longrightarrow}& E } \,.

This shows that the unit e Ae_A is a homomorphism of monoids (A,μ A,e A)(E,μ E,e Ee A)(A,\mu_A, e_A) \longrightarrow (E, \mu_E, e_E\circ e_A).

Now for the converse direction, assume that (A,μ A,e A)(A,\mu_A, e_A) and (E,μ E,e E)(E, \mu_E, e'_E) are two commutative monoids in (𝒞,,1)(\mathcal{C}, \otimes, 1) with e E:AEe_E \;\colon\; A \to E a monoid homomorphism. Then EE inherits a left AA-module structure by

ρ:AEe AidEEμ EE. \rho \;\colon\; A \otimes E \overset{e_A \otimes id}{\longrightarrow} E \otimes E \overset{\mu_E}{\longrightarrow} E \,.

By commutativity and associativity it follows that μ E\mu_E coequalizes the two induced morphisms EAEAAEEE \otimes A \otimes E \underoverset{\longrightarrow}{\longrightarrow}{\phantom{AA}} E \otimes E. Hence the universal property of the coequalizer gives a factorization through some μ E/A:E AEE\mu_{E/A}\colon E \otimes_A E \longrightarrow E. This shows that (E,μ E/A,e E)(E, \mu_{E/A}, e_E) is a commutative AA-algebra.

Finally one checks that these two constructions are inverses to each other, up to isomorphism.

When thinking of commutative monoids in some tensor category as formal duals to certain spaces, as in def. , then we are interested in forming Cartesian products and more generally fiber products of these spaces. Dually this is given by [fcoproducts] of commutative monoids and commutative RR-algebras. The following says that these may be computed just as the tensor product of modules:

Proposition

Let 𝒞\mathcal{C} be a symmetric monoidal category such that

  1. it has reflexive coequalizers

  2. which are preserved by the tensor product functors A():𝒞𝒞A \otimes (-) \colon \mathcal{C} \to \mathcal{C} for all objects AA in 𝒞\mathcal{C}.

Then for f:ABf \colon A \to B and g:ACg \colon A \to C two morphisms in the category CMon(𝒞)CMon(\mathcal{C}) of commutative monoids in 𝒞\mathcal{C} (def. ), the underlying object in 𝒞\mathcal{C} of the pushout in CMon(𝒞)CMon(\mathcal{C}) coincides with the tensor product in the monoidal category AAMod (according to prop. ):

U(B AC)B AC. U\left(B \sqcup_A C\right) \simeq B \otimes_A C \,.

Here BB and CC are regarded as equipped with the canonical AA-module structure induced by the morphisms ff and gg, respectively.

This appears for instance as (Johnstone, page 478, cor. 1.1.9).

Remark

In every tensor category (def. ) the conditions in prop. are satisfied.

Proof

By definition, every tensor category is an abelian category (def. ). The coequalizer of two parallel morphisms f,gf,g in an abelian category is isomorphic to the cokernel of the difference fgf-g (formed in the abelian group struture on the hom-space). Hence all coequalizers exist, in particlar the split coequalizers required in prop. .

Moreover, by definition every tensor category is a rigid monoidal category. This implies that it is also a closed monoidal categories, by prop. , and this means that the functors A()A \otimes (-) are left adjoint functors, and such preserve all colimits.

Proposition

Let 𝒜\mathcal{A} be a tensor category, and let RCMon(𝒜)R \in CMon(\mathcal{A}) be a commutative monoid in 𝒜\mathcal{A}.

Then for A 1,A 2A_1, A_2 two RR-algebas according to def. , regarded as affine schemes Spec(A 1),Spec(A 2)Aff(RMod(𝒞))Spec(A_1), Spec(A_2) \in Aff(R Mod(\mathcal{C})) according to prop. and def. the Cartesian product of Spec(A 1)Spec(A_1) with Spec(A 2)Spec(A_2) exists in Aff(RMod(𝒞))Aff(R Mod(\mathcal{C})) and is the formal dual of the tensor product algebra A 1 RA 2A_1 \otimes_R A_2 according to example :

Spec(A 1)×Spec(A 2)Spec(A 1 RA 2). Spec(A_1) \times Spec(A_2) \simeq Spec(A_1 \otimes_R A_2) \,.
Proof

By prop. the formal dual of the statement is given by prop. , which does apply, according to remark .

Proposition

Let 𝒞\mathcal{C} be a symmetric monoidal category, let A 1,A 2CMon(𝒞)A_1, A_2 \in CMon(\mathcal{C}) be two commutative monoids in 𝒞\mathcal{C} (def. ) and

ϕ:A 1A 2 \phi \;\colon\; A_1 \longrightarrow A_2

a homomorphism commutative monoids (def. ).

Then there is a pair of adjoint functors between the categories of modules (def. )

A 1Mod(𝒞)ϕ *ϕ *A 2Mod(𝒞) A_1 Mod(\mathcal{C}) \underoverset {\underset{\phi^\ast}{\longleftarrow}} {\overset{\phi_\ast}{\longrightarrow}} {} A_2 Mod(\mathcal{C})

where

  1. the right adjoint, called restriction of scalars, sends an A 2A_2-module (N,ρ)(N, \rho) to the A 1A_1-module (N,ρ)(N,\rho') whose action is given by precomposition with ϕ\phi:

    A 1NϕidA 2NρN. A_1 \otimes N \stackrel{\phi \otimes id}{\longrightarrow} A_2 \otimes N \stackrel{\rho}{\longrightarrow} N \,.
  2. the left adjoint, called extension of scalars sends an A 1A_1-module (N,ρ)(N,\rho) to the tensor product

    ϕ *(N)A 2 A 1N \phi_\ast(N) \coloneqq A_2 \otimes_{A_1} N

    (where we are regarding A 2A_2 as a commutative monoid in A 1A_1-modules via prop. ) and equipped with the evident action induced by the multiplication in A 2A_2:

A 2ϕ *(N)=A 2A 2 A 1Nμ A 2 A 1NA 2 A 1N=ϕ *(N). A_2 \otimes \phi^\ast(N) = A_2 \otimes A_2 \otimes_{A_1} N \stackrel{\mu_{A_2} \otimes_{A_1} N }{\longrightarrow} A_2 \otimes_{A_1} N = \phi^\ast(N) \,.
Proof

By prop. the adjunction in question has the form

A 1Mod(𝒞)UFA 2Mod((A 1Mod, A 1,A 1)) A_1 Mod(\mathcal{C}) \underoverset {\underset{U}{\longleftarrow}} {\overset{F}{\longrightarrow}} {} A_2 Mod( (A_1 Mod, \otimes_{A_1}, A_1) )

and hence the statement follows with prop. .

