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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeNov 8th 2010

    I am going to polish the entry local system now.

    The following is long forgotten discussion that had been sitting in a query box there. Everybody involved should check what of that still needs further discussion and then have that discussion here on the forum.


    Urs: I am hoping that maybe David Speyer, whose expositional blog entry is linked to below, or maybe somebody else would enjoy filling in some material here.

    Bruce: Could it perhaps be “On a topological space (why do we need connected?) this is the same as a sheaf of flat sections of a finite-dimensional vector bundle equipped with flat connection;”. I guess by “flat connection” in this general topological context we would mean simply a functor from the homotopy groupoid to the category of vector spaces?

    Zoran Škoda: connected because otherwise you do not have even the same dimension of the typical stalk of teh lcoally constant sheaf. Maybe there is a fancy wording with groupoids avoiding this, but when you have a representation on a single space, you need connectedness.

    Ronnie Brown I do not have time to write more tonight but mention that there is a section of the paper

    • (with P.J.HIGGINS), “The classifying space of a crossed complex”, Math. Proc. Camb. Phil. Soc. 110 (1991) 95–120.

    on local systems, where a module over the fundamental groupoid of a space is regarded as a special case of a crossed complex. This seems convenient for the singular theories but has not been developed in a Cech setting. The homotopy classification theorem

    [X,C][ΠX *,C] [X, \mathcal{B}C] \cong [\Pi X_* ,C]

    where X *X_* is the skeletal filtration of the CW-complex XX, CC is a crossed complex, and C\mathcal{B}C is the classifying space of CC, thus includes the local coefficient version of the classical Eilenberg-Mac Lane theory.

    Tim: Quoting an exercise in Spanier (1966) on page 58:

    A local system on a space XX is a covariant functor from the fundamental groupoid of XX to some category.

    A reference is given to a paper by Steenrod: Homology with local coefficients, Annals 44 (1943) pp. 610 - 627.

    Perhaps the entry could reflect the origins of the idea. The current one seems to me to be much too restrictive. There are other applications of the idea than the ones at present indicated, although of course those are important at the moment. Reference to vector bundles is not on the horizon in Spanier!!!!.

    Local systems with other codomains than vector spaces are used in rational homotopy theory.

    Urs: I am all in favor of emphasizing that “local system” is nothing but a functor from a fundamental groupoid. That’s of course right up my alley, compare the discussion with David Ben-Zvi at the “Journal Club”. Whoever finds the time to write something along these lines here should do so (and in clude in particular the reference Ronnie Brown gives above).

    BUT at the same time it seems to me that many practitioners will by defualt think of the explicitly sheaf-theoretic notion when hearing “local syetem” which the entry currently states. We should emphasize this explicitly, something like: “while in general a local system is to be thought of as a representation of a fundamental groupoid, often the term is understood exclusively in its realization within abelian sheaf theory as follows …”


    (to be continued in next comment)

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeNov 8th 2010
    • (edited Nov 8th 2010)

    (continuation from previous comment)


    Tim: How about wording such as:

    In general a local system is defined to be … (ref. Spanier (1966) and earlier Steenrod (1943).) …. . (Perhaps a classical example would help here.)

    A particularly important case of this is when the functor takes its values in the category of vector spaces or slightly more generally abelian groups. For instance given a locally constant sheaf on a manifold then there is naturally a local system that encodes valuable information in a neat way. In this entry we will primarily discuss this latter more restrictive sense, at least to start with. Later we will look at other applications and instances of the more general case.

    (I tend to not use the word representation when functor is meant and I would discourage saying that a local system IS a locally constant sheaf of whatever, as they are very different types of stuff (I almost get the ’I spy evil’ reaction when I see that!).

    Zoran Skoda: Keep both definitions, and say when they are equivalent, sheaf theoretic and fundamental group one. Why I favour the sheaf theoretic definition as primary is its generality: it works over a site, it is related to combinatorial versions (local system of cohomological coefficients on a simplicial complex or on a poset), as well as local systems on stratified spaces: I do not know how the fundamental group is defined in those cases to yield the same notion; and as aside issue I also do not know if there is a usage of local systems on topological spaces which are not linearly connected. Besides, representations of fundamental group in f.d. vector spaces have also another name: monodromies (monodromy representations).

    Urs: it seems everybody is waiting for everybody else to make the first step. So I did it now. Please see the above changes and please feel free to improve as you see the need!

    Tim: The usual meaning of ’primary’ would be relating to time so Steenrod would have it there! Homology and cohomolgy with local coefficients is known from way back and ’local system’ I thought was short for ’local system of coefficients’. I understand what you mean about generality but would disagree on your last comment. Your objection to the version in Spanier seems to be that someone else thought of another name later, yet the sheaf theoretic definition is saying that local system is another name for a locally constant sheaf of f.d. v. spaces so …. .

    Can someone tell me how old the term ’monodromy’ is? I know that Ehresmann used it so perhaps Cartan? I digress.

    The main aim should be to have a clear description of the idea and a definition or definitions with some discussion of their interrelationships.

    Zoran Skoda: so how will you do the local system on a site ? For a general site with a terminal object it is hard to have a satisfactory notion of fundamental group (though it works for topoi – with regular epi topology assumed) while locally free sheaf still makes sense: you can use a cover of terminal object. The word monodromy is usually associated to the case of ordinary differential equations and Riemann-Hilbert problem, so I believe that it existed around 1900, though I may be wrong.


    (to be continued in next comment)

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeNov 8th 2010

    (continuation from previous comment)


    Tim: My own approach would, I think, be to rephrase things along the line of standard treatments of descent theory from a simplicial viewpoint. I have not thought about this so this may get garbled a bit. Classically you can do local systems on a triangulation of a manifold without reference to the fundamental group(oid), and again classically open covers of a manifold are linked to triangulations by the Cech nerve, (see simplicial local system. The analogue for a general topos would be a hypercovering (I suppose) so it should be feasible to adapt the definition to that setting. (This is probably either well known or wrong!) The fundamental group should be nowhere in sight. Paths are not relevant in this, and, of course, locally constant sheaves or their generalisations are just around the corner. (This is all analytic continuation but does not use paths only (generalisations of) open sets.)

    Local system was, as I said earlier, originally short for ’local system of coefficients’, I believe, i.e. for cohomology or homology.

    My main point is that a local system is not the same as a locally constant sheaf. It is more like a diagram defining such a sheaf, rather than the sheaf itself. If that terminology is used then it is sloppy terminology. This does not make it ’wrong’, just like so much maths, ’systematic abuse of terminology’, and it should only be indulged in with great care and consideration.

