nLab Deligne's theorem on tensor categories

Idea

Deligne’s theorem on tensor categories (Deligne 02, recalled as theorem below) establishes Tannaka duality between

1. linear tensor categories in characteristic zero subject to just a mild size constraint (subexponential growth, def. below),

2. supergroups (“supersymmetries”), realizing these tensor categories as categories of representations of these supergroups.

As explained there?, by tensor category we assume a symmetric braiding.

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 $\mathbb{Z}/2$-graded algebras, or in fact $\mathbb{Z}/n$-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 $k$-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 $k$-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:

• PhysicsForums Insights, Supersymmetry and Deligne’s theorem

For Tannaka duality of Hopf algebras

By Tannaka duality, rigid 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 algebra	category of modules
$A$	$Mod_A$
$R$-algebra	$Mod_R$-2-module
sesquialgebra	2-ring = monoidal presentable category with colimit-preserving tensor product
bialgebra	strict 2-ring: monoidal category with fiber functor
Hopf algebra	rigid monoidal category with fiber functor
hopfish algebra (correct version)	rigid monoidal category (without fiber functor)
weak Hopf algebra	fusion category with generalized fiber functor
quasitriangular bialgebra	braided monoidal category with fiber functor
triangular bialgebra	symmetric monoidal category with fiber functor
quasitriangular Hopf algebra (quantum group)	rigid braided monoidal category with fiber functor
triangular Hopf algebra	rigid symmetric monoidal category with fiber functor
supercommutative Hopf algebra (supergroup)	rigid symmetric monoidal category with fiber functor and Schur smallness
form Drinfeld double	form Drinfeld center
trialgebra	Hopf monoidal category

2-Tannaka duality for module categories over monoidal categories

monoidal category	2-category of module categories
$A$	$Mod_A$
$R$-2-algebra	$Mod_R$-3-module
Hopf monoidal category	monoidal 2-category (with some duality and strictness structure)

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

monoidal 2-category	3-category of module 2-categories
$A$	$Mod_A$
$R$-3-algebra	$Mod_R$-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:

• Tensor categories

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

• Commutative algebra in tensor categories and Affine super-spaces

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:

• (Super-)Groups as (Super-)commutative 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:

• Linear representations as comodules

Tensor products and Super vector spaces

Definition

For $k$ a field, we write $Vect_k$ for the category whose

• objects are $k$-vector spaces;

• morphisms are $k$-linear functions between these.

When the ground field $k$ 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 $\mathcal{C}$ be a category.

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

2. Say that $\mathcal{C}$ 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 $\mathcal{C}$ 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 $k$-linear category, but notice that conventions differ as to which extra properties beyond Vect-enrichment to require on a linear category:

Definition

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

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

2. its hom-sets have the structure of $k$-vector spaces (generally $k$-modules) such that composition of morphisms in $\mathcal{C}$ 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 $k$-linear category is an abelian category with the additional structure of a Vect-enriched category (generally $k$Mod-enriched) such that the underlying Ab-enrichment according to def. is obtained from the $Vect$-enrichment under the forgetful functor $Vect \to Ab$.

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

Example

The category Vect${}_k$ of vector spaces (def. ) is a $k$-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 $V,W$ two $k$-vector spaces, the vector space structure on the hom-set $Hom_{Vect}(V,W)$ of linear maps $\phi \colon V \to W$ is given by “pointwise” multiplication and addition of functions:

$$(c_1 \phi_1 + c_2 \phi_2) \;\colon\, v \;\mapsto\; c_1 \phi_1(v) + c_2 \phi_2(v)$$

for all $c_1, c_2 \in k$ and $\phi_1, \phi_2 \in Hom_{Vect}(V,W)$.

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

Definition

Given two vector spaces over some field $k$, $V_1, V_2 \in Vect_k$, their tensor product of vector spaces is the vector space denoted

$$V_1 \otimes_k V_2 \;\in\; Vect$$

whose elements are equivalence classes of tuples of elements $(v_1,v_2)$ with $v_i \in V_i$, for the equivalence relation given by

$$(c v_1 , v_2) \;\sim\; (v_1, c v_2)$$

$$(v_1 + v'_1 , v_2) \; \sim \; (v_1,v_2) + (v'_1, v_2)$$

$$(v_1 , v_2 + v'_2) \; \sim \; (v_1,v_2) + (v_1, v'_2)$$

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

   $$V_1 \times V_2 \longrightarrow V_1 \otimes V_2$$

   (out of the Cartesian product of the underlying sets)

2. any other bilinear map of the form

   $$V_1 \times V_2 \longrightarrow V_3$$

   factors through the above bilinear map via a unique linear map

   $$\array{ V_1 \times V_2 &\overset{bilinear}{\longrightarrow}& V_3 \\ \downarrow & \nearrow_{\mathrlap{\exists ! \, linear}} \\ V_1 \otimes_k V_2 }$$

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 $\mathcal{C}$ equipped with

1. a functor

   $$\otimes \;\colon\; \mathcal{C} \times \mathcal{C} \longrightarrow \mathcal{C}$$

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

2. an object

   $$1 \in \mathcal{C}$$

   called the unit object or tensor unit,

3. a natural isomorphism

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

   called the associator,

4. a natural isomorphism

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

   called the left unitor, and a natural isomorphism

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

   called the right unitor,

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

1. triangle identity:

   $$\array{ & (x \otimes 1) \otimes y &\stackrel{a_{x,1,y}}{\longrightarrow} & x \otimes (1 \otimes y) \\ & {}_{\rho_x \otimes 1_y}\searrow && \swarrow_{1_x \otimes \lambda_y} & \\ && x \otimes y && }$$

2. the pentagon identity:

   $$\array{ && (w \otimes x) \otimes (y \otimes z) \\ & {}^{\mathllap{\alpha_{w \otimes x, y, z}}}\nearrow && \searrow^{\mathrlap{\alpha_{w,x,y \otimes z}}} \\ ((w \otimes x ) \otimes y) \otimes z && && (w \otimes (x \otimes (y \otimes z))) \\ {}^{\mathllap{\alpha_{w,x,y}} \otimes id_z }\downarrow && && \uparrow^{\mathrlap{ id_w \otimes \alpha_{x,y,z} }} \\ (w \otimes (x \otimes y)) \otimes z && \underset{\alpha_{w,x \otimes y, z}}{\longrightarrow} && w \otimes ( (x \otimes y) \otimes z ) }$$

As expected, we have the following basic example:

Example

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

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

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

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

  $$\alpha_{V_{1}, V_2, V_3} \;\colon\; ((v_1, v_2), v_3) \mapsto (v_1, (v_2,v_3))$$

• the left unitor is the map that on representing tuples is given by

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

  and the right unitor is similarly given by

  $$r_V \colon (v,k) \mapsto k v \,.$$

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

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

A homomorphism

$$\phi \;\colon\; V \longrightarrow W$$

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

$$\phi_g \;\colon\; V_g \longrightarrow W_g$$

for all $g \in G$, such that

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

This defines a category, denoted $Vect^G$. 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 $G$-grading which is obtained by multiplying degree labels in $G$:

$$(V \otimes W)_g \;\coloneqq\; \underset{{g_1, g_2 \in G} \atop {g_1 g_2 = g}}{\oplus} V_{g_1} \otimes_k V_{g_2} \,.$$

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

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

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

1. $\ell_1 = r_1 \;\colon\; 1 \otimes 1 \overset{\simeq}{\longrightarrow} 1$;

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

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

   and

   $$\array{ x \otimes (y \otimes 1) & & \\ {}^\mathllap{\alpha^{-1}_{1, x, y}} \downarrow & \searrow^\mathrlap{id_x \otimes r_y} & \\ (x \otimes y) \otimes 1 & \underset{r_{x \otimes y}}{\longrightarrow} & x \otimes y } \,;$$