Remark

In the dual interpretation of RR-modules as generalized vector bundles (namely: quasicoherent sheaves) over Spec(R)Spec(R) (def. ) then ϕ:A 1A 2\phi \colon A_1 \to A_2 becomes a map of spaces

Spec(ϕ):Spec(A 2)Spec(A 1) Spec(\phi) \colon Spec(A_2) \longrightarrow Spec(A_1)

and then extension of scalars according to prop. corresponds to the pullback of vector bundles from Spec(A 1)Spec(A_1) to Spec(A 2)Spec(A_2).

Super-Groups as super-commutative Hopf algebras

Above we have considered affine spaces Spec(A)Spec(A) (def. ) in symmetric monoidal categories 𝒞\mathcal{C}. Now we discuss what it means to equip these with the stucture of group objects, hence to form affine groups in 𝒞\mathcal{C}.

A (possibly) familiar example arises in differential geometry, where one considers groups whose underlying set is promoted to a smooth manifold and all whose operations (product, inverses) are smooth functions. These are of course the Lie groups.

A linear representation of a Lie group GG on a vector space VV is a smooth function

ρ:G×VV \rho \;\colon\; G \times V \longrightarrow V

such that

  1. (linearity) for all elements gGg \in G the function

    ρ(g):VV \rho(g) \colon V \longrightarrow V

    is a linear function

  2. (unitality) for eGe \in G the neutral element then ρ(e)\rho(e) is the identity function;

  3. (action property) for g 1,g 2Gg_1, g_2 \in G any two elements, then acting with them consecutively is the same as acting with their product:

    ρ(g 2)ρ(g 1)=ρ(g 2g 1). \rho(g_2) \circ \rho(g_1) = \rho(g_2 g_1) \,.

But here we need to consider groups with more general geometric structure. The key to the generalization is to regard spaces dually via their algebras of functions.

In the above example, write C (X)C^\infty(X) for the smooth algebra of smooth functions on a smooth manifold XX. The assignment

C ():SmthMfdSmthAlg op C^\infty(-) \;\colon\; SmthMfd \hookrightarrow SmthAlg_{\mathbb{R}}^{op}

is the embedding of smooth manifolds into formal duals of R-algebras from prop. .

Moreover, the functor C ()C^\infty(-) sends Cartesian products of smooth manifolds to “completed tensor products” c\otimes^c of function algebras (namely to the coproduct of smooth algebras, see there)

C (X×Y)C (X) cC (Y). C^\infty(X \times Y) \simeq C^\infty(X) \otimes^c C^\infty(Y) \,.

Together this means that if X=GX = G is equipped with the structure of a group object, then the product operation in the group induces a “coproduct” operation on its smooth algebra of smooth functions:

product : G×G G C (G) cC (G) C (G) : product *=coproduct. \array{ product &\colon& G \times G &\longrightarrow& G \\ && C^\infty(G) \otimes^c C^\infty(G) &\longleftarrow& C^\infty(G) &\colon& product^\ast = coproduct } \,.

Now the associativity of the group product translates into a corresponding dual property of its dual, called “co-associativity”, and so forth. The resulting algebraic structure is called a Hopf algebra.

While the explicit definition of a Hopf algebra may look involved at first sight, Hopf algebras are simply formal duals of groups. Since this perspective is straightforward, we may just as well consider it in the generality of groupoids.

A simple illustrative archetype of the following construction of commutative Hopf algebroids from homotopy commutative ring spectra is the following situation:

For XX a finite set consider

X×X×X =(pr 1,pr 3) X×X s=pr 1 t=pr 2 X \array{ X \times X \times X \\ \downarrow^{\mathrlap{\circ = (pr_1, pr_3)}} \\ X \times X \\ {}^{\mathllap{s = pr_1}}\downarrow \uparrow \downarrow^{\mathrlap{t = pr_2}} \\ X }

as the (“codiscrete”) groupoid with XX as objects and precisely one morphism from every object to every other. Hence the composition operation \circ, and the source and target maps are simply projections as shown. The identity morphism (going upwards in the above diagram) is the diagonal.

Then consider the image of this structure under forming the free abelian groups [X]\mathbb{Z}[X], regarded as commutative rings under pointwise multiplication.

Since

[X×X][X][X] \mathbb{Z}[X \times X] \simeq \mathbb{Z}[X] \otimes \mathbb{Z}[X]

this yields a diagram of homomorphisms of commutative rings of the form

([X][X]) [X]([X][X]) [X][X] [X] \array{ (\mathbb{Z}[X] \otimes \mathbb{Z}[X] ) \otimes_{\mathbb{Z}[X]} (\mathbb{Z}[X] \otimes \mathbb{Z}[X]) \\ \uparrow^{\mathrlap{} } \\ \mathbb{Z}[X] \otimes \mathbb{Z}[X] \\ \uparrow \downarrow \uparrow \\ \mathbb{Z}[X] }

satisfying some obvious conditions. Observe that here

  1. the two morphisms [X][X][X]\mathbb{Z}[X] \rightrightarrows \mathbb{Z}[X] \otimes \mathbb{Z}[X] are ffef \mapsto f \otimes e and feff \mapsto e \otimes f, respectively, where ee denotes the unit element in [X]\mathbb{Z}[X];

  2. the morphism [X][X][X]\mathbb{Z}[X] \otimes \mathbb{Z}[X] \to \mathbb{Z}[X] is the multiplication in the ring [X]\mathbb{Z}[X];

  3. the morphism

    [X][X][X][C][C]([X][X]) [X]([X][X]) \mathbb{Z}[X] \otimes \mathbb{Z}[X] \longrightarrow \mathbb{Z}[X] \otimes \mathbb{Z}[C] \otimes \mathbb{Z}[C] \overset{\simeq}{\longrightarrow} (\mathbb{Z}[X] \otimes \mathbb{Z}[X] ) \otimes_{\mathbb{Z}[X]} (\mathbb{Z}[X] \otimes \mathbb{Z}[X])

    is given by fgfegf \otimes g \mapsto f \otimes e \otimes g.

We now say this again, in generality:

Definition

Let 𝒜\mathcal{A} be a tensor category (def. ). A commutative Hopf algebroid in 𝒜\mathcal{A} is an internal groupoid in the opposite category CMon(𝒜) opCMon(\mathcal{A})^{op} of commutative monoids in 𝒜\mathcal{A}, regarded with its cartesian monoidal category structure according to prop. .

(e.g. Ravenel 86, def. A1.1.1)

We unwind def. . For RCMon(𝒜)R \in CMon(\mathcal{A}), write Spec(R)Spec(R) for same same object, but regarded as an object in CMon(𝒜) opCMon(\mathcal{A})^{op}.