    (Another case which is more serious, I likewise object to U(1)-gerbes being called just gerbes. This is historically wrong, can confuse a beginning researcher, and also can have a devastating effect on the future of young researchers, when well known ’experts’ insist, for instance, that ’nice’ gerbes are all abelian,(implying that other types are uninteresting, nasty and unimportant) as that condemns workers in non-Abelian cohomology who study non-Abelian gerbes to lack of grant funding etc. (I know I have been there!!! so my vehemence is well founded.))

    Your comment on monodromy reinforced my feeling about it. I have not got Steenrod’s paper, so wonder if it shows that he was aware of the link. His fibre bundles book was still in the future … interesting historical question there.

    Zoran Skoda: I like your comment and historical remarks (by the way, the present homological algebra books like Methods…by Gelfand-Manin (p.28) distinguishes homology coefficient systems (on simplicial sets) and cohomology coefficient systems (maybe some remark within simplicial local system is due). As far as using internal nerve of a (hyper)cover in arbitrary site (I emphasise site, not topos) one can try defining fund. groupoid along such terms, that is implicit in the work of Pataraia on internal cosimplicial objects, which I never studied enough, and includes some conditions; in any case it does not look elementary to me. As far as historical info we should keep looking for it (including original usage of “monodromy”); it is instructive and shows some curious geometrical insights in old papers.


    (end of old query box content)

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeNov 8th 2010

    Went through this cluster of links, polished slightly here and there and added links back and forth:

    A locally constant sheaf / \infty-stack is also called a local system.

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeNov 8th 2010

    I was going to post the following, but strictly speaking this needs a bit more discussion of the lax pullback/Grothendieck construction point. But I am out of time for tonight, will have to continue another time. But maybe it is good if I post the following nevertheless.


    I am through with rewriting local system and locally constant infinity-stack.

    I rewrote it starting with plain \infty-topos theory and then later remark about the case of ordinary sheaves. There is (still) plenty of room for expansion on this special case, of course, but then, there is also a good list of references for standard material.

    So here is a main point that is kind of obvious but which we did not have explicitly stated anywhere before.

    Write 𝒮:=FinGrpdGrpd\mathcal{S} := Fin \infty Grpd \in \infty Grpd.

    • A locally constant infinity-stack on an object XX in an \infty-topos H\mathbf{H} is equivalently

      • a morphism ˜:XLConst𝒮\tilde \nabla : X \to LConst \mathcal{S};

      • the object in the over-topos H/X\mathbf{H}/X obtained by the (,1)(\infty,1)-Grothendieck construction, i.e. the pullback

        P LConst𝒵 X ˜ LConst𝒮 \array{ P &\to& LConst \mathcal{Z} \\ \downarrow && \downarrow \\ X &\stackrel{\tilde \nabla}{\to}& LConst \mathcal{S} }

        of the universal fibration.

    This (PX)H/X(P \to X) \in \mathbf{H}/X is the genuine “\infty-sheaf on XX” in that H/X\mathbf{H}/X is the little \infty-topos over XX.

    Now by general abstraction, cohomology with coefficients in the localy system of coefficients deserves to be nothing but

    H(X,˜):=π 0H /X(X,P ˜) H(X,\tilde \nabla) := \pi_0 \mathbf{H}_{/X} (X, P_{\tilde \nabla})

    which is to be read as nothing but the nonabelian sheaf cohomology on XX with coefficients in the locally constant \infty-sheaf P ˜XP_{\tilde \nabla} \to X.

    And indeed, if you unwind the definitions and use the universality of the comma pullback, you find that a cocycle in here is in H\mathbf{H} a diagram

    * X ˜ LConst𝒮 \array{ && * \\ & \nearrow &\Downarrow& \searrow \\ X &\underset{\tilde \nabla}{\to}& LConst \mathcal{S} }
    • CommentRowNumber6.
    • CommentAuthorMike Shulman
    • CommentTimeNov 8th 2010

    Why the restriction to finite things everywhere? Surely being finite is not an essential aspect of “local constancy,” although perhaps the traditional term “local system” implies some finite-dimensionality.

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeNov 8th 2010
    • (edited Nov 8th 2010)

    At least there needs to be some cardinality bound such that the collection of \infty-groupoids of bounded size still forms a small \infty-groupoid.

    • CommentRowNumber8.
    • CommentAuthorMike Shulman
    • CommentTimeNov 8th 2010

    I would regard that as a problem with “defining” a locally constant object to be a section of a constant stack, rather than using an “intrinsic” definition of locally constant object. But one could also always talk about sections of the large constant stack of small ∞-groupoids by going up a universe.

    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTimeNov 8th 2010

    Sure. I added a brief remark to local system.

    • CommentRowNumber10.
    • CommentAuthorDavidRoberts
    • CommentTimeNov 8th 2010

    I agree with Mike - there should be no restriction to finite things. It is a theorem (Polesello and Waschkies) that locally constant stacks are sections of the constant 2-stack (or more appropriately, given Tim’s copied comments above - it is a theorem that local systems Π 2(X)Gpd\Pi_2(X) \to Gpd are equivalent to sections of the constant 2-stack with fibre GpdGpd). I don’t think it should be the definition.

    • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeNov 8th 2010

    sections of the constant 2-stack with fibre GpdGpd

    It’s just that taken at face value there is no 2-stack with values in GpdGpd. Since GpdGpd is not an object in 2Grpd2 Grpd but in 2GRPD2 GRPD. But of course one can deal with this.

    • CommentRowNumber12.
    • CommentAuthorUrs
    • CommentTimeNov 9th 2010
    • (edited Nov 9th 2010)

    I don’t think it should be the definition.

    As you notice, in low categorical degree it is a theorem that both definitions are equivalent.

    So then when we generalize to higher categorical degree, we have the usual choice of which of several equivalent definitions of the lower dimensional one to pick. We want to pick the one with the nicest abstract properties. So that’s what I do here: since in low categorical degree a locally constant (n-1)-stack is proven to be precisely a section of a constant nn-stack, and since the notion of sections of constant nn-stacks has a very good general abstract formulation, I declare this to be the general definition.

    I think this is good and well established practice of how to proceed with generalizing concepts to higher category theory. Now, with any alternative definition of locally constant \infty-stack one still can and wants to check if or if not it is equivalent to this one.

    • CommentRowNumber13.
    • CommentAuthorMike Shulman
    • CommentTimeNov 9th 2010

    We want to pick the one with the nicest abstract properties.

    I don’t think that’s necessarily always the rule. Sometimes we should instead pick the one which adheres as closely as possible to the desired meaning. If there is a notion which disagrees with that but has better abstract properties, then maybe we should give that other one a different name. Ideally, something called “locally constant” should be defined in such a way as to make clear in what sense it is “locally constant.”

