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Every infinity-functor (equivalently sh-functor, essentially equivalently A-infinity functor) from anywhere to anywhere else can be regarded as a an “action/representation up to homotopy”.
I agree. Things are probably clearer seen the other way round: a (classical) linear representation of a group $G$ is a functor from the delooping groupoid $\mathbf{B}G$ to $Vect$, so it is a very special case of an oo-functor between an oo-groupoid and an oo-category of oo-vector spaces.
Yes. I have turned something like an expanded version of my AlgTop-posting into the detailed Idea-section of an $n$Lab entry: infinity-representation.
I listed some references, but there are many more. Please help to fill them all in!
I have recently defined a ´homotopy action´ in terms of Segal´s special $\Delta$-spaces. It is a map of simplicial spaces $A_* \rightarrow B_*$ such that $B_*$ is a special $\Delta$-space and some combination of maps from that diagram are homotopy equivalences. As with Segal’s work, everything in the diagram is ’built out of products’. The simple example is the simplicial map $Bar_*(G,X) \rightarrow B_*(G)$ constructed given an action $a:G\times X \rightarrow X$ of a topological group $G$ on a space X.
I look at it as a sort of ’relative special $\Delta$-space’. Has anyone here seen a similar thing, e.g. maps of simplicial spaces into an $(\infty,1)-category$(viewed as a Segal space)?
Jim made an edit at infinity-representation: where I mentioned permutation representations with coefficients in $Set$ he mentioned that in topology one wants coefficients in $Top$.
I have added a remark that this case is discussed in detail further below and to make things clear I also added a remark that we have an equivalence of oo-categories $\infty Grpd \simeq Top$.
One can also view a homotopy action of a group $G$ on a group $M$ as given by a factor set. But as shown by Philip and I in 1982,(SLNM 962, Gummersbach) this can also be regarded as a morphism of crossed complexes $k: F^{st}(G) \to AUT(M)$ from the standard free crossed resolution of $G$ to the crossed complex which consists essentially of the crossed module $\xi:M \to Aut(M)$. This generalises in several ways. Dedecker replaced the crossed module $AUT(M)$ by any crossed module $M \to P$. Note also that $F^{st}(G)$ is the fundamental crossed complex of the skeletal filtration of the nerve of $G$. But that crossed resolution may be replaced by any other homotopy equivalent free crossed resolution, and still get the ‘same’ homotopy classes of morphism.
If $G$ is a topological group one may make the standard resolution into a topological crossed complex.
If $X$ is a space, can one construct from $X$ a topological crossed module, say $TopAUT(X)$, so as to analogously get homotopy actions of $G$?
Hi Jim,
what I can offer is this:
homotopy actions of general higher groups and higher group stacks are discussed in section 4.1 of
with more details in section 3.3.11 and section 3.3.13 of
(this material is partly reflected in the nLab entry infinity-action). Discussion of interesting examples are scattered over the document, for instance
section 3.6.13.5 discusses the homotopy action of higher pre-quantum operators on higher pre-quantum states;
section 5.4.3 discusses the homotopy action of $\mathbf{B}U(1)$ on the K-theory moduli stack.
later there is the action of $\mathbf{B}^6 U(1)$ on the moduli stack of String-principal 2-bundles, etc.
A list of fundamental classes of examples is example 3.3.172. I should maybe expand that list. If you are interested.
@Jim What I can mention is the proof contained in an early note of Eric Friedlander of the link between fibrations and homotopy crossed modules. This is essentially a part of the action of the paths on the base on the fibres. (Note I say paths on the base not the fundamental groupoid.) Friedlander attributes the proof to Deligne. This would be about 1973, but I do not have the paper to hand, so cannot be sure. About the same time, in his paper Homotopy limits and colimits , Math. Z., 134, (1973, Rainer Vogt talked about homotopy coherent actions of the loop space. This is one of the earliest EXPLICIT mentions of some notion of homotopy action that I met as before that except perhaps for Peter May, and of course, Boardman, and yourself, not many people had the machinery to talk about such things.
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