geometric representation theory
representation, 2-representation, ∞-representation
Grothendieck group, lambda-ring, symmetric function, formal group
principal bundle, torsor, vector bundle, Atiyah Lie algebroid
Eilenberg-Moore category, algebra over an operad, actegory, crossed module
symmetric monoidal (∞,1)-category of spectra
equivalences in/of $(\infty,1)$-categories
In full generality, an $\infty$-representation is an ∞-action of some higher algebraic structure on some object in a higher category up to coherent homotopy. One also speaks of representation up to homotopy or maybe sh-representations .
We motivate the general notion of $\infty$-representation by recalling first some category theoretic aspects of the ordinary notion of representation, and then leading over to the analogous (∞,1)-category theoretic notions.
Recall that for $G$ a discrete group with delooping groupoid $\mathbf{B}G = (G \stackrel{\to}{\to})$, and $k$Vect the category of vector spaces (over some base field $k$), ordinary linear representations of $G$ are equivalently functors
Such a functor takes the single object of $\mathbf{B}G$ to some vector space $V$ and takes every morphism $(* \stackrel{g}{\to} * )$ in $\mathbf{B}G$ labeled by an element $g \in G$ to a linear automorphism $\rho(g) : V \to V$ such that composition and the identity is respected. We have an equivalence of categories
Here the category $Vect$ could be replaced by other categories and they need not be abelian categories or otherwise linear for the above to make sense. For instance if we take instead the category Set itself, then a functor
is what is called a permutation representation. In topology one is interested in representations in Top
(However, rarely is it sufficient to regard these just as functors to the 1-category $Top$. Instead, in order to speak about topological fiber bundles and fibrations, one needs to regard Top here as an (∞,1)-category and regard $\rho : \mathbf{B}G \to Top$ as an (∞,1)-functor. This we come to below in ∞-Representations)
Moreover, we may replace $\mathbf{B}G$ by a more general groupoid. For $K$ any groupoid, a functor
is called a linear representation of $K$. This now picks not just a single vector space $V \in Vect$, but one vector space $V_x$ for each object $x \in K$. And to each morphism $(x \stackrel{g}{\to} y)$ in $K$ is assigns a linear map $\rho(g) : V_x \to V_y$.
For instance if $K = \Pi_1(X)$ is the fundamental groupoid of a manifold $X$, then a representation
is a vector bundle over $X$ with flat connection on a bundle.
And we do not even need to assume that $K$ here is a groupoid. For instance if $D$ is a directed graph (or quiver) and $F(D)$ its path category, then a functor
is called a quiver-representation of $D$.
One could in principle therefore speak of a functor
as a “quiver permutation representation”, but there does not seem to be much use of this terminology in practice. The examples do show, however, that there is considerable overlap between the notion of representation and of functor .
Still a bit more generally, we can speak of representations that preserve extra structure, such as smooth structure. For instance for $G$ a Lie group we have that $\mathbf{B}G$ is a Lie groupoid: an object in the (2,1)-topos of (2,1)-sheaves over the site $C =$ CartSp or $C =$ Diff.
We may also promote the category Vect to this $(2,1)$-topos, by replacing it by the stack $VectBund$, which assigns to each test manifold $U \in C$ the groupoid of smooth vector bundles over $U$. Then a morphism
in the (2,1)-topos $Sh_{(2,1)}(C)$ is a smooth representation of $G$, in that the linear automorphisms $\rho(g) : V \to V$ depend smoothly on the point $g \in G$.
We can recover the underlying ordinary representation by applying the global section functor $\Gamma : Sh_{(2,1)}(C) \to Grpd$. This is given by evaluating every thing on the terminal object ${} \in C, which here is just the ordinary point, regarded as a smmoth manifold.
This yields the underlying bare representation
Conversely, one finds that extending such a bare representation from the point to all test spaces in $C$ amounts to equipping it with smooth structure.
As before, this is not restricted to connected objects: we may replace $\mathbf{B}G$ here with any Lie groupoid. For instance for $X$ a smooth manifold and $\mathbf{P}_1(X)$ is smooth path groupoid a representation
as a morphism in the 2-topos is a vector bundle on $X$ with connection on a bundle. Or if we consider non-linear representations, a representation
is a $G$-principal bundle with connection on a bundle on $X$. See parallel transport for more details and references.