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

Remark

Just as for an associative algebra it is sufficient to demand $1 a = a$ and $a 1 = a$ and $(a b) c = a (b c)$ 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 $X \otimes (Y \otimes Z)$ is actually equal to $(X \otimes Y)\otimes Z$, 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 $\mathcal{C}$ (def. ) equipped with a natural isomorphism

$$\tau_{x,y} \;\colon\; x \otimes y \to y \otimes x$$

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

$$\array{ (x \otimes y) \otimes z &\stackrel{a_{x,y,z}}{\to}& x \otimes (y \otimes z) &\stackrel{\tau_{x,y \otimes z}}{\to}& (y \otimes z) \otimes x \\ \downarrow^{\tau_{x,y}\otimes Id} &&&& \downarrow^{a_{y,z,x}} \\ (y \otimes x) \otimes z &\stackrel{a_{y,x,z}}{\to}& y \otimes (x \otimes z) &\stackrel{Id \otimes \tau_{x,z}}{\to}& y \otimes (z \otimes x) }$$

and

$$\array{ x \otimes (y \otimes z) &\stackrel{a^{-1}_{x,y,z}}{\to}& (x \otimes y) \otimes z &\stackrel{\tau_{x \otimes y, z}}{\to}& z \otimes (x \otimes y) \\ \downarrow^{Id \otimes \tau_{y,z}} &&&& \downarrow^{a^{-1}_{z,x,y}} \\ x \otimes (z \otimes y) &\stackrel{a^{-1}_{x,z,y}}{\to}& (x \otimes z) \otimes y &\stackrel{\tau_{x,z} \otimes Id}{\to}& (z \otimes x) \otimes y } \,,$$

where $a_{x,y,z} \colon (x \otimes y) \otimes z \to x \otimes (y \otimes z)$ denotes the components of the associator of $\mathcal{C}^\otimes$.

Definition

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

$$\tau_{x,y} \colon x \otimes y \to y \otimes x$$

satisfies the condition:

$$\tau_{y,x} \circ \tau_{x,y} = 1_{x \otimes y}$$

for all objects $x, y$

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 $G$-graded vector spaces from example , the case where $G = \mathbb{Z}/2$ is the cyclic group of order two.

Example

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

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

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

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

$$f \;\colon\; V_1 \longrightarrow V_2$$

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

$$f_{even} \;\colon\; (V_1)_{even} \longrightarrow (V_1)_{even}$$

$$f_{odd} \;\colon\; (V_1)_{odd} \longrightarrow (V_1)_{odd}$$

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

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

The tensor product of $\mathbb{Z}/2$-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 $\mathbb{Z}/2$. Hence

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

and

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

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

Proposition

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

$$V_1 \otimes V_2 \stackrel{\tau_{V_1,V_2}}{\longrightarrow} V_2 \otimes V_1$$

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

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

   $$\tau^{triv}_{V_1, V_2} \;\colon\; (v_1,v_2) \mapsto (v_2, v_1)$$

2. the super-braiding which is the natural linear function given on tuples $(v_1,v_2)$ of homogeneous degree (i.e. $v_i \in (V_i)_{\sigma_i} \hookrightarrow V_i$, for $\sigma_i \in \mathbb{Z}/2$) by

   $$\tau^{super}_{V_1, V_2} \;\colon\; (v_1, v_2) \mapsto (-1)^{deg(v_1) deg(v_2)} \, (v_2,v_1) \,.$$

Proof

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

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

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

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

Consider furthermore the groupoid core (non-full subcategory including all the isomorphisms)

$$Line(\mathcal{C}, \otimes , 1)_{iso} \;\in\; 2 Grp$$

The tensor product now makes this a 2-group, known as the “Picard groupoid” of $\mathcal{C}$. 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

$$B_\otimes Line(\mathcal{C})_{iso}$$

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

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

$$\pi_0(Line(\mathcal{C})_{iso}) \simeq \pi_1(B Line(\mathcal{C})_{iso})$$

the fundamental group of the delooping space.

Now a symmetric braiding on $Line(\mathcal{C})_{iso}$ is precisely the structure that makes it a symmetric 2-group which is equivalently the structure of a second delooping $B^2 Line(\mathcal{C})$ (for the braiding) and then a third delooping $B^3 Line(\mathcal{C})$ (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, $Line(Vect^{\mathbb{Z}/2})$ has precisely two isomorphism classes of objects, namely the ground field $k$ itself, regarded as being in even degree and regarded as being in odd degree. We write $k^{1\vert 0}$ and $k^{0 \vert 1}$ for these, respectively. By the rules of the tensor product of graded vector spaces we have

$$k^{1\vert 0} \otimes_k k^{1\vert 0} \simeq k^{1\vert 0}$$

$$k^{1\vert 0} \otimes_k k^{0\vert 1} \simeq k^{0\vert 1}$$

and

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

In other words

$$\pi_0(Line(Vect^{\mathbb{Z}/2})_{iso}) \simeq \mathbb{Z}/2 \,.$$

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

$$1 = k^{1 \vert 0} \simeq k^{0\vert 1} \otimes_k k^{0 \vert 1} \stackrel{\tau^{super}_{k^{0\vert 1}, k^{0 \vert 1}}}{\longrightarrow} k^{0\vert 1} \otimes_k k^{0\vert 1} \simeq k^{1 \vert 0} = 1$$

Due to the symmetry condition (def. ) we have

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

which implies that

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

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

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

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

Definition

The symmetric monoidal category (def. )

• whose underlying monoidal category is that of $\mathbb{Z}/2$-graded vector spaces (example );

• whose braiding (def. ) is the unique non-trivial symmtric grading $\tau^{super}$ from prop. is called the category of super vector spaces

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

Remark

The non-full symmetric monoidal subcategory

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

of

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

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

$$(\widetilde{Line}(sVect), \otimes, k, \tau^{super}) \;\simeq\; \trunc_1 \Omega \mathbb{S} \,.$$

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

$$Vect_k \hookrightarrow sVect_k$$

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

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

hence if there are natural bijections

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

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

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

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

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

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

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

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

Proof

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

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

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

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

Definition

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

1. a functor

   $$F \;\colon\; \mathcal{C} \longrightarrow \mathcal{D} \,,$$

2. a morphism

   $$\epsilon \;\colon\; 1_{\mathcal{D}} \longrightarrow F(1_{\mathcal{C}})$$

3. a natural transformation

   $$\mu_{x,y} \;\colon\; F(x) \otimes_{\mathcal{D}} F(y) \longrightarrow F(x \otimes_{\mathcal{C}} y)$$

   for all $x,y \in \mathcal{C}$

satisfying the following conditions:

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

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

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

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

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

   and

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

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

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

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

$$\array{ F(x) \otimes_{\mathcal{C}} F(y) &\overset{\tau^{\mathcal{D}}_{F(x), F(y)}}{\longrightarrow}& F(y) \otimes_{\mathcal{D}} F(x) \\ {}^{\mathllap{\mu_{x,y}}}\downarrow && \downarrow^{\mathrlap{\mu_{y,x}}} \\ F(x \otimes_{\mathcal{C}} y ) &\underset{F(\tau^{\mathcal{C}}_{x,y} )}{\longrightarrow}& F( y \otimes_{\mathcal{C}} x ) } \,.$$

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

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

$$\array{ F_1(x) \otimes_{\mathcal{D}} F_1(y) &\overset{f(x)\otimes_{\mathcal{D}} f(y)}{\longrightarrow}& F_2(x) \otimes_{\mathcal{D}} F_2(y) \\ {}^{\mathllap{(\mu_1)_{x,y}}}\downarrow && \downarrow^{\mathrlap{(\mu_2)_{x,y}}} \\ F_1(x\otimes_{\mathcal{C}} y) &\underset{f(x \otimes_{\mathcal{C}} y ) }{\longrightarrow}& F_2(x \otimes_{\mathcal{C}} y) }$$

and

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

We write $MonFun(\mathcal{C},\mathcal{D})$ for the resulting category of lax monoidal functors between monoidal categories $\mathcal{C}$ and $\mathcal{D}$, similarly $BraidMonFun(\mathcal{C},\mathcal{D})$ for the category of braided monoidal functors between braided monoidal categories, and $SymMonFun(\mathcal{C},\mathcal{D})$ 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 $(\mathcal{C},\otimes_{\mathcal{C}}, 1_{\mathcal{C}})$ and $(\mathcal{D},\otimes_{\mathcal{D}}, 1_{\mathcal{D}} )$ are symmetric monoidal categories (def. ) then a braided monoidal functor (def. ) between them is often called a symmetric monoidal functor.

