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    • CommentRowNumber1.
    • CommentAuthorTim_Porter
    • CommentTimeJan 8th 2011
    • (edited Jan 8th 2011)
    Jim has started a discussion on the ALg. Top. list and suggests it move here. The latest point was:
    On 1/8/11 6:52 AM, Urs Schreiber wrote:
    > Jim wrote about "actions/representations up to homotopy"
    >> I would welcome additions to the list
    >> both in nomenclature and references
    > I think it is a long list:
    > 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 think the word action implies an action on an object
    representation usually does but could pehaps be interpreted more laxly
    > Often one will want to demand for an oo-representation that the
    > codomain is an oo-category of oo-vector spaces. A typical choice is
    > CH, presented by the model structure on chain complexes. I believe
    > what is known as "representations up to homotopy" in the literature
    > currently are oo-functors
    > K --> CH,
    > where K is a groupoid, One could take any oo-groupoid. To make this a
    > smooth or algebraic or otherwise geometric representation up to
    > homotopy, regard K and CH as stacks on the corresponding site.

    Again I would expect K to act on something in CH
    so K --> End(C) C\in CH
    > David Ben-Zvi has a series of articles that put all this in the
    > context of representation theory.

    If you get this, could you provide more precise coordiantes.

    I'm not sure alg-top will let us contintue this discussion there.
    Should we move to the n-forum?

    There was also my:

    On 1/8/11 1:53 AM, Timothy Porter wrote:
    > Jim,
    > Happy new year.
    > One point is : what do you mean by 'actions up to homotopy',
    I was quoting from some authors, hoping to point out the ambiguity.
    Yours is unambiguous once you specify actions up to coherent homotopy of __________
    groups? dg algebras? A_\infty
    > is what I would call 'actions up to coherent homotopy'.

    Have you used that in print?

    > That, of course, then leads to a lot of structure, obstructions etc. and questions: `How coherent? and so on. This may be subsummed under your sh-actions perhaps.

    That's indeed one example.

    > Tim

    • CommentRowNumber2.
    • CommentAuthordomenico_fiorenza
    • CommentTimeJan 8th 2011
    • (edited Jan 8th 2011)

    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 GG is a functor from the delooping groupoid BG\mathbf{B}G to VectVect, so it is a very special case of an oo-functor between an oo-groupoid and an oo-category of oo-vector spaces.

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeJan 8th 2011

    Yes. I have turned something like an expanded version of my AlgTop-posting into the detailed Idea-section of an nnLab entry: infinity-representation.

    I listed some references, but there are many more. Please help to fill them all in!

    • CommentRowNumber4.
    • CommentAuthorMatan
    • CommentTimeJan 9th 2011
    • (edited Jan 9th 2011)

    I have recently defined a ´homotopy action´ in terms of Segal´s special Δ\Delta-spaces. It is a map of simplicial spaces A *B *A_* \rightarrow B_* such that B *B_* is a special Δ\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)B *(G)Bar_*(G,X) \rightarrow B_*(G) constructed given an action a:G×XXa:G\times X \rightarrow X of a topological group GG on a space X.

    I look at it as a sort of ’relative special Δ\Delta-space’. Has anyone here seen a similar thing, e.g. maps of simplicial spaces into an (,1)category(\infty,1)-category(viewed as a Segal space)?

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeJan 9th 2011

    Jim made an edit at infinity-representation: where I mentioned permutation representations with coefficients in SetSet he mentioned that in topology one wants coefficients in TopTop.

    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 GrpdTop\infty Grpd \simeq Top.

    • CommentRowNumber6.
    • CommentAuthorronnie
    • CommentTimeJan 10th 2011

    One can also view a homotopy action of a group GG on a group MM 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)AUT(M)k: F^{st}(G) \to AUT(M) from the standard free crossed resolution of GG to the crossed complex which consists essentially of the crossed module ξ:MAut(M)\xi:M \to Aut(M). This generalises in several ways. Dedecker replaced the crossed module AUT(M)AUT(M) by any crossed module MPM \to P. Note also that F st(G)F^{st}(G) is the fundamental crossed complex of the skeletal filtration of the nerve of GG. 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 GG is a topological group one may make the standard resolution into a topological crossed complex.

    If XX is a space, can one construct from XX a topological crossed module, say TopAUT(X)TopAUT(X), so as to analogously get homotopy actions of GG?

    • CommentRowNumber7.
    • CommentAuthorjim_stasheff
    • CommentTimeNov 17th 2012
    I'm now trying to write a brief history of the first level and infty notions of action up to homotopy. All contributions most welcome. Is the first occurance in the action of $\pi_1(X)$ on $\pi_1(X,A) for A \subset X?
    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeNov 17th 2012
    • (edited Nov 17th 2012)

    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

    • Thomas Nikolaus, U.S., Danny Stevenson, Principal \infty-bundles - General theory (arxiv:1207.0248)

    with more details in section 3.3.11 and section 3.3.13 of

    • U.S. Differential cohomology in a cohesive topos (pdf)

    (this material is partly reflected in the nLab entry infinity-action). Discussion of interesting examples are scattered over the document, for instance

    • section discusses the homotopy action of higher pre-quantum operators on higher pre-quantum states;

    • section 5.4.3 discusses the homotopy action of BU(1)\mathbf{B}U(1) on the K-theory moduli stack.

    • later there is the action of B 6U(1)\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.

    • CommentRowNumber9.
    • CommentAuthorTim_Porter
    • CommentTimeNov 18th 2012

    @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.

    • CommentRowNumber10.
    • CommentAuthorjim_stasheff
    • CommentTimeNov 18th 2012
    Rainer's I knew of but thanks for the prompt. Will contact Eric directly.