    In this case, I think both definitions turn out to be equivalent. So I think that when introducing the concept as a definition, there is no harm in using the definition with better intuition, and only then remarking that it is equivalent to another version.

    • CommentRowNumber14.
    • CommentAuthorMike Shulman
    • CommentTimeNov 10th 2010

    By the way, giving an intrinsic definition of “locally constant” rather than as “a section of a constant stack” has the abstract advantage that it doesn’t require any arbitrary cardinality bound on the size of the fibers.

    • CommentRowNumber15.
    • CommentAuthorUrs
    • CommentTimeNov 10th 2010
    • (edited Nov 10th 2010)

    Let’s see if we can make some progress.

    In the case where we do have everything we want, both notions should agree, so we can take that case as the testing case.

    So let me assume a globally and locally \infty-connected \infty-topos H\mathbf{H} and

    XLConst𝒮 X \to LConst \mathcal{S}

    a morphism. i still think the corresponding locally constant stack regarded as an object of H/X\mathbf{H}/X should be the pullback

    P LConst𝒵 X LConst𝒮 \array{ P &\to& LConst \mathcal{Z} \\ \downarrow && \downarrow \\ X &\to& LConst \mathcal{S} }

    for 𝒵𝒮\mathcal{Z} \to \mathcal{S} the universal fibration.

    So how would we recognize this as being “locally constant”? Like this: say a cover of XX “by contractible patches” is a morphism UXU \to X such that

    1. UXU \to X is an effective epimorphism

    2. Π(U)\Pi(U) is 0-truncated (a set).

    The second conditon encodes that each connected component of UU is contractible. We are to think of this is U= iU iU = \coprod_i U_i of contractible patches U iU_i.

    Then pick such a cover pp and pull back to it

    p *P P LConst𝒵 U p X LConst𝒮. \array{ p^* P &\to & P &\to& LConst \mathcal{Z} \\ \downarrow && \downarrow && \downarrow \\ U &\stackrel{p}{\to}& X &\to& LConst \mathcal{S} } \,.

    The total rectangle here is still a pullback. Now use that (ΠLConst)(\Pi \dashv LConst) and that LConstLConst is full and faithful to deduce that the adjunct

    Π(p *P) 𝒵 Π(U) 𝒮 \array{ \Pi (p^* P) &\to& \mathcal{Z} \\ \downarrow && \downarrow \\ \Pi(U) &\to& \mathcal{S} }

    is still a pullback. Since by assumption Π(U)\Pi(U) is a set, the bottom morphism picks a bunch of finite \infty-groupoids i\mathcal{F}_i – which I think we can deduce must all be equivalent – and by the property of 𝒵\mathcal{Z} is follows that

    Π(p *P)Π(U)×Π(U×LConst) \Pi(p^* P) \simeq \Pi(U) \times \mathcal{F} \simeq \Pi(U \times LConst \mathcal{F})

    (Hm, maybe I am assuming now also that Π\Pi preserves finite products.) So we conclude that

    p *PU×LConst. p^*P \simeq U \times LConst \mathcal{F} \,.

    so this says that PP looks locally like a contractible times a constant \infty-stack. Which should be the defining property of locally constant objects in H/X\mathbf{H}/X. I guess.

    • CommentRowNumber16.
    • CommentAuthorUrs
    • CommentTimeNov 10th 2010
    • (edited Nov 10th 2010)

    And now conversely, suppose that PXP \to X is an object in H/X\mathbf{H}/X such that there is an effective epimorphism UXU \to X and a pullback diagram

    U×LConst P U p X. \array{ U \times LConst \mathcal{F} &\to & P \\ \downarrow &&\downarrow \\ U &\stackrel{p}{\to}& X } \,.

    We want to conclude that then there is a morphism XLConst𝒮X \to LConst \mathcal{S} such that we have a pasting of pullbacks

    p *P P LConst𝒵 U p X LConst𝒮. \array{ p^* P &\to & P &\to& LConst \mathcal{Z} \\ \downarrow && \downarrow && \downarrow \\ U &\stackrel{p}{\to}& X &\to& LConst \mathcal{S} } \,.

    For this now use the assumption that UXU \to X is an effective epi, hence that

    U× XU× XUU× XUUX \cdots U \times_X U \times_X U \stackrel{\to}{\stackrel{\to}{\to}} U \times_X U \stackrel{\to}{\to} \to U \to X

    is a colimiting cone under the Cech nerve. This implies that the morphism ULConst𝒮U \to LConst \mathcal{S} that classifies U×LConstU \times LConst \mathcal{F} factors through XX and we get

    p *P P LConst𝒵 U p X LConst𝒮. \array{ p^* P &\to & P & & LConst \mathcal{Z} \\ \downarrow && \downarrow && \downarrow \\ U &\stackrel{p}{\to}& X &\to& LConst \mathcal{S} } \,.

    Now we need to show that the right square fills. For that use that colimits in H\mathbf{H} are universal.

    • CommentRowNumber17.
    • CommentAuthorMike Shulman
    • CommentTimeNov 11th 2010

    Yes, let’s work this out. First of all, I definitely want a definition that makes sense for any (∞,1)-topos at all, not just a locally and/or globally connected one. Part of the point of considering these things is to be use them to define the “fundamental pro-∞-groupoid” or “fundamental Galois topos” that represents the shape of an ill-behaved topos. And not wanting to restrict to locally ∞-connected toposes means that we shouldn’t expect to be able to restrict to covers by contractible objects, either; we’re forced to allow arbitrary covers.

    In a 1-topos, one definition of “locally constant object” is an object XX such that there exists a well-supported object UU (i.e. U1U\to 1 is an (effective) epi) such that U×XU× ΔIΔSU\times X \cong U \times_{\Delta I} \Delta S over UU, for some map SIS\to I in Set and UΔIU\to \Delta I in the topos \mathcal{E}, where I am writing Δ=LConst\Delta= L Const for brevity. Of course if \mathcal{E} is locally connected, then UΔIU\to \Delta I is equivalently Π 0(U)I\Pi_0(U)\to I, so we can think of this as saying we have an I-indexed family of sets S, and then to each connected component of U we assign one of those sets and take it to be the constant fiber over that connected component.

    A different, and perhaps better, way of thinking about it is to note that /ΔI I\mathcal{E}/\Delta I \simeq \mathcal{E}^I, by extensivity of \mathcal{E}, so that UΔIU\to \Delta I is equivalently an II-indexed family (U i)(U_i) of objects of \mathcal{E}. Saying that U= iU iU = \sum_i U_i is well-supported then means that the U iU_i jointly cover 11\in\mathcal{E}, and the isomorprhism U×XU× ΔIΔSU\times X \cong U \times_{\Delta I} \Delta S over UU is equivalently an isomorphism U i×XU i×ΔS iU_i\times X \cong U_i \times \Delta S_i over each U iU_i—so that XX actually becomes literally constant over each U iU_i.