Proposition

An internal category in CMon(𝒜) opCMon(\mathcal{A})^{op} is a diagram in CMon(𝒜) opCMon(\mathcal{A})^{op} of the form

Spec(Γ)×Spec(A)Spec(Γ) Spec(Γ) s i t Spec(A), \array{ Spec(\Gamma) \underset{Spec(A)}{\times} Spec(\Gamma) \\ \downarrow^{\mathrlap{\circ}} \\ Spec(\Gamma) \\ {}^{\mathllap{s}}\downarrow \; \uparrow^{\mathrlap{i}} \downarrow^{\mathrlap{t}} \\ Spec(A) } \,,

(where the fiber product at the top is over ss on the left and tt on the right) such that the pairing \circ defines an associative composition over Spec(A)Spec(A), unital with respect to ii. This is an internal groupoid if it is furthemore equipped with a morphism

inv:Spec(Γ)Spec(Γ) inv \;\colon\; Spec(\Gamma) \longrightarrow Spec(\Gamma)

acting as assigning inverses with respect to \circ.

The key fact to use now is prop. : the tensor product of commutative monoids exhibits the cartesian monoidal category structure on CMon(𝒜) opCMon(\mathcal{A})^{op}, :

Spec(R 1)×Spec(R 3)Spec(R 2)Spec(R 1 R 3R 2). Spec(R_1) \underset{Spec(R_3)}{\times} Spec(R_2) \simeq Spec(R_1 \otimes_{R_3} R_2) \,.

This means that def. is equivalently a diagram in CMon(𝒜)CMon(\mathcal{A}) of the form

ΓAΓ Ψ Γ η L ϵ η R A \array{ \Gamma \underset{A}{\otimes} \Gamma \\ \uparrow^{\mathrlap{\Psi}} \\ \Gamma \\ {}^{\mathllap{\eta_L}}\uparrow \downarrow^{\mathrlap{\epsilon}} \; \uparrow^{\mathrlap{\eta_R}} \\ A }

as well as

c:ΓΓ c \; \colon \; \Gamma \longrightarrow \Gamma

and satisfying formally dual conditions, spelled out as def. below. Here

  • η L,etaR\eta_L, \etaR are called the left and right unit maps;

  • ϵ\epsilon is called the co-unit;

  • Ψ\Psi is called the comultiplication;

  • cc is called the antipode or conjugation

Remark

Generally, in a commutative Hopf algebroid, def. , the two morphisms η L,η R:AΓ\eta_L, \eta_R\colon A \to \Gamma from remark need not coincide, they make Γ\Gamma genuinely into a bimodule over AA, and it is the tensor product of bimodules that appears in remark . But it may happen that they coincide:

An internal groupoid 𝒢 1ts𝒢 0\mathcal{G}_1 \stackrel{\overset{s}{\longrightarrow}}{\underset{t}{\longrightarrow}} \mathcal{G}_0 for which the domain and codomain morphisms coincide, s=ts = t, is euqivalently a group object in the slice category over 𝒢 0\mathcal{G}_0.

Dually, a commutative Hopf algebroid Γη Rη LA\Gamma \stackrel{\overset{\eta_L}{\longleftarrow}}{\underset{\eta_R}{\longleftarrow}} A for which η L\eta_L and η R\eta_R happen to coincide is equivalently a commutative Hopf algebra Γ\Gamma over AA.

Writing out the formally dual axioms of an internal groupoid as in remark yields the following equivalent but maybe more explicit definition of commutative Hopf algebroids, def.

Definition

A commutative Hopf algebroid is

  1. two commutative rings, AA and Γ\Gamma;

  2. ring homomorphisms

    1. (left/right unit)

      η L,η R:AΓ\eta_L,\eta_R \colon A \longrightarrow \Gamma;

    2. (comultiplication)

      Ψ:ΓΓAΓ\Psi \colon \Gamma \longrightarrow \Gamma \underset{A}{\otimes} \Gamma;

    3. (counit)

      ϵ:ΓA\epsilon \colon \Gamma \longrightarrow A;

    4. (conjugation)

      c:ΓΓc \colon \Gamma \longrightarrow \Gamma

such that

  1. (co-unitality)

    1. (identity morphisms respect source and target)

      ϵη L=ϵη R=id A\epsilon \circ \eta_L = \epsilon \circ \eta_R = id_A;

    2. (identity morphisms are units for composition)

      (id Γ Aϵ)Ψ=(ϵ Aid Γ)Ψ=id Γ(id_\Gamma \otimes_A \epsilon) \circ \Psi = (\epsilon \otimes_A id_\Gamma) \circ \Psi = id_\Gamma;

    3. (composition respects source and target)

      1. Ψη R=(id Aη R)η R\Psi \circ \eta_R = (id \otimes_A \eta_R) \circ \eta_R;

      2. Ψη L=(η L Aid)η L\Psi \circ \eta_L = (\eta_L \otimes_A id) \circ \eta_L

  2. (co-associativity) (id Γ AΨ)Ψ=(Ψ Aid Γ)Ψ(id_\Gamma \otimes_A \Psi) \circ \Psi = (\Psi \otimes_A id_\Gamma) \circ \Psi;

  3. (inverses)

    1. (inverting twice is the identity)

      cc=id Γc \circ c = id_\Gamma;

    2. (inversion swaps source and target)

      cη L=η Rc \circ \eta_L = \eta_R; cη R=η Lc \circ \eta_R = \eta_L;

    3. (inverse morphisms are indeed left and right inverses for composition)

      the morphisms α\alpha and β\beta induced via the coequalizer property of the tensor product from ()c()(-) \cdot c(-) and c()()c(-)\cdot (-), respectively

      ΓAΓ ΓΓ coeq Γ AΓ ()c() α Γ \array{ \Gamma \otimes A \otimes \Gamma & \underoverset {\longrightarrow} {\longrightarrow} {} & \Gamma \otimes \Gamma & \overset{coeq}{\longrightarrow} & \Gamma \otimes_A \Gamma \\ && {}_{\mathllap{(-)\cdot c(-)}}\downarrow & \swarrow_{\mathrlap{\alpha}} \\ && \Gamma }

      and

      ΓAΓ ΓΓ coeq Γ AΓ c()() β Γ \array{ \Gamma \otimes A \otimes \Gamma & \underoverset {\longrightarrow} {\longrightarrow} {} & \Gamma \otimes \Gamma & \overset{coeq}{\longrightarrow} & \Gamma \otimes_A \Gamma \\ && {}_{\mathllap{c(-)\cdot (-)}}\downarrow & \swarrow_{\mathrlap{\beta}} \\ && \Gamma }

      satisfy

      αΨ=η Lϵ\alpha \circ \Psi = \eta_L \circ \epsilon

      and

      βΨ=η Rϵ\beta \circ \Psi = \eta_R \circ \epsilon .

e.g. (Ravenel 86, def. A1.1.1)

By internalizing all of the above from VectVect to sVectsVect, we obtain the concept of supergroups:

Definition

An affine algebraic supergroup GG is equivalently

We will often just say “supergroup” for short in the following. If HH is the corresponding supercommutative Hopf algebra then we also write Spec(H)Spec(H) for this supergroup.