Example

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

$$\mathcal{A} \longrightarrow Vect$$

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

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

Proposition

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

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

and

$$\mu^{G \circ F}_{c_1,c_2} \;\colon\; G(F(c_1)) \otimes_{\mathcal{E}} G(F(c_2)) \overset{\mu^{G}_{F(c_1), F(c_2)}}{\longrightarrow} G( F(c_1) \otimes_{\mathcal{D}} F(c_2) ) \overset{G(\mu^F_{c_1,c_2})}{\longrightarrow} G(F( c_1 \otimes_{\mathcal{C}} c_2 )) \,.$$

We now discuss one more extra property on monoidal categories

Definition

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

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

consists of

1. a morphism of the form

   $$ev_A\;\colon\;A^\ast \otimes A \longrightarrow 1$$

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

2. a morphism of the form

   $$i_A \;\colon\; 1 \longrightarrow A \otimes A^\ast$$

   called the unit or coevaluation map

such that

• (triangle identity) the following diagrams commute

  $$\array{ A^\ast \otimes (A \otimes A^\ast) &\overset{id_{A^\ast} \otimes i_A}{\longleftarrow}& A^\ast \otimes 1 \\ {}^{\mathllap{\alpha^{-1}_{A^\ast,A, A^\ast}}}_{\mathllap{\simeq}}\downarrow && \downarrow^{\mathrlap{\ell^{-1}_A \circ r_A}}_{\mathrlap{\simeq}} \\ (A^\ast \otimes A) \otimes A^\ast &\underset{ev_A \otimes id_{A^\ast}}{\longrightarrow}& 1 \otimes A^\ast }$$

  and

  $$\array{ (A \otimes A^\ast) \otimes A &\overset{i_A \otimes id_A}{\longleftarrow}& 1 \otimes A \\ {}^{\mathllap{\alpha_{A,A^\ast, A}}}_{\mathllap{\simeq}}\downarrow && \downarrow^{\mathrlap{r_A^{-1}\circ \ell_A}}_{\mathrlap{\simeq}} \\ A \otimes (A^\ast \otimes A) &\underset{id_A \otimes ev_A}{\longrightarrow}& A \otimes 1 }$$

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

We say that $A^\ast$ is the right dual object to $A$. Similarly a left dual for $A$ is an object $A^\ast$ and the structure of $A$ as a right dual of $A^\ast$. If $(\mathcal{C}, \otimes, 1)$ 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 $(\mathcal{C}, \otimes, 1)$ every object has a left and right dual, then it is called a rigid monoidal category.

Example

Let

$$FinDimVect_k \hookrightarrow Vect_k$$

be the full subcategory FinDimVect of that of all vector spaces (over the given ground field $k$) 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

$$(FinDimVect_k, \otimes = \otimes_k, 1 = k ) \hookrightarrow (Vect_k, \otimes_k, k) \,.$$

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

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

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)

$$ev_{V} \;\colon\; V^\ast \otimes_k V \simeq hom(V,k) \otimes_k k \overset{}{\longrightarrow} k$$

$$ev \;\colon\; (V \stackrel{\phi}{\to} k, v) \;\;\mapsto \;\; \phi(v) \,.$$

The co-evaluation map

$$i_V \;\colon\; k \longrightarrow V \otimes V^\ast$$

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

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

$$ev \;\colon\; (e^i, e_j) \mapsto e^i(e_j) = \delta^i_j$$

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

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

In this perspective the triangle identities are the statements that

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

and

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

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

Example

Similarly, the full subcategory

$$(sFinDimVect, \otimes_k, k, \tau^{super}) \hookrightarrow (sVect, \otimes_k, k, \tau^{super})$$

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 $V$ has dimension

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

if its even part has dimension $p$ and its odd part has dimension $q$:

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

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:

$$V = V_{even} \oplus V_{odd} \;\;\;\; \Rightarrow \;\;\;\; V^\ast = (V_{even}^\ast) \oplus (V_{odd}^\ast) \,.$$

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

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

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

Proof

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

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

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 $k$-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 $k$ a field, then a $k$-tensor category $\mathcal{A}$ 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 $\otimes \colon \mathcal{A} \times \mathcal{A} \longrightarrow \mathcal{A}$ is in both arguments separately

   1. $k$-linear (def. );

   2. exact.

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

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

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

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

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

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

Proposition

(Jordan-Hölder theorem)

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

(e.g. EGNO 15, theorem 1.5.4)

Definition

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

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

Definition

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

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

2. Each object $a$ is of finite length;

3. For any two objects $a$, $b$ of $C$, the hom-object ($k$-vector space) $\hom(a, b)$ has finite dimension;

Example

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

Also category of finite dimensional super vector spaces is a finite tensor category. This has two isomorphism classes of simple objects, $k = k^{1 \vert 0}$ regarded in even degree, and $k^{0\vert 1}$ 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 $k$-tensor category (def. ) is called finitely $\otimes$-generated if there exists an object $E \in \mathcal{A}$ such that every other object $X \in \mathcal{A}$ is a subquotient of a direct sum of tensor products $E^{\otimes^n}$, for some $n \in \mathbb{N}$:

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

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

(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 $\mathcal{A}$ (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 $X$ there exists a natural number $N_X$ such that $X$ is of length at most $N_X$, and that also all tensor product powers of $X$ are of length bounded by the corresponding powers of $N_X$:

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

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

The evident example is the following:

Example

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

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

Categories that do not satisfy sub-exponential growth have come to be known as Deligne categories, see e.g. Hu 2024.

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

1. all objects have finite length;

2. all hom spaces are of finite dimension over $k$

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

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

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

   1. which stable under forming subquotients

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

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

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

   where $X_i,X_j \in \mathcal{A}$, by the above.

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

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

Example

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

$$Vect \simeq Ind(FinDimVect) \,.$$

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

$$sVect \simeq Ind(sFinDimVect) \,.$$

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

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

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 $X,Y$ there is a natural bijection between the smooth functions $X \to Y$ and the $\mathbb{R}$-algebra homomorphisms $C^\infty(X)\leftarrow C^\infty(Y)$.

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 $A$, then we may think of it as representing a space $Spec(A)$ which is such that it has $A$ 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 $(\mathcal{C}, \otimes, 1)$ (def ), then a monoid internal to $(\mathcal{C}, \otimes, 1)$ is

1. an object $A \in \mathcal{C}$;

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

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

such that

1. (associativity) the following diagram commutes

   $$\array{ (A\otimes A) \otimes A &\underoverset{\simeq}{a_{A,A,A}}{\longrightarrow}& A \otimes (A \otimes A) &\overset{A \otimes \mu}{\longrightarrow}& A \otimes A \\ {}^{\mathllap{\mu \otimes A}}\downarrow && && \downarrow^{\mathrlap{\mu}} \\ A \otimes A &\longrightarrow& &\overset{\mu}{\longrightarrow}& A } \,,$$

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

2. (unitality) the following diagram commutes:

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

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

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

• (commutativity) the following diagram commutes

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

A homomorphism of monoids $(A_1, \mu_1, e_1)\longrightarrow (A_2, \mu_2, f_2)$ is a morphism

$$f \;\colon\; A_1 \longrightarrow A_2$$

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

$$\array{ A_1 \otimes A_1 &\overset{f \otimes f}{\longrightarrow}& A_2 \otimes A_2 \\ {}^{\mathllap{\mu_1}}\downarrow && \downarrow^{\mathrlap{\mu_2}} \\ A_1 &\underset{f}{\longrightarrow}& A_2 }$$

and

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

Write $Mon(\mathcal{C}, \otimes,1)$ for the category of monoids in $\mathcal{C}$, and $CMon(\mathcal{C}, \otimes, 1)$ 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 $Vect$ is an ordinary commutative algebra. Moreover, in both cases the homomorphisms of monoids agree with usual algebra homomorphisms. Hence there are equivalences of categories.

$$Mon(Vect_k) \simeq Alg_k$$

$$CMon(Vect_k) \simeq CAlg_k \,.$$

Example

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

$$Alg_k^G \simeq Mon(Vect_k^G) \,.$$

This means that a $G$-graded algebra is

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

2. an associative algebra structure on the underlying vector space $A$

such that for two elements of homogeneous degree, i.e. $a_1 \in A_{g_1} \hookrightarrow A$ and $a_2 \in A_{g_2} \hookrightarrow A$ then their product is in degre $g_1 g_2$

$$a_{g_1} a_{g_2} \in A_{g_1 g_2} \hookrightarrow A \,.$$

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 $sCAlg_k$ for the category of supercommutative superalgebras with the induced homomorphisms between them:

$$sCAlg_k \;\coloneqq\; CMon(sVect_k) \,.$$

Unwinding what this means, then a supercommutative superalgebra $A$ is

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

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

   $$a b \; = \; (-1)^{deg(a) deg(b)}\, b a \,.$$

Remark

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

$$sAlg_k \coloneqq Mon(sVect) \,.$$

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 $\mathbb{Z}/2$-graded vector spaces only via their braiding (example ), this yields equivalently just the $\mathbb{Z}/2$-graded algebras froom example :

$$sAlg_k \simeq Alg_k^{\mathbb{Z}/2} \,.$$

Hence the heart of superalgebra is super-commutativity.

Example

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

$$\theta_i \theta_j = - \theta_j \theta_i$$

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

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

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

Example

Given a homotopy commutative ring spectrum $E$ (i.e., via the Brown representability theorem, a multiplicative generalized cohomology theory), then its stable homotopy groups $\pi_\bullet(E)$ 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

$$\array{ CAlg_k &\hookrightarrow& sCAlg_k \\ = && = \\ CMon(Vect_k) &\hookrightarrow& CMon(sVect_k) }$$

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

$$i \;\colon\; Vectk \hookrightarrow sVect_k$$

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”):

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

Here

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

2. the left adjoint $(-)/(-)_{even}$ sends a supercommutative superalgebra to the quotient by the ideal which is generated by its odd part $A \mapsto A/(A_{odd})$ (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 $i$ is evident. To see the adjunctions observe their characteristic natural bijections between hom-sets: If $A_{ordinary}$ is an ordinary commutative algebra regarded as a superalgeba $i(A_{ordinary})$ concentrated in even degree, and if $B$ is any superalgebra,

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

   $$Hom_{sCAlg_k}(i(A_{ordinary}), B) \simeq Hom_{CAlg_k}(A_{ordinary,B_{even}}) \,,$$

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

   $$Hom_{sCalg_k}(B, i(A_{ordinary})) \simeq Hom_{Alg_k}(B/B_{odd}, A_{ordinary}) \,.$$

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

Definition

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

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

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

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

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

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

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

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

Definition

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

$$Aff(sVect_k) = scAlg_k^{op} = CMon(sVect_k)^{op}$$

are affine super schemes over $k$.

Example

For $A \in CAlg_{\mathbb{R}}$ 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. )

$$Aff(Vect_k) \hookrightarrow Aff(sVect_k) \,.$$

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

$$\mathbb{R}^{p \vert 0} \in Aff(sVect_k) \,.$$

Example

The formal dual space, according to def. (example ) to a Grassmann algebra $\wedge^\bullet_{\mathbb{R}}(\mathbb{R}^q)$ (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 $q$-th power.

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

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

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

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

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

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

Example

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

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

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 $A$ (def. ), its parity involution is the algebra automorphism

$$par \;\colon\; A \overset{\simeq}{\longrightarrow} A$$

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

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

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

$$\ell_1 = r_1 \;\colon\; 1 \otimes 1 \overset{\simeq}{\longrightarrow} 1 \,.$$

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

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

Example

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

$$e \;\colon\; 1 \overset{\simeq}{\longrightarrow} 1 \otimes 1 \overset{e_1 \otimes e_2}{\longrightarrow} E_1 \otimes E_2$$

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

$$E_1 \otimes E_2 \otimes E_1 \otimes E_2 \overset{id \otimes \tau_{E_2, E_1} \otimes id}{\longrightarrow} E_1 \otimes E_1 \otimes E_2 \otimes E_2 \overset{\mu_1 \otimes \mu_2}{\longrightarrow} E_1 \otimes E_2$$

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

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

Similarly one checks that for $E_1 = E_2 = E$ then the unit maps

$$E \simeq E \otimes 1 \overset{id \otimes e}{\longrightarrow} E \otimes E$$

$$E \simeq 1 \otimes E \overset{e \otimes 1}{\longrightarrow} E \otimes E$$

and the product map

$$\mu \;\colon\; E \otimes E \longrightarrow E$$

and the braiding

$$\tau_{E,E} \;\colon\; E \otimes E \longrightarrow E \otimes E$$

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

Monoids are preserved by lax monoidal functors:

Proposition

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

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

$$\mu_{F(A)} \;\colon\; F(A) \otimes_{\mathcal{C}} F(A) \overset{}{\longrightarrow} F(A \otimes_{\mathcal{C}} A) \overset{F(\mu_A)}{\longrightarrow} F(A)$$

(where the first morphism is the structure morphism of $F$) and setting

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

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

This construction extends to a functor

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

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

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

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

Proof

This follows immediately from combining the associativity and unitality (and symmetry) constraints of $F$ with those of $A$.

Modules in tensor categories and Super vector bundles

Above (in def. ) we considered spaces $X$ from a dual perspective, as determined by their algebras of functions $\mathcal{O}(X)$. 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 $X$. Here we discuss the corresponding algebraic incarnation of these, namely as modules over algebras of functions.

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

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

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

First of all, since $V \to X$ is a vector bundle, we have fiber-wise the vector space operations. This means that given two elements $c_1, c_2 \in \mathbb{R}$ in the real numbers, and given two sections $\sigma_1$ and $\sigma_2$, we may form in each fiber $V_x$ the linear combination $c_1 \sigma_1(x) + c_2 \sigma_2(x)$. This hence yields a new section $c_1 \sigma_1 + c_2 \sigma_2$. 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 $c \in \mathbb{R}$ 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 $f \in C^\infty(X)$ in the $\mathbb{R}$-algebra of smooth functions on $X$, takes a smooth section $\sigma$ of $V$ to a new smooth section $f \cdot \sigma$. This operation enjoys some evident properties. It is bilinear in the real vector spaces $C^\infty(X)$ and $\Gamma_X(V)$, and it satisfies the “action property”

$$(f g) \cdot \sigma = f\cdot (g \cdot \sigma)$$

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

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

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

The smooth Serre-Swan theorem states sufficient conditions on $X$ 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 $X$ a smooth manifold, then the construction which sends a smooth vector bundle $V \to X$ to its $C^\infty(X)$-module $\Gamma_X(V)$ of sections is an equivalence of categories

$$VectBund_X^{fin} \stackrel{\simeq}{\longrightarrow} C^\infty(X) Mod_{proj}^{fin\,gen}$$

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

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

We now state the relevant definitions and constructions formally.