    Now suppose we translate that straightforwardly into an (,1)(\infty,1)-topos, and define a locally constant object to be an XX such that there exists a well-supported object UU (i.e. U1U\to 1 is an effective epi) such that U×XU× ΔIΔSU\times X \simeq U \times_{\Delta I} \Delta S over UU, for some maps SIS\to I in Gpd\infty Gpd and UΔIU\to \Delta I in \mathcal{E}. This is a bit different from your definition, but for the above reasons I think it’s better especially in the non-locally-contractible case.

    In the special case when U=1U=1, this says that XX is the fiber of ΔSΔI\Delta S \to \Delta I along a global section of ΔI\Delta I. Clearly a special case of this is when I=Gpd κI = \infty Gpd_\kappa is the \infty-groupoid of \infty-groupoids bounded in size by some cardinal κ\kappa and SS is the universal fibration over it, so any pullback of the constant stack on the universal fibration (your definition) is easily locally constant in this sense.

    Conversely, suppose given a locally constant object with UU, SS, and II as defined above. Then there exists a cardinal κ\kappa such that SIS\to I is the pullback of the universal fibration over Gpd κ\infty Gpd_\kappa along some map IGpd κI\to \infty Gpd_\kappa. Since Δ\Delta preserves pullbacks, we may as well assume that I=Gpd κI= \infty Gpd_\kappa and SS is the universal fibration. Now, since U×XU\times X over UU is a pullback of the “global object” XX, it has descent data over the kernel of U1U\to 1, and hence (via the equivalence) so does U× ΔIΔSU \times_{\Delta I} \Delta S.

    Here I want to do the same thing you did, but I don’t understand how you get from this to a saying that the map UΔIU\to \Delta I has descent data over the kernel and hence factors through 11. That seems to require that knowing the pullback functor Hom (U,Δ(Gpd κ))Core(/U)Hom_{\mathcal{E}}(U,\Delta (\infty Gpd_\kappa)) \to Core(\mathcal{E}/U) is fully faithful, which seems plausible, but I don’t know how to prove it. Is this a known fact?

    • CommentRowNumber18.
    • CommentAuthorUrs
    • CommentTimeNov 11th 2010
    • (edited Nov 11th 2010)

    Yes, let’s work this out.

    Yes, let’s do that.

    This is a bit different from your definition,

    I think what we wrote coincides.

    There is only a slight difference in notation and setup: I wrote your \mathcal{E} as 𝒯/B\mathcal{T}/B in order to be able to connect back to a situation in 𝒯\mathcal{T} in cases where that is gros and suitably connected. (And I’ll write BB for the base object now, not to collide with your use of XX.)

    For given that Δ 𝒯/B\Delta_{\mathcal{T}/B} is given by S(Δ 𝒯S)×BS \mapsto (\Delta_{\mathcal{T}} S) \times B your formula

    U×XU× Δ IΔ S U \times X \simeq U \times_{\Delta_{\mathcal{E}} I} \Delta_{\mathcal{E}} S

    in \mathcal{E} identifies with my formula

    U× BXU× Δ 𝒯IΔ 𝒯S U \times_B X \simeq U \times_{\Delta_{\mathcal{T}} I} \Delta_{\mathcal{T}} S

    in 𝒯\mathcal{T}. Notably we may restrict to the case that B=*B = * and =𝒯\mathcal{E} = \mathcal{T}. But allowing more general over-toposes here is supposed to allow us to conclude in the end that if \mathcal{E} is the little topos of an object in a \infty-connected one, then finite locally constant objects in it are classified by maps into Δ 𝒯𝒮\Delta_{\mathcal{T}} \mathcal{S} in 𝒯\mathcal{T}.

    Now I need to think more about the other points.

    • CommentRowNumber19.
    • CommentAuthorUrs
    • CommentTimeNov 11th 2010
    • (edited Nov 11th 2010)

    which seems plausible, but I don’t know how to prove it. Is this a known fact?

    I think what we need is the statement about the classification of associated infinity-bundles. The article by Wendt referenced at that link shows this for \infty-toposes over 1-sites: if ΔFXB\Delta F \to X \to B is a fiber sequence that is locally trivial (in our sense here) then it is classified by a morphism BBAUT(ΔF)B \to \mathbf{B}AUT(\Delta F).

    For our case FF is a κ\kappa-bounded \infty-groupoid and since Δ\Delta is full and faithful (in the \infty-connected 𝒯\mathcal{T}, compare my remarks above) we have that this is a morphism BΔ(BAUT(F))B \to \Delta ( \mathbf{B} AUT(F)). But here BAUT(F)CoreGrpd κ\mathbf{B} AUT(F) \hookrightarrow Core \infty Grpd_\kappa is precisely the sub-\infty-groupoid of CoreGrpd κCore \infty Grpd_\kappa on FF, so we conclude that we have a pullback

    X Δ𝒵 B ΔCoreGrpd κ \array{ X &\to& \Delta \mathcal{Z} \\ \downarrow && \downarrow \\ B &\to& \Delta Core \infty Grpd_\kappa }

    as desired.

    There should be a general abstract version of Wendt’s argument that works in \infty-toposes over general \infty-sites.

    But I need to stop thinking about this for the moment, because I have to bring some more \infty-operad theory into place for our seminar tomorrow, and I have already spent way too much time doing other things here… But let’s get back to this a little later.

    • CommentRowNumber20.
    • CommentAuthorMike Shulman
    • CommentTimeNov 11th 2010

    There is only a slight difference in notation and setup

    I guess I misunderstood your definition of “locally constant object.” I thought you were requiring U to be contractible, and asking for a decomposition as a product U×ΔSU\times \Delta S rather than a pullback U× ΔIΔSU\times_{\Delta I} \Delta S.

    (Sorry about mixing up the meaning of XX; I wrote my comment before I had a chance to read yours.)

    if \mathcal{E} is the little topos of an object in a \infty-connected one, then finite locally constant objects in it are classified by maps into Δ 𝒯𝒮\Delta_{\mathcal{T}} \mathcal{S} in 𝒯\mathcal{T}.

    I think that should follow automatically if we just do everything for a general topos \mathcal{E}, and then at the end specialize to 𝒯/B\mathcal{T}/B, rather than needing to carry through the extra notation everywhere. For if finite locally constant objects in any \mathcal{E} are classified by global sections of Δ (CoreGpd ω)\Delta_{\mathcal{E}}(Core\infty Gpd_\omega), then those in 𝒯/B\mathcal{T}/B are classified by global sections in 𝒯/B\mathcal{T}/B of Δ 𝒯/B(CoreGpd ω)=B×Δ 𝒯(CoreGpd ω)\Delta_{\mathcal{T}/B}(Core\infty Gpd_\omega) = B \times \Delta_{\mathcal{T}}(Core\infty Gpd_\omega), which are equivalent to maps BΔ 𝒯(CoreGpd ω)B\to \Delta_{\mathcal{T}}(Core\infty Gpd_\omega) in 𝒯\mathcal{T} by adjunction.