The following asks that the parity involution (def. ) on a supergroup is an inner automorphism:

Definition

An inner parity of a supergroup GG, def. is an element ϵG even\epsilon \in G_{even} such that

  1. it is involutive i.e. ϵ 2=1\epsilon^2 = 1

  2. its adjoint action on GG is the parity involution of def. .

Dually this mean that an inner pariy is an algebra homomorphism ϵ *:𝒪(G)k\epsilon^\ast \colon\mathcal{O}(G) \to k such that

  1. the composite

    𝒪(G)Ψ𝒪(G) k𝒪(G)ϵ * kϵ *k kkk \mathcal{O}(G) \stackrel{\Psi}{\longrightarrow} \mathcal{O}(G) \otimes_k \mathcal{O}(G) \stackrel{\epsilon^\ast \otimes_k \epsilon^\ast}{\longrightarrow} k \otimes_k k \simeq k

    is the counit of the Hopf algebra (hence the formal dual of the neutral element)

  2. the parity involution 𝒪(G)𝒪(G)\mathcal{O}(G) \stackrel{\simeq}{\longrightarrow} \mathcal{O}(G) conincides with the composite

    𝒪(G)(id kΨ)Ψ𝒪(G) k𝒪(G) k𝒪(G)ϵ * kid k(cϵ *)k k𝒪(G) kk \mathcal{O}(G) \stackrel{(id \otimes_k \Psi) \circ \Psi}{\longrightarrow} \mathcal{O}(G) \otimes_k \mathcal{O}(G) \otimes_k \mathcal{O}(G) \stackrel{\epsilon^\ast \otimes_k id \otimes_k (c \circ \epsilon^\ast)}{\longrightarrow} k \otimes_k \mathcal{O}(G) \otimes_k k

(Deligne 02, 0.3)

Example

For GG an ordinary affine algebraic group, regarded as a supergroup with trivial odd-graded part, then every element ϵZ(G)\epsilon \in Z(G) in the center defines an inner parity, def. .

(Deligne 02, 0.4 i))

Remark

In view of remak , specifying an involutive central element in an ordinary group is a faint shadow of genuine supergroup structure. In fact such pairs are being referred to as “supergroups” in (Müger 06).

Demanding the existence of inner parity is not actually a restriction of the theory:

Example

For HH any supergroup, def. , and 2={id,par}\mathbb{Z}_2 = \{id,par\} acting on it by parity involution, def. then the semidirect product group 2G\mathbb{Z}_2 \ltimes G has inner parity, def. , given by ϵpar 2 2G\epsilon \coloneqq par \in \mathbb{Z}_2 \hookrightarrow \mathbb{Z}_2 \ltimes G.

(Deligne 02, 0.4 ii))

Linear super-representations as Comodules

Definition

Given a commutative Hopf algebroid Γ\Gamma over AA (def. ) in some tensor category (def. ), then a left comodule over Γ\Gamma is

  1. an AA-module object NN in 𝒜\mathcal{A} (def. ) i;

  2. an AA-module homomorphism (co-action)

    Ψ N:NΓ AN\Psi_N \;\colon\; N \longrightarrow \Gamma \otimes_A N;

such that

  1. (co-unitality)

    (ϵ Aid N)Ψ N=id N(\epsilon \otimes_A id_N) \circ \Psi_N = id_N;

  2. (co-action property)

    (Ψ Aid N)Ψ N=(id Γ AΨ N)Ψ N(\Psi \otimes_A id_N) \circ \Psi_N = (id_\Gamma \otimes_A \Psi_N)\circ \Psi_N.

A homomorphism between comodules N 1N 2N_1 \to N_2 is a homomorphism of underlying AA-modules making commuting diagrams with the co-action morphism. Write

ΓCoMod(𝒜) \Gamma CoMod(\mathcal{A})

for the resulting category of (left) comodules over Γ\Gamma. Analogously there are right comodules.

Example

For (Γ,A)(\Gamma,A) a commutative Hopf algebroid, then AA becomes a left Γ\Gamma-comodule (def. ) with coaction given by the right unit

Aη RΓΓ AA. A \overset{\eta_R}{\longrightarrow} \Gamma \simeq \Gamma \otimes_A A \,.
Proof

The required co-unitality property is the dual condition in def.

ϵη R=id A \epsilon \circ \eta_R = id_A

of the fact in def. that identity morphisms respect sources:

id:Aη RΓΓ AAϵ AidA AAA id \;\colon\; A \overset{\eta_R}{\longrightarrow} \Gamma \simeq \Gamma \otimes_A A \overset{\epsilon \otimes_A id}{\longrightarrow} A \otimes_A A \simeq A

The required co-action property is the dual condition

Ψη R=(id Aη R)η R \Psi \circ \eta_R = (id \otimes_A \eta_R) \circ \eta_R

of the fact in def. that composition of morphisms in a groupoid respects sources

A η R Γ η R Ψ ΓΓ AA id Aη R Γ AΓ. \array{ A &\overset{\eta_R}{\longrightarrow}& \Gamma \\ {}^{\mathllap{\eta_R}}\downarrow && \downarrow^{\mathrlap{\Psi}} \\ \Gamma \simeq \Gamma \otimes_A A &\underset{id \otimes_A \eta_R}{\longrightarrow}& \Gamma \otimes_A \Gamma } \,.
Definition

Given two comodules N 1,N 2N_1, N_2 over a commutative Hopf algebra Γ\Gamma over kk, then their tensor product is the the tensor product of modules N 1 kN 2N_1 \otimes_k N_2 equipped with the following co-action

N 1 kN 2longightarrowΨ1 kΨ 2Γ kN 1 kΓN 2Γ kΓ kN 1 kN 2(()()) kid N 1 kid N 2Γ kN 1 kN 2. N_1 \otimes_k N_2 \overset{\Psi1 \otimes_k \Psi_2}{\longightarrow} \Gamma \otimes_k N_1 \otimes_k \Gamma \otimes N_2 \overset{}{\longrightarrow} \Gamma \otimes_k \Gamma \otimes_k N_1 \otimes_k N_2 \overset{((-)\cdot (-)) \otimes_k id_{N_1} \otimes_k id_{N_2} }{\longrightarrow} \Gamma \otimes_k N_1 \otimes_k N_2 \,.

This is the formal dual of the tensor product of representations, the action on which is induced by

G×V 1×V 2Δ G×idG×G×V 1×V 2G×V 1×G×V 1ρ 1×ρ 2V 1×V 2. G \times V_1 \times V_2 \overset{\Delta_G \times id}{\longrightarrow} G \times G \times V_1 \times V_2 \simeq G \times V_1 \times G \times V_1 \overset{\rho_1 \times \rho_2}{\longrightarrow} V_1 \times V_2 \,.

Under the tensor product of co-modules (def. ), these form a symmetric monoidal category (def. ).