Definition

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

1. an object $N \in \mathcal{C}$;

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

such that

1. (unitality) the following diagram commutes:

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

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

2. (action property) the following diagram commutes

   $$\array{ (A\otimes A) \otimes N &\underoverset{\simeq}{a_{A,A,N}}{\longrightarrow}& A \otimes (A \otimes N) &\overset{A \otimes \rho}{\longrightarrow}& A \otimes N \\ {}^{\mathllap{\mu \otimes N}}\downarrow && && \downarrow^{\mathrlap{\rho}} \\ A \otimes N &\longrightarrow& &\overset{\rho}{\longrightarrow}& N } \,,$$

A homomorphism of left $A$-module objects

$$(N_1, \rho_1) \longrightarrow (N_2, \rho_2)$$

is a morphism

$$f\;\colon\; N_1 \longrightarrow N_2$$

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

$$\array{ A\otimes N_1 &\overset{A \otimes f}{\longrightarrow}& A\otimes N_2 \\ {}^{\mathllap{\rho_1}}\downarrow && \downarrow^{\mathrlap{\rho_2}} \\ N_1 &\underset{f}{\longrightarrow}& N_2 } \,.$$

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

$$A Mod(\mathcal{C}) \,.$$

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

Example

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

$$\ell_C \;\colon\; 1\otimes C \longrightarrow C$$

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

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

of $\mathcal{C}$ 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 $(Ab, \otimes_{\mathbb{Z}}, \mathbb{Z})$ (def. ) is equivalently a ring.

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

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

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

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

Example

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

Proposition

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

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

The $A$-modules of this form are called free modules.

The free functor $F$ constructing free $A$-modules is left adjoint to the forgetful functor $U$ which sends a module $(N,\rho)$ to the underlying object $U(N,\rho) \coloneqq N$.

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

Proof

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

$$f \;\colon\; A\otimes C \longrightarrow N$$

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

$$\array{ A \otimes A \otimes C &\overset{A \otimes f}{\longrightarrow}& A \otimes N \\ {}^{\mathllap{\mu \otimes id}}\downarrow && \downarrow^{\mathrlap{\rho}} \\ A \otimes C &\underset{f}{\longrightarrow}& N } \,.$$

Consider the composite

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

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

$$\array{ A \otimes C &\overset{id \otimes \tilde f}{\longrightarrow}& A \otimes N \\ {}^{\mathllap{id \otimes e \otimes id}}\downarrow && \downarrow^{\mathrlap{=}} \\ A \otimes A \otimes C &\underset{id \otimes f}{\longrightarrow}& A \otimes N } \,.$$

Pasting this square onto the top of the previous one yields

$$\array{ A \otimes C &\overset{id \otimes \tilde f}{\longrightarrow}& A \otimes N \\ {}^{\mathllap{id \otimes e \otimes id}}\downarrow && \downarrow^{\mathrlap{=}} \\ A \otimes A \otimes C &\overset{A \otimes f}{\longrightarrow}& A \otimes N \\ {}^{\mathllap{\mu \otimes id}}\downarrow && \downarrow^{\mathrlap{\rho}} \\ A \otimes C &\underset{f}{\longrightarrow}& N } \,,$$

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

$$f = \rho \circ (id_A \otimes \tilde f) \,.$$

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

Definition

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

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

   $$N_1 \otimes A \otimes N_2 \underoverset {\underset{\rho_{1}\circ (\tau_{N_1,A} \otimes N_2)}{\longrightarrow}} {\overset{N_1 \otimes \rho_2}{\longrightarrow}} {\phantom{AAAA}} N_1 \otimes N_1 \overset{coeq}{\longrightarrow} N_1 \otimes_A N_2$$

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

2. the function module $hom_A(N_1,N_2)$ is, if it exists, the equalizer

   $$hom_A(N_1, N_2) \overset{equ}{\longrightarrow} hom(N_1, N_2) \underoverset {\underset{hom(A \otimes N_1, \rho_2)\circ (A \otimes(-))}{\longrightarrow}} {\overset{hom(\rho_1,N_2)}{\longrightarrow}} {\phantom{AAAAAA}} hom(A \otimes N_1, N_2) \,.$$

   equipped with the left $A$-action that is induced by the left $A$-action on $N_2$ via

   $$\frac{ A \otimes hom(X,N_2) \longrightarrow hom(X,N_2) }{ A \otimes hom(X,N_2) \otimes X \overset{id \otimes ev}{\longrightarrow} A \otimes N_2 \overset{\rho_2}{\longrightarrow} N_2 } \,.$$

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

Proposition

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

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

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

Proof sketch

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

Example

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

$$(A \otimes C_1) \otimes_A (A \otimes C_2) \simeq A \otimes (C_1 \otimes C_2) \,.$$

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

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

Proof

It is sufficient to show that the diagram

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

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

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

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

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

We claim that

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

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

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

$$\array{ A \otimes A &\overset{\mu}{\longrightarrow}& A \\ {}^{\mathllap{id \otimes r^{-1}}}_{\mathllap{\simeq}}\downarrow && \downarrow^{\mathrlap{r^{-1}}}_{\simeq} \\ A \otimes A \otimes 1 &\overset{\mu \otimes id}{\longrightarrow}& A \otimes 1 \\ {}^{\mathllap{id \otimes e}}\downarrow && \downarrow^{\mathrlap{id \otimes e} } \\ A \otimes A \otimes A &\overset{\mu \otimes id}{\longrightarrow}& A \otimes A \\ {}^{\mathllap{id \otimes \mu}}\downarrow && \downarrow^{\mathrlap{f}} \\ A \otimes A &\underset{f}{\longrightarrow}& Q } \,.$$

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

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

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

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

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 $q = \phi$.

Definition

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

Propposition

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

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

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

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

Proof

In one direction, consider a $A$-algebra $E$ with unit $e_E \;\colon\; A \longrightarrow E$ and product $\mu_{E/A} \colon E \otimes_A E \longrightarrow E$. There is the underlying product $\mu_E$

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

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

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

$$(\star) \;\;\;\;\; \;\;\;\;\; \array{ A \otimes A &\overset{id \otimes e_E}{\longrightarrow}& A \otimes E \\ {}^{\mathllap{\mu_A}}\downarrow && \downarrow^{\mathrlap{\rho}} \\ A &\underset{e_E}{\longrightarrow}& E }$$

commutes. Moreover it satisfies the unit property

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

By forgetting the tensor product over $A$, the latter gives

$$\array{ A \otimes E &\overset{e \otimes id}{\longrightarrow}& E \otimes E \\ \downarrow && \downarrow^{\mathrlap{}} \\ A \otimes_A E &\overset{e_E \otimes id}{\longrightarrow}& E \otimes_A E \\ {}^{\mathllap{\simeq}}\downarrow && \downarrow^{\mathrlap{\mu_{E/A}}} \\ E &=& E } \;\;\;\;\;\;\;\; \simeq \;\;\;\;\;\;\;\; \array{ A \otimes E &\overset{e_E \otimes id}{\longrightarrow}& E \otimes E \\ {}^{\mathllap{\rho}}\downarrow && \downarrow^{\mathrlap{\mu_{E}}} \\ E &\underset{id}{\longrightarrow}& E } \,,$$

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

$$\array{ A \otimes A &\overset{id\otimes e_E}{\longrightarrow}& A \otimes E &\overset{e_E \otimes id}{\longrightarrow}& E \otimes E \\ {}^{\mathllap{\mu_A}}\downarrow && {}^{\mathllap{\rho}}\downarrow && \downarrow^{\mathrlap{\mu_{E}}} \\ A &\underset{e_E}{\longrightarrow}& E &\underset{id}{\longrightarrow}& E } \;\;\;\;\;\;\;\;\;\; = \;\;\;\;\;\;\;\;\;\; \array{ A \otimes A &\overset{e_E \otimes e_E}{\longrightarrow}& E \otimes E \\ {}^{\mathllap{\mu_A}}\downarrow && \downarrow^{\mathrlap{\mu_E}} \\ A &\underset{e_E}{\longrightarrow}& E } \,.$$

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

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

$$\rho \;\colon\; A \otimes E \overset{e_A \otimes id}{\longrightarrow} E \otimes E \overset{\mu_E}{\longrightarrow} E \,.$$

By commutativity and associativity it follows that $\mu_E$ coequalizes the two induced morphisms $E \otimes A \otimes E \underoverset{\longrightarrow}{\longrightarrow}{\phantom{AA}} E \otimes E$. Hence the universal property of the coequalizer gives a factorization through some $\mu_{E/A}\colon E \otimes_A E \longrightarrow E$. This shows that $(E, \mu_{E/A}, e_E)$ is a commutative $A$-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 $R$-algebras. The following says that these may be computed just as the tensor product of modules:

Proposition

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

1. it has reflexive coequalizers

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

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

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

Here $B$ and $C$ are regarded as equipped with the canonical $A$-module structure induced by the morphisms $f$ and $g$, 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 $f,g$ in an abelian category is isomorphic to the cokernel of the difference $f-g$ (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 $A \otimes (-)$ are left adjoint functors, and such preserve all colimits.

