Representation theory

Higher algebra

(,1)(\infty,1)-Category theory



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.

Representations as functors

Recall that for GG a discrete group with delooping groupoid BG=(G)\mathbf{B}G = (G \stackrel{\to}{\to}), and kkVect the category of vector spaces (over some base field kk), ordinary linear representations of GG are equivalently functors

ρ:BGkVect. \rho : \mathbf{B}G \to k Vect \,.

Such a functor takes the single object of BG\mathbf{B}G to some vector space VV and takes every morphism (*g*)(* \stackrel{g}{\to} * ) in BG\mathbf{B}G labeled by an element gGg \in G to a linear automorphism ρ(g):VV\rho(g) : V \to V such that composition and the identity is respected. We have an equivalence of categories

Func(BG,kVect)Rep k(G). Func(\mathbf{B}G, k Vect) \simeq Rep_k(G) \,.

Here the category VectVect 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

ρ:BGSet \rho : \mathbf{B}G \to Set

is what is called a permutation representation. In topology one is interested in representations in Top

ρ:BGTop. \rho : \mathbf{B}G \to Top \,.

(However, rarely is it sufficient to regard these just as functors to the 1-category TopTop. Instead, in order to speak about topological fiber bundles and fibrations, one needs to regard Top here as an (∞,1)-category and regard ρ:BGTop\rho : \mathbf{B}G \to Top as an (∞,1)-functor. This we come to below in ∞-Representations)

Moreover, we may replace BG\mathbf{B}G by a more general groupoid. For KK any groupoid, a functor

ρ:KkVect \rho : K \to k Vect

is called a linear representation of KK. This now picks not just a single vector space VVectV \in Vect, but one vector space V xV_x for each object xKx \in K. And to each morphism (xgy)(x \stackrel{g}{\to} y) in KK is assigns a linear map ρ(g):V xV y\rho(g) : V_x \to V_y.

For instance if K=Π 1(X)K = \Pi_1(X) is the fundamental groupoid of a manifold XX, then a representation

ρ:Π(X)vect \rho : \Pi(X) \to vect

is a vector bundle over XX with flat connection on a bundle.

And we do not even need to assume that KK here is a groupoid. For instance if DD is a directed graph (or quiver) and F(D)F(D) its path category, then a functor

F(D)Vect F(D) \to Vect

is called a quiver-representation of DD.

One could in principle therefore speak of a functor

F(D)Set F(D) \to Set

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 .

Structured representations as morphisms in 2-toposes

Still a bit more generally, we can speak of representations that preserve extra structure, such as smooth structure. For instance for GG a Lie group we have that BG\mathbf{B}G is a Lie groupoid: an object in the (2,1)-topos of (2,1)-sheaves over the site C=C = CartSp or C=C = Diff.

We may also promote the category Vect to this (2,1)(2,1)-topos, by replacing it by the stack VectBundVectBund, which assigns to each test manifold UCU \in C the groupoid of smooth vector bundles over UU. Then a morphism

ρ:BGVectBund \rho : \mathbf{B}G \to VectBund

in the (2,1)-topos Sh (2,1)(C)Sh_{(2,1)}(C) is a smooth representation of GG, in that the linear automorphisms ρ(g):VV\rho(g) : V \to V depend smoothly on the point gGg \in G.

We can recover the underlying ordinary representation by applying the global section functor Γ:Sh (2,1)(C)Grpd\Gamma : Sh_{(2,1)}(C) \to Grpd. This is given by evaluating every thing on the terminal object *C{*} \in C, which here is just the ordinary point, regarded as a smmoth manifold.

This yields the underlying bare representation

Γ(ρ):BGVect. \Gamma(\rho) : \mathbf{B}G \to Vect \,.

Conversely, one finds that extending such a bare representation from the point to all test spaces in CC amounts to equipping it with smooth structure.

As before, this is not restricted to connected objects: we may replace BG\mathbf{B}G here with any Lie groupoid. For instance for XX a smooth manifold and P 1(X)\mathbf{P}_1(X) is smooth path groupoid a representation

ρ:P 1(X)VectBund \rho : \mathbf{P}_1(X) \to VectBund

as a morphism in the 2-topos is a vector bundle on XX with connection on a bundle. Or if we consider non-linear representations, a representation

ρ:P 1(X)BG \rho : \mathbf{P}_1(X) \to \mathbf{B}G

is a GG-principal bundle with connection on a bundle on XX. See parallel transport for more details and references.

By just changing the site here, we can implement other geometric structures. For instance for GG an algebraic group we may think of BG\mathbf{B}G as an algebraic stack over something like the fppf-site structure C=CAlg k opC = CAlg_k^{op} on formal duals of commutative kk-algebras or similar.

In this case there is a well-known good generalization of VectBundVectBund: 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

QC:SpecAAMod, QC : Spec A \mapsto A Mod \,,

where on the right we have the groupoid of modules over AA. Over the point this is again just a kk-module hence a vector space and hence a representation

ρ:BGQC \rho : \mathbf{B}G \to QC

of an algebraic group is a representation of GG on a vector space.