Definition

A linear representation of a supergroup GG, def. , with inner parity ϵ\epsilon, def. , is

such that

  • the innr parity element ϵ\epsilon acts as the identity on V evenV_{even} and by multiplicatin with 1-1 on V oddV_{odd}.
Example

For GG an ordinary (affine algebraic) group regarded as a supergroup with trivial odd-graded part, and for ϵ=e\epsilon = e its neutral element taken as the inner parity, then Rep(G,ϵ)Rep(G,\epsilon) in the sense of def. is just the ordinary category of representations of GG.

(Deligne 02, 0.4 i))

Proposition

The category of representations Rep(G,ϵ)Rep(G,\epsilon) of def. of an affine algebraic supergroup GG, def. , with inner parity ϵ\epsilon (def. ) on finite-dimensional super vector spaces (example ) and equippd with the tensor product of comodules from def. is a kk-tensor category (def. ) of subexponential growth (def. ).

(Deligne 02, 1.21)

Moreover, any finite dimensional faithful representation (which always exists, prop.) serves as an \otimes-generator (def. ).

See (this prop.).

Super Fiber functors and their automorphism supergroups

The first step in exhibiting a given tensor category 𝒜\mathcal{A} as being a category of representations is to exhibit its objects as having an underlying representation space of sorts, and then an action represented on that space. Hence a necessary condition on 𝒜\mathcal{A} is that there exists a forgetful functor

ω:𝒜𝒱 \omega \;\colon\; \mathcal{A} \longrightarrow \mathcal{V}

to some other tensor category, such that ω\omega satisfies a list of properties, in particular it should be a symmetric strong monoidal functor.

Such functors are called fiber functors. The idea is that we think of 𝒜\mathcal{A} as a bundle over 𝒱\mathcal{V}, and over each V𝒱V \in \mathcal{V} we find the fiber ω 1(V)\omega^{-1}(V) of that bundle, consisting of all those objects in 𝒜\mathcal{A} whose underlying object in the given VV.

The main point of Tannaka duality of tensor categories is the observation that if 𝒜\mathcal{A} is a category of representations of some group GG, then GG also acts by automorphisms on that fiber functor (i.e. via natural isomorphisms of functors). In good cases then this may be turned around, and the full automorphism group of a fiber functor is identified with the group GG for which the objects in its fibers are representations, this is the process of Tannaka reconstruction.

There are slight variants on what one requires of a fiber functor. For the present purpose we fix the following definition

Definition

Let 𝒜\mathcal{A} and 𝒯\mathcal{T} be two kk-tensor categories (def. ) such that

  1. all objects have finite length;

  2. all hom spaces are of finite dimension over kk.

Let RCMon(Ind(𝒯))R \in CMon(Ind(\mathcal{T})) be a commutative monoid (def. ) in the category of ind-objects in 𝒯\mathcal{T} (prop. ).

Then a fiber functor on 𝒜\mathcal{A} over RR is a functor

ω:𝒜RMod(Ind(𝒯)) \omega \;\colon\; \mathcal{A} \longrightarrow R Mod(Ind(\mathcal{T}))

from 𝒜\mathcal{A} to the category of module objects over RR (def. ) in the category of ind-objects Ind(𝒯)Ind(\mathcal{T}) (def. ), which is

  1. a braided strong monoidal functor;

  2. an exact functor in both variables.

If here 𝒯=\mathcal{T} = sFinVect (def. ), then this is called a super fiber functor.

(Deligne 02, 3.1)

Definition

A tensor category 𝒜\mathcal{A} (def. ) is called

  1. a neutral Tannakian category if it admits a fiber functor (def. ) to Vect kVect_k (example ) (Deligne-Milne 12, def. 2.19)

  2. a neutral super Tannakian category if it admits a fiber functor (def. ) to sVect ksVect_k (def. )

  3. (not needed here) a general Tannakian category if the stack on Aff kAff_k which sends RCRingR \in CRing to the groupoid of fiber functors to RProjRModR Proj \hookrightarrow R Mod (projective modules over RR) is an affine gerbe such that its category of representations is equivalent to 𝒜\mathcal{A} (Deligne-Milne 12, def. 3.7).

Given a super fiber functor ω:𝒜sVect k\omega \colon \mathcal{A} \to sVect_k (def. ) there is an evident notion of its automorphism group: a homomorphism between functors is a natural transformation, and that between monoidal functors is a monoidal natural transformation, according to def. , and this is an automorphism of functors if it is a natural automorphism. We write

Aut(ω)Grp Aut(\omega) \in Grp

for this automorphism group.

So far this is a group without geometric structure (a discrete group). But it is naturally equipped with supergeometry (super-algebraic geometry) exhibited by a rule for what the geometrically parameterized families of its elements are. (For exposition of this perspective see at motivation for sheaves, cohomology and higher stacks).

Concretely, this means that for each supercommutative superalgebra AA with corresponding affine super scheme Spec(A)Spec(A) (def. , def. ) we are to say what the set

Aut̲(ω)(Spec(A))Set \underline{Aut}(\omega)(Spec(A)) \in Set

of Spec(A)Spec(A)-parameterized elements of Aut(ω)Aut(\omega) is. In fact, under parameter-wise multiplication in the group, any such set must inherit group structure, so that we should have not one discrete group, but a system of them, labeled by supercommutative superalgebras:

Aut̲(ω)(Spec(A))Grp. \underline{Aut}(\omega)(Spec(A)) \in Grp \,.

Moreover, if A 1A 2A_1 \longrightarrow A_2 is an algebra homomorphism, hence

Spec(A 2)Spec(A 1) Spec(A_2) \longrightarrow Spec(A_1)

a map of affine super schemes according to def. , then there should be a group homomorphism

Aut̲(ω)(Spec(A 2))Aut̲(ω)(Spec(A 1)) \underline{Aut}(\omega)(Spec(A_2)) \longleftarrow \underline{Aut}(\omega)(Spec(A_1))

that expresses how a Spec(A 1)Spec(A_1)-parameterized family of elements of Aut(ω)Aut(\omega) becomes a Spec(A 1)Spec(A_1)-parameterized family, under this map.

For a minimum of consistency, this assignment must be such that the identity map on Spec(A)Spec(A) induces the identity on Aut̲(ω)(Spec(A))\underline{Aut}(\omega)(Spec(A)), and that the composite of two maps of affine superschemes goes to the correspondng composite group homomorphisms.