    • CommentRowNumber21.
    • CommentAuthorUrs
    • CommentTimeNov 11th 2010

    Mike,

    sure, I was just trying to deduce and study what should be the general definition from and in the special case that we are dealing with the little topos of an object in a, say, cohesive topos, just so as to be able to identify the equivalence of two different definitions in that case.

    For the general case we don’t need all these extra assumptions. But we will then also not have the alternative slick definition.

    More later…

    • CommentRowNumber22.
    • CommentAuthorMike Shulman
    • CommentTimeNov 11th 2010

    But we will then also not have the alternative slick definition.

    What do you mean? Why not?

    • CommentRowNumber23.
    • CommentAuthorMike Shulman
    • CommentTimeNov 12th 2010

    To start with something very basic that I understand, and which is probably obvious to you, I think it’s easy to see that whenever \mathcal{E} is a presheaf (∞,1)-topos, then the pullback functor Hom(1,Δ(CoreGpd))Core()Hom(1,\Delta(Core \infty Gpd)) \to Core(\mathcal{E}) is fully faithful. (Since slices of a presheaf topos are again presheaf topoi, this also applies to the situation relevant above where 1 is replaced by UU.)

    For if =Psh(C)\mathcal{E} = Psh(C) for some small (∞,1)-category CC, then \mathcal{E} is locally ∞-connected, and Π(1)\Pi(1) is the ∞-groupoid reflection of C opC^{op}, call it gpd(C op)gpd(C^{op}). Thus we have

    Hom (1,Δ(CoreGpd))Hom Gpd(gpd(C op),CoreGpd)Hom (,1)Cat(C op,CoreGpd) Hom_{\mathcal{E}}(1,\Delta(Core\infty Gpd)) \simeq Hom_{\infty Gpd}(gpd(C^{op}),Core\infty Gpd) \simeq Hom_{(\infty,1)Cat}(C^{op}, Core \infty Gpd)

    using the adjunctions ΠΔ\Pi \dashv \Delta and that gpdgpd is a reflection of (∞,1)-categories into ∞-groupoids. But this latter ∞-groupoid is the same as (the core of) the full subcategory of Hom (,1)Cat(C op,Gpd)Hom_{(\infty,1)Cat}(C^{op},\infty Gpd) spanned by those presheaves C opGpdC^{op}\to \infty Gpd which take all maps in CC to equivalences. Since =Psh(C)\mathcal{E} = Psh(C) is the category of all presheaves C opGpdC^{op}\to \infty Gpd, this is clearly the core of a full subcategory of it.

    Of course we also notice here the nice characterization of locally constant objects in a presheaf topos: they are the presheaves taking values in the core.

    Now can we somehow transport this along a left exact reflection? I have no idea how to do that…

    • CommentRowNumber24.
    • CommentAuthorUrs
    • CommentTimeNov 12th 2010

    Hm, not sure. Let’s see, with this argument we’d want to use that the pullback that we are computing is the sheafification of the pullback of presheaves and decompose the functor in question as

    Hom Sh(C)(1,ΔCoreGrpd)=Hom PSh(C)(1,ΔCoreGrpd)PSh(C)Sh(C) Hom_{Sh(C)}(1, \Delta Core \infty Grpd) = Hom_{PSh(C)}(1, \Delta Core \infty Grpd) \to PSh(C) \to Sh(C)

    You argued that up to the last morphism this is full and faithful. So the remaining question would be if sheafification is faithful on locally constant presheaves.

    • CommentRowNumber25.
    • CommentAuthorUrs
    • CommentTimeNov 12th 2010
    • (edited Nov 12th 2010)

    I am inclined to think we should go about it instead like this:

    since or when we are interested only in the core of the \infty-category of locally constant \infty-sheaves anyway, we may just as well regard them in terms of the principal ΔAUT(F)\Delta AUT(F)-principal bundles that they are associated to.

    And then we can use the discussion at principal infinity-bundle to conclude that they are classified by maps into BΔAUT(F)\mathbf{B} \Delta AUT(F). And since Δ\Delta preserves looping this is ΔBAUT(F)\Delta \mathbf{B}AUT(F) and this in turn is the image under Δ\Delta of the full subcategory of CoreGrpdCore \infty Grpd on the given FF. So that finally we find this way that AUT(F)AUT(F)-principal \infty-bundles for arbitrary FF are represented by ΔCoreGrpd\Delta Core \infty Grpd.

    Notice that this does reproduce our previous notion of total spaces of these beasts:

    if

    P * X ΔBG \array{ P &\to& * \\ \downarrow && \downarrow \\ X &\to& \Delta \mathbf{B}G }

    is a ΔG\Delta G-principal \infty-bundle and ρ:BGGrpd\rho : \mathbf{B}G \to \infty Grpd an \infty-permutation representation, then the corresponding associated \infty-bundle is the pullback of the universal associated \infty-bundle which is

    P Δρ(*)//G X ΔBG \array{ P &\to& \Delta \rho(*)//G \\ \downarrow && \downarrow \\ X &\to& \Delta \mathbf{B}G }

    Here the action \infty-ggroupoid ρ(*)//G\rho(*)//G is the colimit lim ρ\cdots \simeq \lim_\to \rho and colimits with values in \infty-groupoids are indeed computed by the pullback of the universal fibration, so that we have a pasting diagram of pullbacks

    P Δρ(*)//G Δ𝒵 X ΔBAUT(F) CoreGrpd \array{ P &\to& \Delta \rho(*)//G &\to& \Delta \mathcal{Z} \\ \downarrow && \downarrow && \downarrow \\ X &\to& \Delta \mathbf{B}AUT(F) &\hookrightarrow& Core \infty Grpd }

    Gee, I really have to run now. I’ll be late for the seminar after all…

    • CommentRowNumber26.
    • CommentAuthorMike Shulman
    • CommentTimeNov 12th 2010

    since or when we are interested only in the core of the ∞-category of locally constant ∞-sheaves anyway

    I think we’re interested in more than that. Specifically, by analogy with 1-topos theory, the full (∞,1)-category of the locally constant objects is supposed to be the “Galois topos” of the topos we started with, which we want to identify with the topos of actions of its “fundamental pro-∞-groupoid.”

    Moreover, you seem sort of to be retreating again to the “define things by their classifying maps” approach that I want to get away from. (-: How is this related to the “locally a family of constant objects” definition?