By just changing the site here, we can implement other geometric structures. For instance for $G$ an algebraic group we may think of $\mathbf{B}G$ as an algebraic stack over something like the fppf-site structure $C = CAlg_k^{op}$ on formal duals of commutative $k$-algebras or similar.
In this case there is a well-known good generalization of $VectBund$: instead of just vector bundles we can consider their completion to quasicoherent sheaves. The stack of these is the object in the (2,1)-topos given by
where on the right we have the groupoid of modules over $A$. Over the point this is again just a $k$-module hence a vector space and hence a representation
of an algebraic group is a representation of $G$ on a vector space.
But also here we may allow the represented structure to have more than one object. For instance for $X$ any scheme regarded as an object in $Sh_{(2,1)}(C)$ a representation of $X$ in the context of $QC$ is a morphism
which is equivalently a quasicoherent sheaf of modules on $X$. As before, we may think of this as assigning to each point of $X$ one representation space, only that in a scheme $X$ there are no morphisms that would act on these.
But more generally for $K$ an algebraic stack, a representation
assigns to each point of $K_0$ a representation space, such that these glue together to a quasicoherent sheaf of modules, and to each morphism in $K$ a morphism between the corresponding representation spaces, as before.
Analogous constructions are available for more general sites, effectively we can take $C$ to be the opposite category of $T$-algebras over a Lawvere theory for $T$ any algebraic theory that contains the theory of abelian groups. If for instance we take $T =$ CartSp we are back to the smooth case discussed before.
Also notice that if we take the site to be the point, $C = *$, then sheaves over it are just sets and stacks over it are just bare groupoids, so that we recover the discussion at the very beginning.
We have found above the the term “representation” it to a large extent congruent with the term “morphism in a (2,1)-topos with codomain a stack of modules”.
This way of thinking of representations has an immediate generalization to higher category theory and in particular to (∞,1)-category theory/homotopy theory.
To start with the simple discussion over the point again, a model for a notion of a category of $\infty$-modules that is useful is an (∞,1)-category $Ch_\bullet(k)$ that is presented by a model structure on chain complexes.
If again $G$ is a discrete group, then an (∞,1)-functor (equivalently: a “strong homotopy-functor” or “”homotopy coherent functor, see there for details)
assigns
to the single obect of $\mathbf{B}G$ a chain complex $V_\bullet$;
to a group element $g \in G$ a chain map $\rho(g) : V_\bullet \to V_\bullet$;
to a pair of group elements $g, g'$ a chain homotopy
$\rho(g,g') : \rho(g')\circ\rho(g) \Rightarrow \rho(g' g)$;
to a triple of group elements a homotopy of homotopies between composites of $\rho(g,g'), \rho(g,g'')$ and $\rho(g',g'')$ and so on
(see the diagrams at group cohomology for more details in low degree).
In other words, this is much like representation of $G$ as before on an ordinary vector space, only that now the action property of $\rho$ holds only up to coherent homotopy . Therefore people also speak of representations up to homotopy (AbadCrainic) as well as strong homotopy representations and many other variants.
As before, there is in principle no reason to restrict oneself to representations of groupoids here. For $K$ any ∞-groupoid or even (∞,1)-category (recall the quiver representations) and for $Mod$ any (∞,1)-category of $\infty$-modules (for instance as presented by a model structure on modules over an algebra over an operad) we may call an (∞,1)-functor
an $\infty$-representation of $K$.
If we wish to consider $\infty$-generalizations of permutation representations we can also consider more general codomain $(\infty,1)$-categories here. For instance if we take ∞Grpd itself, then an $\infty$-permutation representation
is known as an (∞,1)-presheaf. For $K$ the delooping of an ordinary group or the orbit category of a topological group, such appear genuinely from the point of view of representations for instance in equivariant cohomology and equivariant homotopy theory. Notice that by the homotopy hypothesis-theorem we have an equivalence of (∞,1)-categories
so that the above is equivalently an (∞,1)-functor
hence literally a representation up to homotopy in the classical sense of homotopy theory.
As before, all this may be lifted from the point into large classes of (∞,1)-toposes to equip the notion of $\infty$-representation with geometric structure (algebraic structure, smooth structure, etc.)