In conclusion, this says that an algebraic supergeometric structure on Aut(ω)Aut(\omega) is the datum of a presheaf of groups, hence of a functor

Aut̲(ω):Aff(sVect) opCMon(sVect)Grp \underline{Aut}(\omega) \;\colon\; Aff(sVect)^{op} \simeq CMon(sVect) \longrightarrow Grp

such that the underlying points are those of Aut(ω)Aut(\omega):

Aut̲(ω)(Spec(k))Aut(ω). \underline{Aut}(\omega)(Spec(k)) \simeq Aut(\omega) \,.
Definition

We say that a functor

Aut̲(ω):Aff(sVect) op=CMon(sVect)Grp \underline{Aut}(\omega) \;\colon\; Aff(sVect)^{op} = CMon(sVect) \longrightarrow Grp

is representable if there exists a supercommutative Hopf algebra HH, hence an affine algebraic group Spec(H)Spec(H) (def. ) and a natural isomorphism with the hom functor into Spec(H)Spec(H):

Aut̲(ω)Hom CMon(sVect)(H,)=Hom Aff(sVect)(,Spec(H)). \underline{Aut}(\omega) \simeq Hom_{CMon(sVect)}(H,-) = Hom_{Aff(sVect)}(-,Spec(H)) \,.
Definition

Let ω:𝒜\omega \colon \mathcal{A} \to \mathcal{B} be a fiber functor (def ).

For ACMon()A \in CMon(\mathcal{B}) a commutative monoid (def. ), write

ω A:𝒜ωA()AMod() \omega_A \;\colon\; \mathcal{A} \stackrel{\omega}{\longrightarrow} \mathcal{B} \stackrel{A \otimes(-)}{\longrightarrow} A Mod(\mathcal{B})

for its image under extension of scalars along 1A1 \to A to AA (prop. ).

With this, the automorphism group of ω\omega

Aut̲(ω)PSh(Aff()) \underline{Aut}(\omega) \in PSh(Aff(\mathcal{B}))

is defined to be the functor which on objects assigns the discrete group of natural automorphisms of the image ω A\omega_A of ω\omega under extension of scalars as above

Aut̲(ω)(Spec(A))Aut(ω A) \underline{Aut}(\omega)(Spec(A)) \coloneqq Aut(\omega_{A})

and which to a homomorphism of algebras

ϕ:A 1A 2 \phi \;\colon\; A_1 \longrightarrow A_2

assigns the action of the extension of scalars-functor along ϕ\phi:

ϕ *:Aut(ω A 1)Aut(ω A 2). \phi_\ast \;\colon\; Aut(\omega_{A_1}) \longrightarrow Aut(\omega_{A_2}) \,.

This is clearly a presheaf, by functoriality of extension of scalars.

Specializing def. to =\mathcal{B} = sVect (def. ), where a commutative monoid is a supercommutative superalgebra (def. ) it reads as follows:

Example

Let ω:𝒜sVect\omega \colon \mathcal{A} \to sVect be a super fiber functor (def ).

For ACMon(sVect)A \in CMon(sVect) a commutative monoid (def. ), write

ω A:𝒜ωsVectA()AMod(sVect) \omega_A \;\colon\; \mathcal{A} \stackrel{\omega}{\longrightarrow} sVect \stackrel{A \otimes(-)}{\longrightarrow} A Mod(sVect)

For ACMon(sVect)A \in CMon(sVect) a supercommutative algebra, write

ω A:𝒜ωsVectA()AMod(sVect) \omega_A \;\colon\; \mathcal{A} \stackrel{\omega}{\longrightarrow} sVect \stackrel{A \otimes(-)}{\longrightarrow} A Mod(sVect)

for its image under extension of scalars to AA (prop. ).

With this, the automorphism super-group of ω\omega

Aut̲(ω)PSh(Aff(sVect)) \underline{Aut}(\omega) \in PSh(Aff(sVect))

is defined by

Aut̲(ω)(Spec(A))Aut(ω A). \underline{Aut}(\omega)(Spec(A)) \coloneqq Aut(\omega_{A}) \,.
Proposition

For kk an algebraically closed field of characteristic zero, and for 𝒜\mathcal{A} a kk-tensor category equipped with a super fiber functor ω\omega, then its automorphism supergroup (def. ) is representable (def. ): there exists a supercommutative Hopf algebra H ωH_\omega and a natural isomorphism

Aut̲(ω)Hom CMon(sVect)(H ω,)=Hom Aff(sVect)(,Spec(H ω)), \underline{Aut}(\omega) \simeq Hom_{CMon(sVect)}(H_\omega,-) = Hom_{Aff(sVect)}(-, Spec(H_\omega)) \,,

which, with the Yoneda embedding understood, we write simply as

Aut̲(ω)Spec(H ω). \underline{Aut}(\omega) \simeq Spec(H_\omega) \,.

(Deligne 90, prop. 8.11)

The following says that in fact all homomorphisms between fiber functors are necessarily isomorphisms:

Lemma

Every monoidal natural transformation (def. ) between two fiber functors (def. ) is an isomorphism (i.e. a natural isomorphism).

(Deligne 90, 8.11 (ii), Deligne 02, lemma 3.2)

Definition

Let 𝒜\mathcal{A} be a tensor category and regard the identity functor on it as a fiber functor (def. ). Then the automorphism group of id 𝒜id_{\mathcal{A}} according to def. is called the fundamental group of 𝒜\mathcal{A}, denoted:

π(𝒜)Aut̲(id 𝒜) \pi(\mathcal{A}) \coloneqq \underline{Aut}(id_{\mathcal{A}})

(Deligne 90, 8.12, 8.13)

Example

The fundamental group (def. ) of the category of super vector spaces sVect (def. ) is /2\mathbb{Z}/2:

π(sVect)/2. \pi(sVect) \simeq \mathbb{Z}/2 \,.

The non-trivial element in π(sVect)\pi(sVect) acts on any super-vector space as the endomorphism which is the identity on even graded elements, and multiplication by (1)(-1) on odd graded elements.

(Deligne 90, 8.14 iv))

Proposition

For 𝒜\mathcal{A} a kk-tensor category equipped with a super fiber functor ω:𝒜sVect\omega \colon \mathcal{A} \to sVect (def. ), then the automorphism supergroup of ω\omega is the image under the super fiber functor ω\omega of the fundamental group of 𝒜\mathcal{A}, according to def. :

Aut̲(ω)ω(π(𝒜)). \underline{Aut}(\omega) \simeq \omega(\pi(\mathcal{A})) \,.

Here on the right we are using that ω\omega is a strong monoidal functor so that it preserves commutative monoids as well as comonoids by prop. , hence preserves commutative Hopf algebras.

(Deligne 90 (8.13.1))

Proposition

Let 𝒜 1,𝒜 2\mathcal{A}_1, \mathcal{A}_2 be two kk-tensor categories and let

η:𝒜 1𝒜 2 \eta \;\colon\; \mathcal{A}_1 \longrightarrow \mathcal{A}_2

be a functor which is kk-linear, monoidal and exact functor. Then there is induced a canonical group homomorphism

π(𝒜 2)η(π(𝒜 1)) \pi(\mathcal{A}_2) \longrightarrow \eta(\pi(\mathcal{A}_1))

from the fundamental group of 𝒜 1\mathcal{A}_1 (def. ) to the image under η\eta of the fundamental group of 𝒜 2\mathcal{A}_2.