Proposition

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

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

$$Spec(A_1) \times Spec(A_2) \simeq Spec(A_1 \otimes_R A_2) \,.$$

Proof

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

Proposition

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

$$\phi \;\colon\; A_1 \longrightarrow A_2$$

a homomorphism commutative monoids (def. ).

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

$$A_1 Mod(\mathcal{C}) \underoverset {\underset{\phi^\ast}{\longleftarrow}} {\overset{\phi_\ast}{\longrightarrow}} {} A_2 Mod(\mathcal{C})$$

where

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

   $$A_1 \otimes N \stackrel{\phi \otimes id}{\longrightarrow} A_2 \otimes N \stackrel{\rho}{\longrightarrow} N \,.$$

2. the left adjoint, called extension of scalars sends an $A_1$-module $(N,\rho)$ to the tensor product

   $$\phi_\ast(N) \coloneqq A_2 \otimes_{A_1} N$$

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

$$A_2 \otimes \phi^\ast(N) = A_2 \otimes A_2 \otimes_{A_1} N \stackrel{\mu_{A_2} \otimes_{A_1} N }{\longrightarrow} A_2 \otimes_{A_1} N = \phi^\ast(N) \,.$$

Proof

By prop. the adjunction in question has the form

$$A_1 Mod(\mathcal{C}) \underoverset {\underset{U}{\longleftarrow}} {\overset{F}{\longrightarrow}} {} A_2 Mod( (A_1 Mod, \otimes_{A_1}, A_1) )$$

and hence the statement follows with prop. .

Remark

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

$$Spec(\phi) \colon Spec(A_2) \longrightarrow Spec(A_1)$$

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

Super-Groups as super-commutative Hopf algebras

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

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 $G$ on a vector space $V$ is a smooth function

$$\rho \;\colon\; G \times V \longrightarrow V$$

such that

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

   $$\rho(g) \colon V \longrightarrow V$$

   is a linear function

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

3. (action property) for $g_1, g_2 \in G$ any two elements, then acting with them consecutively is the same as acting with their product:

   $$\rho(g_2) \circ \rho(g_1) = \rho(g_2 g_1) \,.$$

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 $C^\infty(X)$ for the smooth algebra of smooth functions on a smooth manifold $X$. The assignment

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

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

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

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

Together this means that if $X = G$ 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:

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

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 $X$ a finite set consider

$$\array{ X \times X \times X \\ \downarrow^{\mathrlap{\circ = (pr_1, pr_3)}} \\ X \times X \\ {}^{\mathllap{s = pr_1}}\downarrow \uparrow \downarrow^{\mathrlap{t = pr_2}} \\ X }$$

as the (“codiscrete”) groupoid with $X$ 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 $\mathbb{Z}[X]$, regarded as commutative rings under pointwise multiplication.

Since

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

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

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

satisfying some obvious conditions. Observe that here

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

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

3. the morphism

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

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

We now say this again, in generality:

Definition

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

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

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

Proposition

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

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

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

$$inv \;\colon\; Spec(\Gamma) \longrightarrow Spec(\Gamma)$$

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 $CMon(\mathcal{A})^{op}$, :

$$Spec(R_1) \underset{Spec(R_3)}{\times} Spec(R_2) \simeq Spec(R_1 \otimes_{R_3} R_2) \,.$$

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

$$\array{ \Gamma \underset{A}{\otimes} \Gamma \\ \uparrow^{\mathrlap{\Psi}} \\ \Gamma \\ {}^{\mathllap{\eta_L}}\uparrow \downarrow^{\mathrlap{\epsilon}} \; \uparrow^{\mathrlap{\eta_R}} \\ A }$$

as well as

$$c \; \colon \; \Gamma \longrightarrow \Gamma$$

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

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

• $\epsilon$ is called the co-unit;

• $\Psi$ is called the comultiplication;

• $c$ is called the antipode or conjugation

Remark

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

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

Dually, a commutative Hopf algebroid $\Gamma \stackrel{\overset{\eta_L}{\longleftarrow}}{\underset{\eta_R}{\longleftarrow}} A$ for which $\eta_L$ and $\eta_R$ happen to coincide is equivalently a commutative Hopf algebra $\Gamma$ over $A$.

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, $A$ and $\Gamma$;

2. ring homomorphisms

   1. (left/right unit)

      $\eta_L,\eta_R \colon A \longrightarrow \Gamma$;

   2. (comultiplication)

      $\Psi \colon \Gamma \longrightarrow \Gamma \underset{A}{\otimes} \Gamma$;

   3. (counit)

      $\epsilon \colon \Gamma \longrightarrow A$;

   4. (conjugation)

      $c \colon \Gamma \longrightarrow \Gamma$

such that

1. (co-unitality)

   1. (identity morphisms respect source and target)

      $\epsilon \circ \eta_L = \epsilon \circ \eta_R = id_A$;

   2. (identity morphisms are units for composition)

      $(id_\Gamma \otimes_A \epsilon) \circ \Psi = (\epsilon \otimes_A id_\Gamma) \circ \Psi = id_\Gamma$;

   3. (composition respects source and target)

      1. $\Psi \circ \eta_R = (id \otimes_A \eta_R) \circ \eta_R$;

      2. $\Psi \circ \eta_L = (\eta_L \otimes_A id) \circ \eta_L$

2. (co-associativity) $(id_\Gamma \otimes_A \Psi) \circ \Psi = (\Psi \otimes_A id_\Gamma) \circ \Psi$;

3. (inverses)

   1. (inverting twice is the identity)

      $c \circ c = id_\Gamma$;

   2. (inversion swaps source and target)

      $c \circ \eta_L = \eta_R$; $c \circ \eta_R = \eta_L$;

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

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

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

      and

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

      satisfy

      $\alpha \circ \Psi = \eta_L \circ \epsilon$

      and

      $\beta \circ \Psi = \eta_R \circ \epsilon$.

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

By internalizing all of the above from $Vect$ to $sVect$, we obtain the concept of supergroups:

Definition

An affine algebraic supergroup $G$ is equivalently

• a pointed, one-object internal groupoid in the opposite category $Aff(sVect) = CMon(sVect_k)^{op}$ (def. ) of supercommutative superalgebras from def.

• the formal dual of a super-commutative Hopf algebra, namely a commutative Hopf algebra (prop. , remark ).