    • CommentRowNumber27.
    • CommentAuthorMike Shulman
    • CommentTimeNov 12th 2010

    Hom Sh(C)(1,ΔCoreGrpd)=Hom PSh(C)(1,ΔCoreGrpd)Hom_{Sh(C)}(1, \Delta Core \infty Grpd) = Hom_{PSh(C)}(1, \Delta Core \infty Grpd)

    Wait… those two Δ\Deltas are different! One of them is the constant presheaf, the other is the constant sheaf.

    • CommentRowNumber28.
    • CommentAuthorUrs
    • CommentTimeNov 12th 2010

    Concerning Δ\Deltas:

    Wait…

    Er, silly me.

    Concerning retreats:

    Moreover, you seem sort of to be retreating again to the “define things by their classifying maps” approach

    I wouldn’t think I am. I think instead I am making use of the fact that for principal \infty-bundles I have the equivalence between their total-space definition and their cocycles. At least that’s the claim here. There is a definition “principal GG-action” which talks about total objects PP and actions by group objects GG.

    I am proposing to make use of the fact that every locally constant \infty-sheaf XX with typical fiber FF is (or should be) a ΔAUT(F)\Delta AUT(F)-principal \infty-bundle/torsor.

    How is this related to the “locally a family of constant objects” definition?

    To the extent that a principal \infty-bundle is locally trivial, a ΔAUT(F)\Delta AUT(F)-principal \infty-bundle is given locally trivial, with transition functions in ΔAUT(F) 0\Delta AUT(F)_0, and so on.

    The corresponding (ρ:BAUT(F)CoreGrpd)(\rho : \mathbf{B}AUT(F) \to Core \infty Grpd)-associated bundle hece is locally constant on FF, and so forth.

    Specifically, by analogy with 1-topos theory, the full (∞,1)-category of the locally constant objects is supposed to be the “Galois topos

    Okay, that’s easy enough to get once we have the core of it: just take the full sub-infy\infy-category of the ambient topos on the objects of the core.

    • CommentRowNumber29.
    • CommentAuthorMike Shulman
    • CommentTimeNov 12th 2010

    I think what I’m worried about is that (1) in general, a locally constant object might not have a “typical fiber,” and (2) even if it does, I only see why it would be an AUT(ΔF)AUT(\Delta F)-bundle, not a Δ(AUT(F))\Delta(AUT(F))-bundle. And identifying AUT(ΔF)AUT(\Delta F) with Δ(AUT(F))\Delta(AUT(F)) seems like the same sort of problem as showing that Hom(1,Δ(CoreGpd))Core()Hom(1,\Delta(Core \infty Gpd)) \to Core(\mathcal{E}) is fully faithful.

    • CommentRowNumber30.
    • CommentAuthorUrs
    • CommentTimeNov 12th 2010
    • (edited Nov 12th 2010)

    I only see why it would be an AUT(ΔF)AUT(\Delta F)-bundle, not a Δ(AUT(F))\Delta(AUT(F))-bundle.

    But this is our design criterion for locally constant \infty-sheaves: we want them to be classified by morphisms into ΔCoreGrpd κ\Delta Core \infty Grpd_\kappa. That is Δ [F]BAUT(F)\cdots \simeq \Delta \coprod_{[F]} \mathbf{B}AUT(F).

    • CommentRowNumber31.
    • CommentAuthorMike Shulman
    • CommentTimeNov 12th 2010

    My intuitive design criterion for locally constant sheaves would be that they are locally constant, i.e. there is a cover over which they pull back to a (family of) constant objects, as in #17. This is what I’ve been claiming since the beginning: the definition of “locally constant object” should be one which is locally, space, constant. If they are classified by morphisms into ΔCoreGpd\Delta Core \infty Gpd then I would want to regard that as a nice characterization theorem.

    On the other hand, if this characterization turns out to be false, i.e. the two notions turn out to be different, then I’ll be willing to listen to arguments about which of them is the better ∞-categorical replacement for locally constant objects in a 1-topos. But we aren’t going to figure out whether the characterization is true if we take it as the definition of “locally constant.” Maybe it would clarify things if I phrase the questions in this way:

    • Given a well-supported U, if X is such that we have an equivalence U×XU× ΔIΔSU\times X \simeq U\times_{\Delta I} \Delta S, then we call say that X is U-split.

    • Say that an object is globally classified if it is the pullback of the constant sheaf on the universal fibration along a global section 1ΔCoreGpd1\to \Delta Core \infty Gpd.

    Question 1: is it true that X is globally classified if and only if it is U-split for some well-supported U?

    Question 2: If not, which of the two is the better definition of “locally constant object”?

    • CommentRowNumber32.
    • CommentAuthorUrs
    • CommentTimeNov 13th 2010
    • (edited Nov 13th 2010)

    Okay, good that you are trying to formalize the discussion. I’ll follow up on that by giving some formal statements. Then after that I reply to your questions.

    Fix some \infty-topos \mathcal{E}. Let FGrpd κF \in \infty Grpd_\kappa.

    Claim Principal GG-torsors are represented by BG\mathbf{B}G in that GTor (*,BG)G Tor_{\mathcal{E}} \simeq \mathcal{E}(*, \mathbf{B}G).

    Definition

    1. A principal GG-torsor PP is locally trivial if there exists an effective epi U*U \to * such that U×PU×GU \times P \simeq U \times G.

    2. Write F//AUF(F):=lim (BAUT(F)Grpd κ)F//AUF(F) := \lim_{\to} ( \mathbf{B}AUT(F) \hookrightarrow \infty Grpd_\kappa).

    3. The ΔF\Delta F-bundle P× ΔAUT(F)ΔFP \times_{\Delta AUT(F)} \Delta F associated to a ΔAUT(F)\Delta AUT(F)-torsor PP is *× ΔBAUT(F)Δ(F//AUT(F))* \times_{\Delta \mathbf{B}AUT(F)} \Delta (F// AUT(F)).

    Proposition If PΔAUT(F)TorP \in \Delta AUT(F) Tor is locally trivial with trivializing cover UU then the associated bundle is a locally trivial ΔF\Delta F-bundle in that

    U×(P× ΔAUT(F)ΔF)U×ΔF. U \times (P \times_{\Delta AUT(F)} \Delta F) \simeq U \times \Delta F \,.