But also here we may allow the represented structure to have more than one object. For instance for XX any scheme regarded as an object in Sh (2,1)(C)Sh_{(2,1)}(C) a representation of XX in the context of QCQC is a morphism

ρ:XQC, \rho : X \to QC \,,

which is equivalently a quasicoherent sheaf of modules on XX. As before, we may think of this as assigning to each point of XX one representation space, only that in a scheme XX there are no morphisms that would act on these.

But more generally for KK an algebraic stack, a representation

ρ:KQC \rho : K \to QC

assigns to each point of K 0K_0 a representation space, such that these glue together to a quasicoherent sheaf of modules, and to each morphism in KK a morphism between the corresponding representation spaces, as before.

Analogous constructions are available for more general sites, effectively we can take CC to be the opposite category of TT-algebras over a Lawvere theory for TT any algebraic theory that contains the theory of abelian groups. If for instance we take T=T = CartSp we are back to the smooth case discussed before.

Also notice that if we take the site to be the point, C=*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 (k)Ch_\bullet(k) that is presented by a model structure on chain complexes.

If again GG is a discrete group, then an (∞,1)-functor (equivalently: a “strong homotopy-functor” or “homotopy coherent functor”, see there for details)

ρ:BGCh (k) \rho : \mathbf{B}G \to Ch_\bullet(k)


  • to the single obect of BG\mathbf{B}G a chain complex V V_\bullet;

  • to a group element gGg \in G a chain map ρ(g):V V \rho(g) : V_\bullet \to V_\bullet;

  • to a pair of group elements g,gg, g' a chain homotopy

    ρ(g,g):ρ(g)ρ(g)ρ(gg)\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 ρ(g,g),ρ(g,g)\rho(g,g'), \rho(g,g'') and ρ(g,g)\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 GG as before on an ordinary vector space, only that now the action property of ρ\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 KK any ∞-groupoid or even (∞,1)-category (recall the quiver representations) and for ModMod 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

ρ:KMod \rho : K \to Mod

an \infty-representation of KK.

If we wish to consider \infty-generalizations of permutation representations we can also consider more general codomain (,1)(\infty,1)-categories here. For instance if we take ∞Grpd itself, then an \infty-permutation representation

KGrpd K \to \infty Grpd

is known as an (∞,1)-presheaf. For KK 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

GrpdTop \infty Grpd \simeq Top

so that the above is equivalently an (∞,1)-functor

KTop K \to Top

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 H\mathbf{H} there is a notion of fundamental ∞-groupoid P(X)\mathbf{P}(X) of any object XX. Representation of Π(X)\mathbf{\Pi}(X) define general abstract flat differential cohomology and local systems on XX, 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

QC:SpecAAMod QC : Spec A \mapsto A Mod

on this site, where now however AModA 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 AA.

For XX any ∞-stack then a morphism

ρ:XQC \rho : X \to QC

is a equivalently a quasicoherent ∞-stack of modules on XX, or an \infty-representation with “dg-algebraic structure”.

If one replaces XX here by its de Rham stack X dRX_{dR} then dg-algebraic \infty-representations

ρ:X dRQC \rho : X_{dR} \to QC

are D-modules on XX.

A discussion of these higher categorical structure in representation theory is in (Ben-ZviNadler).


If the codomain ModMod is not an (∞,1)-category that is just a (n,1)-category (all k-morphismss for k>nk \gt n are effectively identities) then an \infty-representation is called an nn-representation. These are representations up to homotopy where from degree nn on all homotopies are actually identities: nn-truncated representations up to homotopy.

As always in higher category theory, the cases for low nn 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=1n=1, 2-Representations P 2(X)2Vect\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 homotopy type theory

In a general abstract context of homotopy type theory we may define \infty-representations as follows.

For H\mathbf{H} an (∞,1)-topos, let GGrp(H)G \in Grp(\mathbf{H}) be a group object in H\mathbf{H}, hence an ∞-group. Then the slice (∞,1)-topos H /BG\mathbf{H}_{/\mathbf{B}G} over its delooping is the (∞,1)-category of ∞-actions of GG

Act(G)H /BG, Act(G) \simeq \mathbf{H}_{/\mathbf{B}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)Act(G).

representation theory and equivariant cohomology in terms of (∞,1)-topos theory/homotopy type theory (FSS 12 I, exmp. 4.4):

homotopy type theoryrepresentation theory
pointed connected context BG\mathbf{B}G∞-group GG
dependent type on BG\mathbf{B}GGG-∞-action/∞-representation
dependent sum along BG*\mathbf{B}G \to \astcoinvariants/homotopy quotient
context extension along BG*\mathbf{B}G \to \asttrivial representation
dependent product along BG*\mathbf{B}G \to \asthomotopy invariants/∞-group cohomology
dependent product of internal hom along BG*\mathbf{B}G \to \astequivariant cohomology
dependent sum along BGBH\mathbf{B}G \to \mathbf{B}Hinduced representation
context extension along BGBH\mathbf{B}G \to \mathbf{B}Hrestricted representation
dependent product along BGBH\mathbf{B}G \to \mathbf{B}Hcoinduced representation
spectrum object in context BG\mathbf{B}Gspectrum 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 (,1)(\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

Last revised on September 11, 2018 at 13:19:46. See the history of this page for a list of all contributions to it.