(Deligne 90, 8. 15. 2)

Super-exterior powers and Schur functors

A finite dimensional vector space VV has the property that a high enough alternating power of it vanishes nV=0\wedge^n V = 0, namely this is the case for all n>dim(V)n \gt dim(V), and hence this vanishing is just another reflection of the finiteness of the dimension of VV. For a super vector space VV of degreewise finite dimension an analog statement is still true, but one needs to form not just alternating powers but also symmetric powers (prop. below), in fact one needs to apply a generalization of both of these constructions, a Schur functor.

The operation of forming symmetric powers and alternating powers makes sense in every tensor category. Moreover, these operations are the two extreme cases of the more general concept of Schur functors: Given any object XX and given any choice of irreducible representation V λV_\lambda of the symmetric group Σ n\Sigma_n, then one consider the subobject S λ(X n)S_\lambda(X^{\otimes^n}) of the nn-fold tensor power that is invariant under this action.

The first step in the proof of the main theorem (theorem below) is the proposition (prop. below) that all objects that have subexponential growth of length (def. ) are actually annihilated by some Schur functor for the symmetric group.

Definition

For (𝒜,)(\mathcal{A},\otimes) a kk-tensor category as in def., for X𝒜X \in \mathcal{A} an object, for nn \in \mathbb{N} and λ\lambda a partition of nn, regarded as a Young diagram and hence as a representation of the symmetric group V λV_\lambda, say that the value of the Schur functor S λS_\lambda on XX is

S λ(X) (V λX n) S n =(1n!gS nρ(g))(V λX n) \begin{aligned} S_{\lambda}(X) & \coloneqq (V_\lambda \otimes X^{\otimes_n})^{S_n} \\ & = \left( \frac{1}{n!} \underset{g\in S_n}{\sum} \rho(g) \right) \left( V_\lambda \otimes X^{\otimes_n} \right) \end{aligned}

where

  • () S n(-)^{S_n} is the subobject of invariants;

  • S nS_n is the symmetric group on nn elements;

  • V λV_\lambda is the irreducible representation of S nS_n corresponding to λ\lambda;

  • ρ\rho is diagonal action of S nS_n on V λX nV_\lambda \otimes X^{\otimes_n}, coming from the canonical permutation action on X nX^{\otimes_n};

  • () S n(-)^{S_n} denotes the subspace of invariants under the action ρ\rho

  • the second expression just rewrites the invariants as the image of all elements under group averaging.

(Deligne 02, 1.4)

Example

For λ=(n)\lambda = (n), then V (n)=kV_{(n)} = k equipped with the trivial action of the symmetric group. In this case the corresponding Schur functor (def. ) forms the nnth symmetric power

S (n)(X)=Sym n(X). S_{(n)}(X) = Sym^n(X) \,.

For the dual case where λ=(1,1,,1)\lambda = (1,1, \cdots, 1) then V (1,1,,1)=kV_{(1,1,\cdots, 1)} = k equipped with the action by multiplication with the signature of a permutation and the corresponding Schur functor forms the alternating power

S (1,1,,1)(X)= nX. S_{(1,1, \cdots, 1)}(X) = \wedge^n X \,.
Proposition

Let V=V evenV oddV = V_{even} \oplus V_{odd} be a super vector space of degreewise finite dimension d even,d oddd_{even}, d_{odd} \in \mathbb{N}. Then there exists a Schur functor S λS_\lambda (def. ) that annihilates VV:

S λ(V)0. S_\lambda(V) \simeq 0 \,.

Specifically, this is the case precisely if the corresponding Young tableau [λ][\lambda] satifies

[λ]{(i,j)|id even,jd odd}. [\lambda] \subset \left\{ (i,j)\;\vert\; i \leq d_{even}, j \leq d_{odd} \right\} \,.

(Deligne 02, corollary 1.9)

Statement of the theorem

Theorem

Every kk-tensor category 𝒜\mathcal{A} (def. ) such that

  1. kk is an algebraically closed field of characteristic zero (e.g. the field of complex numbers)

  2. 𝒜\mathcal{A} is of subexponential growth according to def.

then 𝒜\mathcal{A} is a neutral super Tannakian category (def. ) and there exists

  1. an affine algebraic supergroup GG (def. ) whose algebra of functions 𝒪(G)\mathcal{O}(G) is a finitely generated kk-algebra.

  2. a tensor-equivalence of categories

    𝒜Rep(G,ϵ). \mathcal{A} \simeq Rep(G,\epsilon) \,.

    between 𝒜\mathcal{A} and the category of representations of GG of finite dimension, according to def. and prop. .

(Deligne 02, theorem 0.6)

Proof

We outline key steps of the proof of theorem , given in Deligne 02.

Throughout, let kk be an algebraically closed field of characteristic zero (for instance the complex numbers).

The proof proceeds in three main steps:

  1. Proposition states that in a kk-tensor category an object XX is of subexponential growth (def. ) precisely if there exists a Schur functor that annihilates it, hence if some power of XX, skew-symmetrized in sme variables and symmetrized in others, vanishes.

    This proposition is where the symmetric group and its permutation action on tensor powers appears, from just a kind of finite-dimensionality assumption.

  2. Proposition in turn says that if every object in 𝒜\mathcal{A} is annihilated by some Schur functor, then there exists a super fiber functor on 𝒜\mathcal{A} over some supercommutative superalgebra RR, hence then every object of 𝒜\mathcal{A} has underlying it a super vector space with some extra structure.

    This proposition is where superalgebra proper appears.

  3. Proposition states that every kk-tensor category equipped with a super fiber functor ω:𝒜sVect\omega \colon \mathcal{A} \to sVect, is equivalent to the category of super-representations of the automorphism supergroup of ω\omega.

    This proposition is the instance of general Tannaka reconstruction applied to the case of fiber functors with values in super vector spaces. This is where the “supersymmetry” supergroup is extracted.

Proposition

For 𝒜\mathcal{A} a kk-tensor category (def. ), then the following are equivalent:

  1. the category 𝒜\mathcal{A} has subexponential growth (def. );

  2. for every object X𝒜X \in \mathcal{A} there exists nn \in \mathbb{N} and a partition λ\lambda of nn such that the corresponding value of the Schur functor, def. , on XX vanishes: S λ(X)=0S_\lambda(X) = 0.

(Deligne 02, prop. 05)

Proposition

If for every object of a kk-tensor category 𝒜\mathcal{A} (def. ) there exists a Schur functor (def. ) that annihilates it, then there exists a super fiber functor (def. ) over kk, hence then 𝒜\mathcal{A} is a neutral super Tannakian category (def. ).

ω:𝒜sVect. \omega \;\colon\; \mathcal{A} \longrightarrow sVect \,.