There are then analogs of the above relation between representations of path groupoids and connections on bundles. For more on this see higher parallel transport. Quite generally, in every locally ∞-connected (∞,1)-topos $\mathbf{H}$ there is a notion of fundamental ∞-groupoid $\mathbf{P}(X)$ of any object $X$. Representation of $\mathbf{\Pi}(X)$ define general abstract flat differential cohomology and local systems on $X$, generally also in nonabelian cohomology (see there for some more properties and examples).
For instance dg-geometry is the study of the (∞,1)-topos over an (∞,1)-site of formal duals of dg-algebras. Again there is the canonical ∞-stack
on this site, where now however $A Mod$ denotes the ∞-groupoid (or (∞,1)-category if we do a more comprehensive discussion) of chain complexes equipped with the structure of a module ove the dg-algebra $A$.
For $X$ any ∞-stack then a morphism
is a equivalently a quasicoherent ∞-stack of modules on $X$, or an $\infty$-representation with “dg-algebraic structure”.
If one replaces $X$ here by its de Rham stack $X_{dR}$ then dg-algebraic $\infty$-representations
are D-modules on $X$.
A discussion of these higher categorical structure in representation theory is in (Ben-ZviNadler).
If the codomain $Mod$ is not an (∞,1)-category that is just a (n,1)-category (all k-morphismss for $k \gt n$ are effectively identities) then an $\infty$-representation is called an $n$-representation. These are representations up to homotopy where from degree $n$ on all homotopies are actually identities: $n$-truncated representations up to homotopy.
As always in higher category theory, the cases for low $n$ are more restrictive but typically admit a more tractable detailed analysis and construction.
2-representations of 2-groups and Lie 2-groups on various variants of 2-vector spaces have been considered for instance in (Schreiber, BaezBaratinFreidelWise, and other places).
In analogy to the case for $n=1$, 2-Representations $\mathbf{P}_2(X) \to 2 Vect$ of the smooth path 2-groupoid of a smooth manifold describe connections on a 2-bundle. See there for more details.
In a general abstract context of homotopy type theory we may define $\infty$-representations as follows.
For $\mathbf{H}$ an (∞,1)-topos, let $G \in Grp(\mathbf{H})$ be a group object in $\mathbf{H}$, hence an ∞-group. Then the slice (∞,1)-topos $\mathbf{H}_{/\mathbf{B}G}$ over its delooping is the (∞,1)-category of ∞-actions of $G$
hence of possibly “non-linear” $\infty$-representations. (See at ∞-action for details). A genuine (linear) $\infty$-representation is then an abelian ∞-group object in $Act(G)$.
representation, $\infty$-representation
representation theory and equivariant cohomology in terms of (∞,1)-topos theory/homotopy type theory:
homotopy type theory | representation theory |
---|---|
pointed connected context $\mathbf{B}G$ | ∞-group $G$ |
dependent type | ∞-action/∞-representation |
dependent sum along $\mathbf{B}G \to \ast$ | coinvariants/homotopy quotient |
context extension along $\mathbf{B}G \to \ast$ | trivial representation |
dependent product along $\mathbf{B}G \to \ast$ | homotopy invariants/∞-group cohomology |
dependent product of internal hom along $\mathbf{B}G \to \ast$ | equivariant cohomology |
dependent sum along $\mathbf{B}G \to \mathbf{B}H$ | induced representation |
context extension along $\mathbf{B}G \to \mathbf{B}H$ | |
dependent product along $\mathbf{B}G \to \mathbf{B}H$ | coinduced representation |
spectrum object in context $\mathbf{B}G$ | spectrum with G-action (naive G-spectrum) |
2-representations of 2-groups and Lie 2-groups such as the string 2-group on 2-vector spaces are disscussed in
References for 2- and 3-representatons of path n-groupoids are at higher parallel transport.
$\infty$-Representations of groupoids and Lie algebroids on $(\infty,1)$-categories of chain complexes are discussed under the term representations up to homotopy in
A discussion of quasicoherent $\infty$-stacks and D-modules in the context of representation theory is for instance in
The character theory of a complex group (arXiv:0904.1247)
Loop Spaces and Representations (arXiv:1004.5120)