We will often just say “supergroup” for short in the following. If $H$ is the corresponding supercommutative Hopf algebra then we also write $Spec(H)$ 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 $G$, def. is an element $\epsilon \in G_{even}$ such that

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

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

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

1. the composite

   $$\mathcal{O}(G) \stackrel{\Psi}{\longrightarrow} \mathcal{O}(G) \otimes_k \mathcal{O}(G) \stackrel{\epsilon^\ast \otimes_k \epsilon^\ast}{\longrightarrow} k \otimes_k k \simeq k$$

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

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

   $$\mathcal{O}(G) \stackrel{(id \otimes_k \Psi) \circ \Psi}{\longrightarrow} \mathcal{O}(G) \otimes_k \mathcal{O}(G) \otimes_k \mathcal{O}(G) \stackrel{\epsilon^\ast \otimes_k id \otimes_k (c \circ \epsilon^\ast)}{\longrightarrow} k \otimes_k \mathcal{O}(G) \otimes_k k$$

(Deligne 02, 0.3)

Example

For $G$ an ordinary affine algebraic group, regarded as a supergroup with trivial odd-graded part, then every element $\epsilon \in Z(G)$ 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 $H$ any supergroup, def. , and $\mathbb{Z}_2 = \{id,par\}$ acting on it by parity involution, def. then the semidirect product group $\mathbb{Z}_2 \ltimes G$ has inner parity, def. , given by $\epsilon \coloneqq par \in \mathbb{Z}_2 \hookrightarrow \mathbb{Z}_2 \ltimes G$.

(Deligne 02, 0.4 ii))

Linear super-representations as Comodules

Definition

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

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

2. an $A$-module homomorphism (co-action)

   $\Psi_N \;\colon\; N \longrightarrow \Gamma \otimes_A N$;

such that

1. (co-unitality)

   $(\epsilon \otimes_A id_N) \circ \Psi_N = id_N$;

2. (co-action property)

   $(\Psi \otimes_A id_N) \circ \Psi_N = (id_\Gamma \otimes_A \Psi_N)\circ \Psi_N$.

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

$$\Gamma CoMod(\mathcal{A})$$

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

Example

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

$$A \overset{\eta_R}{\longrightarrow} \Gamma \simeq \Gamma \otimes_A A \,.$$

Proof

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

$$\epsilon \circ \eta_R = id_A$$

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

$$id \;\colon\; A \overset{\eta_R}{\longrightarrow} \Gamma \simeq \Gamma \otimes_A A \overset{\epsilon \otimes_A id}{\longrightarrow} A \otimes_A A \simeq A$$

The required co-action property is the dual condition

$$\Psi \circ \eta_R = (id \otimes_A \eta_R) \circ \eta_R$$

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

$$\array{ A &\overset{\eta_R}{\longrightarrow}& \Gamma \\ {}^{\mathllap{\eta_R}}\downarrow && \downarrow^{\mathrlap{\Psi}} \\ \Gamma \simeq \Gamma \otimes_A A &\underset{id \otimes_A \eta_R}{\longrightarrow}& \Gamma \otimes_A \Gamma } \,.$$

Definition

Given two comodules $N_1, N_2$ over a commutative Hopf algebra $\Gamma$ over $k$, then their tensor product is the the tensor product of modules $N_1 \otimes_k N_2$ equipped with the following co-action

$$N_1 \otimes_k N_2 \overset{\Psi1 \otimes_k \Psi_2}{\longightarrow} \Gamma \otimes_k N_1 \otimes_k \Gamma \otimes N_2 \overset{}{\longrightarrow} \Gamma \otimes_k \Gamma \otimes_k N_1 \otimes_k N_2 \overset{((-)\cdot (-)) \otimes_k id_{N_1} \otimes_k id_{N_2} }{\longrightarrow} \Gamma \otimes_k N_1 \otimes_k N_2 \,.$$

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

$$G \times V_1 \times V_2 \overset{\Delta_G \times id}{\longrightarrow} G \times G \times V_1 \times V_2 \simeq G \times V_1 \times G \times V_1 \overset{\rho_1 \times \rho_2}{\longrightarrow} V_1 \times V_2 \,.$$

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

Definition

A linear representation of a supergroup $G$, def. , with inner parity $\epsilon$, def. , is

• a comodule $V$ (def. ) in sVect (def. ) for the formally dual commutative Hopf algebra $\mathcal{O}(G)$.

such that

• the innr parity element $\epsilon$ acts as the identity on $V_{even}$ and by multiplicatin with $-1$ on $V_{odd}$.

Example

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

(Deligne 02, 0.4 i))

Proposition

The category of representations $Rep(G,\epsilon)$ of def. of an affine algebraic supergroup $G$, def. , with inner parity $\epsilon$ (def. ) on finite-dimensional super vector spaces (example ) and equippd with the tensor product of comodules from def. is a $k$-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 $\mathcal{A}$ 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 $\mathcal{A}$ is that there exists a forgetful functor

$$\omega \;\colon\; \mathcal{A} \longrightarrow \mathcal{V}$$

to some other tensor category, such that $\omega$ 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 $\mathcal{A}$ as a bundle over $\mathcal{V}$, and over each $V \in \mathcal{V}$ we find the fiber $\omega^{-1}(V)$ of that bundle, consisting of all those objects in $\mathcal{A}$ whose underlying object in the given $V$.

The main point of Tannaka duality of tensor categories is the observation that if $\mathcal{A}$ is a category of representations of some group $G$, then $G$ 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 $G$ 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 $\mathcal{A}$ and $\mathcal{T}$ be two $k$-tensor categories (def. ) such that

1. all objects have finite length;

2. all hom spaces are of finite dimension over $k$.

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

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

$$\omega \;\colon\; \mathcal{A} \longrightarrow R Mod(Ind(\mathcal{T}))$$

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

1. a braided strong monoidal functor;

2. an exact functor in both variables.

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

(Deligne 02, 3.1)

Definition

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

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

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

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

Given a super fiber functor $\omega \colon \mathcal{A} \to sVect_k$ (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

$$Aut(\omega) \in Grp$$

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 $A$ with corresponding affine super scheme $Spec(A)$ (def. , def. ) we are to say what the set

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

of $Spec(A)$-parameterized elements of $Aut(\omega)$ 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:

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

Moreover, if $A_1 \longrightarrow A_2$ is an algebra homomorphism, hence

$$Spec(A_2) \longrightarrow Spec(A_1)$$

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

$$\underline{Aut}(\omega)(Spec(A_2)) \longleftarrow \underline{Aut}(\omega)(Spec(A_1))$$

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

For a minimum of consistency, this assignment must be such that the identity map on $Spec(A)$ induces the identity on $\underline{Aut}(\omega)(Spec(A))$, 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 $Aut(\omega)$ is the datum of a presheaf of groups, hence of a functor

$$\underline{Aut}(\omega) \;\colon\; Aff(sVect)^{op} \simeq CMon(sVect) \longrightarrow Grp$$

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

$$\underline{Aut}(\omega)(Spec(k)) \simeq Aut(\omega) \,.$$

Definition

We say that a functor

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

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

$$\underline{Aut}(\omega) \simeq Hom_{CMon(sVect)}(H,-) = Hom_{Aff(sVect)}(-,Spec(H)) \,.$$

Definition

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

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

$$\omega_A \;\colon\; \mathcal{A} \stackrel{\omega}{\longrightarrow} \mathcal{B} \stackrel{A \otimes(-)}{\longrightarrow} A Mod(\mathcal{B})$$

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

With this, the automorphism group of $\omega$

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

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

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

and which to a homomorphism of algebras

$$\phi \;\colon\; A_1 \longrightarrow A_2$$

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

$$\phi_\ast \;\colon\; Aut(\omega_{A_1}) \longrightarrow Aut(\omega_{A_2}) \,.$$

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 $\omega \colon \mathcal{A} \to sVect$ be a super fiber functor (def ).