    Proof : Consider the diagram

    U×ΔF P× ΔAUT(F)ΔF//AUT(F) Δ(F//AUF(F)) Δ𝒵 κ U * BΔAUT(F) ΔCoreGrpd κ, \array{ U \times \Delta F &\to& P \times_{\Delta AUT(F)} \Delta F//AUT(F) &\to& \Delta (F//AUF(F)) &\to& \Delta \mathcal{Z}_\kappa \\ \downarrow && \downarrow && \downarrow && \downarrow \\ U &\to& * &\to& \mathbf{B}\Delta AUT(F) &\to& \Delta Core \infty Grpd_\kappa } \,,

    where on the far right we have the universel fibration. The diagram on the right is a pullback by the characterization of colimits of \infty-groupoids as pullbacks of the universal fibration and the fact that Δ\Delta preserves finite limits (in particulal looping and delooping). The middle square is a pullback by definition of the associated bundle. Now let the leftmost diagram be a pullback to get its restriction to the cover in the top left. By the pasting law this is U× ΔGrpd κΔ𝒵U \times_{\Delta \infty Grpd_\kappa} \Delta \mathcal{Z}. By the assumption that on UU the torsor PP and hence its cocycle *BΔAUT(F)* \to \mathbf{B} \Delta AUT(F) are trivial and again using that Δ\Delta preserves finite limits this is

    U×ΔF ΔF Δ𝒵 U Δ* ΔCoreGrpd κ, \array{ U \times \Delta F &\to& \Delta F &\to & \Delta \mathcal{Z} \\ \downarrow && \downarrow && \downarrow \\ U &\to& \Delta * &\stackrel{}{\to}& \Delta Core \infty Grpd_\kappa } \,,

    QED.

    Remark If ** has several direct summands it is straightforward to generalize this discussion to one where FF may be different on each summand.

    Propososal Call an object XX locally constant if it is equivalent to a ΔF\Delta F-bundle associated to an ΔAUT(F)\Delta AUT(F)-torsor.

    Remark In the case that all AUT(F)AUT(F)-torsors are locally trivial, this means that every locally constant object locally looks like U×ΔFU \times \Delta F. But the fact that it is associated to a ΔAUT(F)\Delta AUT(F)-torsor also implies that there is descent data on this local data, and this descent data is itself “locally constant” in that it takes values in ΔAUT(F)\Delta AUT(F).

    Replies to your questions

    1. I don’t see the “if and only if”. I see an implication as above. It is not even clear to me that in full generality we have a right to expext the existence of sufficiently nice UUs at all.

    2. The associtated-bundle definition makes sense without any assumption on existence of good UUs. But if good UUs exist, it reproduces objects that look locally like U×ΔFU \times \Delta F plus locally constant descent data. So that seems to be good to me.

    • CommentRowNumber33.
    • CommentAuthorMike Shulman
    • CommentTimeNov 13th 2010

    I don’t see the “if and only if”. I see an implication as above.

    That’s why it was a question, not a statement. (-: I don’t quite see where niceness of U comes in, though.

    The associtated-bundle definition makes sense without any assumption on existence of good Us.

    So does the U-split definition, as a definition. Nothing in the definition requires U to be “good.” And if the reason the associated-bundle definition is good is that it reproduces the other definition when good Us exist, that sounds to me like an argument that the other definition is the one we actually want in general.

    The reason I numbered the questions is that I think we need to answer Question 1 before starting on Question 2. And I mean “answer” in the sense of either prove the iff, or produce a counterexample—and ideally characterize the cases in which the iff holds, and how you get counterexamples.

    • CommentRowNumber34.
    • CommentAuthorDavidRoberts
    • CommentTimeNov 14th 2010

    Just a quick thought:

    keep in mind the example of GG-bundles on non-paracompact spaces. If they are not trivialised by a numerable cover, then they can’t be classified by a map to the usual classifying spaces, because the universal bundles thereon are trivialised over a numerable cover. More generally, if one has a universal bundle of some sort, which is UU-split for a smaller class of covers than generic objects admit (cf (numerable cover) \subset (open covers) ), then they may be locally split, but not globally classified. I don’t know if this example can be worked around, or if it applies to other cases, but I thought I throw it in there.

    • CommentRowNumber35.
    • CommentAuthorMike Shulman
    • CommentTimeNov 14th 2010

    @David: interesting point! My first inclination is that that has more to do with the difference between the “usual” classifying spaces and the corresponding classifying topoi. At least in the 1-dimensional case when G is discrete, I think the classifying topos classifies all bundles, not just numerable ones. But perhaps you are suggesting that in the ∞-case, even the universal bundle over the classifying topos might be trivialized over a cover with some special property that doesn’t pull back?

    • CommentRowNumber36.
    • CommentAuthorDavidRoberts
    • CommentTimeNov 14th 2010

    No, I was wondering that perhaps the universal bundle is trivialised over a cover with special properties full stop. Say we had a universal family that was Zariski trivial - it wouldn’t classify families that are only etale-trivial because the Zariski trivialisation should pull back along the classifying map. The situation is perhaps a bit artificial, I admit…

    It is perhaps purposely looking at the wrong problem for the classifying object. I know that every topos is the classifying space for a localic groupoid, so perhaps if we have a non-sober topological group with rather trivial soberification its classifying topos will have the same inability to classify the naive objects we might at first consider. We can consider its classifying space which will classify bundles in the usual sense. I’m not sure if isomorphism classes of these could be different to universal classes of bundles with structure group its soberification.

    Actually the classifying topos classifies either numerable bundles or all bundles, depending on the site one uses to define the category of sheaves that is the classifying topos (the site of all open covers or numerable open covers).

    • CommentRowNumber37.
    • CommentAuthorDavidRoberts
    • CommentTimeNov 14th 2010
    • (edited Nov 14th 2010)

    -deleted (duplicate post)

    • CommentRowNumber38.
    • CommentAuthorMike Shulman
    • CommentTimeNov 14th 2010

    Actually the classifying topos classifies either numerable bundles or all bundles, depending on the site one uses to define the category of sheaves that is the classifying topos (the site of all open covers or numerable open covers).

    I don’t understand. When G is discrete, what I call the “classifying topos” of G is the category of G-sets. Where do open covers come in?

    • CommentRowNumber39.
    • CommentAuthorMike Shulman
    • CommentTimeNov 14th 2010

    Say we had a universal family that was Zariski trivial - it wouldn’t classify families that are only etale-trivial because the Zariski trivialisation should pull back along the classifying map.

    What would it mean for a bundle over the classifying topos to be “Zariski trivial”? A general topos doesn’t have an etale or Zariski topology, it just is. I must be misunderstanding what you mean somehow.

    • CommentRowNumber40.
    • CommentAuthorDavidRoberts
    • CommentTimeNov 14th 2010

    I wasn’t thinking of a classifying topos - just a boring old ’classifying scheme’, which is probably a stupid, non-existent example. I may be barking up the wrong tree, you know…

    • CommentRowNumber41.
    • CommentAuthorDavidRoberts
    • CommentTimeNov 14th 2010

    @Mike #38 - I was thinking of a topological group for some reason, and something more complicated - isn’t there are classifying topos by taking the levelwise sheaves on the nerve and taking some sort of homotopy colimit of the corresponding simplicial topos?