(Deligne 02, prop. 2.1 “résultat clé de l’article”, together with prop. 4.5)

Proof idea

First (Deligne 02, middle of p. 16) consider the tensor category

s𝒜(𝒜 /2,τ super) s \mathcal{A} \coloneqq (\mathcal{A}^{\mathbb{Z}/2}, \tau_{super} )

which is that of /2\mathbb{Z}/2-graded objects of 𝒜\mathcal{A}, and whose braiding is given on objects X,YX,Y of homogeneous degree by that of 𝒜\mathcal{A} multiplied with (1) deg(X)deg(Y)(-1)^{deg(X) deg(Y)}.

Write 11 and 1¯\overline{1} for the tensor unit of 𝒜\mathcal{A}, regarded in even degree and in odd degree in s𝒜s \mathcal{A}, respectively.

For ACMon(𝒜)A \in CMon(\mathcal{A}) a commutative monoid, write

𝒜1Mod(𝒜)U() AFAMod(𝒜) \mathcal{A} \simeq 1 Mod(\mathcal{A}) \underoverset{\underset{U}{\longleftarrow}}{\overset{(-)_A \coloneqq F}{\longrightarrow}} {} A Mod(\mathcal{A})

for the extension of scalars operation A()A \otimes(-), left adjoint to restriction of scalars (prop. ).

Show then that the condition that an object XX is annihilated by some Schur functor is equivalent to the existence of an algebra AA such that

X A1 p1¯ q X_A \simeq 1^p \oplus \overline{1}^q

for some p,qp,q \in \mathbb{N}, hence that each such object is AA-locally a super vector space.

(Deligne 02, prop. 2.9).

Moreover, for each short exact sequence

0XYZ0 0 \to X \to Y \to Z \to 0

in s𝒜s \mathcal{A}, there exists an algebra AA such that

0X AY AZ A0 0 \to X_A \to Y_A \to Z_A \to 0

is a split exact sequence, (hence every short exact sequence is locally split).

(Deligne 90, 7.14, Deligne 02, rappel 2.102))

Now (Deligne 02, middle of p. 17) let AA be the commutative monoid which is the tensor product of commutative monoids (example ) over all isomorphism classes of objects and of short exact sequences in 𝒜\mathcal{A} of choices of commutative monoids for which these objects/exact sequencs are locally split, as above.

Then for an AA-module NN, write ρ(N)\rho(N) for the subobject of NN inside sVectInd1,1¯s𝒜sVect \simeq Ind\langle 1, \overline{1}\rangle \hookrightarrow s \mathcal{A}.

Check (Deligne 02, bottom of p. 17) that ρ(A)\rho(A) inherits the structure of a commutative monoid, and that ρ(N)\rho(N) inherits the structure of a module over ρ(N)\rho(N).

Set

Rρ(A). R \coloneqq \rho(A) \,.

Hence for every object XX, then

ω(X)ρ(X A) \omega(X) \coloneqq \rho(X_A)

has the structure of an RR-module. By AA-local splitness of all short exact sequence, ω\omega is an exact functor.

Hence

ω:s𝒜RMod(sVect) \omega \;\colon\; s \mathcal{A} \longrightarrow R Mod(sVect)

is a super fiber functor on s𝒜s\mathcal{A} over RR. This restricts to a super fiber functor over RR on 𝒜\mathcal{A}, regarded as the sub-category of even-graded objects in s𝒜s \mathcal{A}:

𝒜s𝒜ωRMod(sVect), \mathcal{A} \hookrightarrow s \mathcal{A} \overset{\omega}{\longrightarrow} R Mod(sVect) \,,

Finally check (Deligne 02, prop. 4.5) that if a kk-tensor category 𝒜\mathcal{A} (def. ) admits a super fiber functor (def. ) over a supercommutative superalgebra RR over kk

𝒜RMod(sVect) \mathcal{A} \longrightarrow R Mod(sVect)

then it also admits a super fiber functor over kk itself, i.e. a fiber functor to sVect

𝒜kMod(sVect)sVect. \mathcal{A} \longrightarrow k Mod(sVect) \simeq sVect \,.

This is argued by expressing RR as an inductive limit

R=lim βR β R = \underset{\longrightarrow}{\lim}_\beta R_\beta

over supercommutative superalgebras R βR_\beta of finite type over kk and observing (…) that there exists β\beta such that ω β\omega_\beta is still a fiber functor and such that there exists an algebra homomorphism R βkR_\beta \to k.

Finally then the fiber functor in question is

ω β R βk:𝒜sVect k. \omega_\beta \otimes_{R_\beta} k \;\colon\; \mathcal{A} \longrightarrow sVect_k \,.
Proposition

For every kk-tensor category 𝒜\mathcal{A} (def. ) and a super fiber functor over kk (def. )

ω:𝒜sVect k \omega \;\colon\; \mathcal{A} \longrightarrow sVect_k

then ω\omega induces an equivalence of categories

𝒜Rep(Aut̲(ω),ϵ) \mathcal{A} \stackrel{\simeq}{\longrightarrow} Rep( \underline{Aut}(\omega),\epsilon)

of 𝒜\mathcal{A} with the category of finite dimensional representations, according to def. and prop. , of the automorphism supergroup Aut̲(ω)\underline{Aut}(\omega) (example. , prop. ) of the super fiber functor, where ϵ\epsilon is te image of the unique nontrivial element in

Aut̲(sVect)/2 \underline{Aut}(sVect) \simeq \mathbb{Z}/2

(according to example ) under the group homomorphism

π(sVect)ω(π(𝒜))Aut̲(ω) \pi(sVect) \longrightarrow \omega(\pi(\mathcal{A})) \simeq \underline{Aut}(\omega)

from prop. and using the isomorphism from prop. .

This is the main Tannaka reconstruction theorem (Deligne 90, 8.17) specialized to super fiber functors (Deligne 90, 8.19).

References

The theorem is due to

  • Pierre Deligne, Catégorie Tensorielle, Moscow Math. Journal 2 (2002) no. 2, 227-248. (pdf)

building on the general results on Tannakian categories in

which are reviewed and further generalized in

Review is in

Further discussion in view of the theory of triangular Hopf algebras is in

Tannaka duality for ordinary compact groups regarded as supergroups (hence equipped with “inner parity”, def. , here just being an involutive central element) is discussed in

as a proof of Doplicher-Roberts reconstruction

Commutative algebra internal to symmetric monoidal categories is discussed in

and specifically for commutative Hopf algebroids in

(These authors are motivated by the application of the general theory of algebra in monoidal categories to “higher algebra” (“brave new algebra”) in stable homotopy theory. This happens to also be a version of supercommutative superalgebra, see at Introduction to Stable homotopy theory the section 1-2 Homotopy commutative ring spectra.)

For an attempt to generalize Deligne’s theorem to positive characteristic, see

Discussion relating to 2-rings and the spin-statistics theorem is in

Last revised on September 15, 2018 at 09:37:12. See the history of this page for a list of all contributions to it.