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

$$\omega_A \;\colon\; \mathcal{A} \stackrel{\omega}{\longrightarrow} sVect \stackrel{A \otimes(-)}{\longrightarrow} A Mod(sVect)$$

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

$$\omega_A \;\colon\; \mathcal{A} \stackrel{\omega}{\longrightarrow} sVect \stackrel{A \otimes(-)}{\longrightarrow} A Mod(sVect)$$

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

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

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

is defined by

$$\underline{Aut}(\omega)(Spec(A)) \coloneqq Aut(\omega_{A}) \,.$$

Proposition

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

$$\underline{Aut}(\omega) \simeq Hom_{CMon(sVect)}(H_\omega,-) = Hom_{Aff(sVect)}(-, Spec(H_\omega)) \,,$$

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

$$\underline{Aut}(\omega) \simeq Spec(H_\omega) \,.$$

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

$$\pi(\mathcal{A}) \coloneqq \underline{Aut}(id_{\mathcal{A}})$$

(Deligne 90, 8.12, 8.13)

Example

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

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

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

(Deligne 90, 8.14 iv))

Proposition

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

$$\underline{Aut}(\omega) \simeq \omega(\pi(\mathcal{A})) \,.$$

Here on the right we are using that $\omega$ 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 $\mathcal{A}_1, \mathcal{A}_2$ be two $k$-tensor categories and let

$$\eta \;\colon\; \mathcal{A}_1 \longrightarrow \mathcal{A}_2$$

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

$$\pi(\mathcal{A}_2) \longrightarrow \eta(\pi(\mathcal{A}_1))$$

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

(Deligne 90, 8. 15. 2)

Super-exterior powers and Schur functors

A finite dimensional vector space $V$ has the property that a high enough alternating power of it vanishes $\wedge^n V = 0$, namely this is the case for all $n \gt dim(V)$, and hence this vanishing is just another reflection of the finiteness of the dimension of $V$. For a super vector space $V$ 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 $X$ and given any choice of irreducible representation $V_\lambda$ of the symmetric group $\Sigma_n$, then one consider the subobject $S_\lambda(X^{\otimes^n})$ of the $n$-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 $(\mathcal{A},\otimes)$ a $k$-tensor category as in def., for $X \in \mathcal{A}$ an object, for $n \in \mathbb{N}$ and $\lambda$ a partition of $n$, regarded as a Young diagram and hence as a representation of the symmetric group $V_\lambda$, say that the value of the Schur functor $S_\lambda$ on $X$ is

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

where

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

• $S_n$ is the symmetric group on $n$ elements;

• $V_\lambda$ is the irreducible representation of $S_n$ corresponding to $\lambda$;

• $\rho$ is diagonal action of $S_n$ on $V_\lambda \otimes X^{\otimes_n}$, coming from the canonical permutation action on $X^{\otimes_n}$;

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

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

(Deligne 02, 1.4)

Example

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

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

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

$$S_{(1,1, \cdots, 1)}(X) = \wedge^n X \,.$$

Proposition

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

$$S_\lambda(V) \simeq 0 \,.$$

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

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

(Deligne 02, corollary 1.9)

Statement of the theorem

Theorem

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

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

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

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

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

2. a tensor-equivalence of categories

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

   between $\mathcal{A}$ and the category of representations of $G$ 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 $k$ 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 $k$-tensor category an object $X$ is of subexponential growth (def. ) precisely if there exists a Schur functor that annihilates it, hence if some power of $X$, 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 $\mathcal{A}$ is annihilated by some Schur functor, then there exists a super fiber functor on $\mathcal{A}$ over some supercommutative superalgebra $R$, hence then every object of $\mathcal{A}$ has underlying it a super vector space with some extra structure.

   This proposition is where superalgebra proper appears.

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

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

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

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

(Deligne 02, prop. 05)

Proposition

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

$$\omega \;\colon\; \mathcal{A} \longrightarrow sVect \,.$$

(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

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

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

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

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

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

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

Show then that the condition that an object $X$ is annihilated by some Schur functor is equivalent to the existence of an algebra $A$ such that

$$X_A \simeq 1^p \oplus \overline{1}^q$$

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

(Deligne 02, prop. 2.9).

Moreover, for each short exact sequence

$$0 \to X \to Y \to Z \to 0$$

in $s \mathcal{A}$, there exists an algebra $A$ such that

$$0 \to X_A \to Y_A \to Z_A \to 0$$

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 $A$ 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 $\mathcal{A}$ of choices of commutative monoids for which these objects/exact sequencs are locally split, as above.

Then for an $A$-module $N$, write $\rho(N)$ for the subobject of $N$ inside $sVect \simeq Ind\langle 1, \overline{1}\rangle \hookrightarrow s \mathcal{A}$.

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

Set

$$R \coloneqq \rho(A) \,.$$

Hence for every object $X$, then

$$\omega(X) \coloneqq \rho(X_A)$$

has the structure of an $R$-module. By $A$-local splitness of all short exact sequence, $\omega$ is an exact functor.

Hence

$$\omega \;\colon\; s \mathcal{A} \longrightarrow R Mod(sVect)$$

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

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

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

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

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

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

This is argued by expressing $R$ as an inductive limit

$$R = \underset{\longrightarrow}{\lim}_\beta R_\beta$$

over supercommutative superalgebras $R_\beta$ of finite type over $k$ and observing (…) that there exists $\beta$ such that $\omega_\beta$ is still a fiber functor and such that there exists an algebra homomorphism $R_\beta \to k$.

Finally then the fiber functor in question is

$$\omega_\beta \otimes_{R_\beta} k \;\colon\; \mathcal{A} \longrightarrow sVect_k \,.$$

Proposition

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

$$\omega \;\colon\; \mathcal{A} \longrightarrow sVect_k$$

then $\omega$ induces an equivalence of categories

$$\mathcal{A} \stackrel{\simeq}{\longrightarrow} Rep( \underline{Aut}(\omega),\epsilon)$$

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

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

(according to example ) under the group homomorphism

$$\pi(sVect) \longrightarrow \omega(\pi(\mathcal{A})) \simeq \underline{Aut}(\omega)$$

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, pdf)

building on the general results on Tannakian categories in

• Pierre Deligne, Catégories Tannakiennes, Grothendieck Festschrift, vol. II, Birkhäuser Progress in Math. 87 (1990) pp. 111-195 (pdf)

which are reviewed and further generalized in

• Pierre Deligne, James Milne, Tannakian categories, 2012 (pdf)

Review is in

• Victor Ostrik, Tensor categories (after P. Deligne) (arXiv:math/0401347)

• Pavel Etingof, Shlomo Gelaki, Dmitri Nikshych, Victor Ostrik, section 9.11 in Tensor categories, Mathematical Surveys and Monographs, Volume 205, American Mathematical Society, 2015 (pdf)

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

• Pavel Etingof, Shlomo Gelaki, The classification of finite-dimensional triangular Hopf algebras over an algebraically closed field of characteristic 0 (arXiv:0202258)

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

• Michael Müger, Abstract Duality Theory for Symmetric Tensor $*$-Categories appendix (pdf), in Hans Halvorson, Algebraic quantum field theory (arXiv:math-ph/0602036), in J. Butterfield & J. Earman (eds.) Handbook of Philosophy and Physics

as a proof of Doplicher-Roberts reconstruction

Commutative algebra internal to symmetric monoidal categories is discussed in

• Anthony Elmendorf, Igor Kriz, Michael Mandell, Peter May, Rings, modules and algebras in stable homotopy theory, AMS 1997, 2014

• Mark Hovey, Brooke Shipley, Jeff Smith, Symmetric spectra, J. Amer. Math. Soc. 13 (2000), 149-208 (arXiv:math/9801077)

and specifically for commutative Hopf algebroids in

• Doug Ravenel, chapter 2 and appendix A.1 of Complex cobordism and stable homotopy groups of spheres, 1986 (pdf)

(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

• Victor Ostrik, On symmetric fusion categories in positive characteristic, (arXiv:1503.01492)

This was realised for Frobenius exact tensor categories in positive characteristic in:

• Kevin Coulembier, Pavel Etingof, Victor Ostrik, On Frobenius exact symmetric tensor categories $[$arXiv:2107.02372$]$

where $Vect$ and $sVect$ are replaced by more exotic targets.

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

• Theo Johnson-Freyd, Spin, statistics, orientations, unitarity, Algebraic and Geometric Topology (arXiv:1507.06297)

On Deligne categories:

• Serina Hu. An Introduction to Deligne Categories (2024). (arXiv:2404.08689).