    More sensibly, just remind me, the classifying map of a G-bundle (G discrete) on XX is a map Sh(X)GsetSh(X) \to G-set (geometric morphism?) If that is the case the Gr. topology one uses on XX determines what sort of cover the bundle is trivialised by.

    • CommentRowNumber42.
    • CommentAuthorDavidRoberts
    • CommentTimeNov 14th 2010

    I think it is probably the case that the classifying topos gets around all my arguments. I was working on the assumption (perhaps mistakenly) that we could think about all this internal to an (oo,1)-topos.

    • CommentRowNumber43.
    • CommentAuthorMike Shulman
    • CommentTimeNov 14th 2010

    Yes, you can take levelwise sheaves on the nerve of a topological group and then take a hocolim / codescent object. If you do that as a 1-topos, then you get the category Cont(G) of continuous G-sets (i.e. discrete topological spaces with a continuous G-action), which doesn’t classify G-bundles in the sense that people usually mean for a topological group. For instance, if G is connected, then a continuous G-set necessarily has trivial action, so Cont(G)=Set. I suspect, however, that if you take the hocolim as an (∞,1)-topos, then (at least when G is nice, such as locally contractible) you’ll get the presheaf (∞,1)-topos on BΠ (G)B \Pi_\infty(G)—the delooping of the ∞-group Π (G)\Pi_\infty(G)—which should classify G-torsors in the sense that one usually wants.

    I wouldn’t think that you’d want to change that by replacing sheaves on G by sheaves on its numerable opens; isn’t the point about numerable covers more about the space X over which the bundle lives?

    If that is the case the Gr. topology one uses on X determines what sort of cover the bundle is trivialised by.

    Yes, of course, the notion of “G-bundle on X” depends on the topology of X, whether X is a space or more generally a topos. And a given spacelike datum (such as a topological space, scheme, etc.) can give rise to more than one topos, and hence more than one notion of bundle. But I think the classifying topos of G should classify all bundles in the sense that once you’ve defined a topos, and hence a notion of bundle, then the classifying topos classifies those.

    I was working on the assumption (perhaps mistakenly) that we could think about all this internal to an (oo,1)-topos.

    I guess that’s not an unreasonable assumption, since up until now we’ve just been talking about a notion of “locally constant object” in a particular topos. However, if G is an ∞-groupoid, then its presheaf (∞,1)-topos [G,Gpd][G,\infty Gpd] has the nice property that for any other topos E, geometric morphisms E[G,Gpd]E\to [G,\infty Gpd] are equivalent to global sections of the constant stack ΔG\Delta G in E. So that’s why I expect ΔG\Delta G to also classify all G-bundles (defined in the appropriate way relative to the topology we used to construct the topos E itself).

    • CommentRowNumber44.
    • CommentAuthorDavidRoberts
    • CommentTimeNov 14th 2010

    isn’t the point about numerable covers more about the space X over which the bundle lives?

    yes. Argh - I was terribly confused.

    I think I’ll bow out of this discussion - I’m a bit out of my depth. :)

    • CommentRowNumber45.
    • CommentAuthorMarc Hoyois
    • CommentTimeMar 4th 2012

    I’d like to revive this old discussion, since I was just thinking about this the other day. Like Mike I am really interested in topoi that are not locally \infty-connected, specifically the étale topos of a scheme.

    A local system on a topos TT, as defined on the nLab, is an element of Sh(T)(coreGrpd κ)Sh(T)(core\infty Grpd_\kappa), where Sh(T)Sh(T) is the shape of TT. If (X i)(X_i) is a cofiltered system corepresenting Sh(T)Sh(T), then any local system on TT is represented by a local system on some X iX_i (i.e. an object of Grpd/X i\infty Grpd/X_i).

    Mike, in post #17 you propose a definition of locally constant on which I’m not quite sold. One thing about your definition that bothers me is that given a local system, it is trivially a locally constant sheaf with U=1U=1, so there’s no need for any cover at all. I think we can all agree that the following is the most naive definition possible: XX is locally constant if there exists an effective epimorphism αU α1\coprod_\alpha U_\alpha\to 1 such that XX becomes constant over each U αU_\alpha. This is the definition in Appendix A to Higher Algebra (although Jacob Lurie says that this is “well-behaved” only for locally \infty-connected topoi).

    We can use descent to show that any locally constant sheaf in the naive sense is a local system. When the topos is locally \infty-connected, the converse is the content of Theorem A.1.15 in Higher Algebra. In this case the effective epi αU α1\coprod_\alpha U_\alpha\to 1 that will work comes from a contractible cover of Π(1)\Pi(1) (= the shape). I’m hoping that this argument can be adapted to the non-locally \infty-connected case (but of course the cover αU α\coprod_\alpha U_\alpha will now depend on the local system).

    Here’s how it would go, very roughly. Let X:IGrpdX:I\to \infty Grpd be a cofiltered system corepresenting the shape of TT. Morally IGrpdI\to \infty Grpd is the pullback of Grpd *Grpd\infty Grpd_\ast\to\infty Grpd along the shape (really II is a small final subcategory of what you get this way). Start with a local system LL on TT. This is given by a local system on X iX_i for some iIi\in I. Choose a contractible cover αK αX i\coprod_\alpha K_\alpha\to X_i. Going around the pullback square defining X:IGrpdX:I\to \infty Grpd gives us a canonical point of Sh(T)(X i)Sh(T)(X_i), i.e., a morphism 1LConst(X i)1\to LConst(X_i) in TT. The local system LL should then be the image of the local system over X iX_i under the composition

    f:Grpd/X iT/LConst(X i)Tf\colon \infty Grpd/X_i\to T/LConst(X_i)\to T,

    where the last functor is pullback along 1LConst(X i)1\to LConst(X_i). Set U α=f(K α)U_\alpha=f(K_\alpha). Then αU α1\coprod_\alpha U_\alpha\to 1 is a cover of 11. The functor ff followed by pullback to U αU_\alpha factors through Grpd/K αGrpd\infty Grpd/K_\alpha\simeq \infty Grpd, so L×U αL\times U_\alpha is constant, as desired. Sorry this is still very sketchy.

    Thoughts?

    • CommentRowNumber46.
    • CommentAuthorMike Shulman
    • CommentTimeMar 4th 2012

    Hi Marc,

    I haven’t read your comment carefully yet, but I should point out that the discussion in this thread continued for a while on the blog. Is your question answered there?

    • CommentRowNumber47.
    • CommentAuthorUrs
    • CommentTimeMar 4th 2012
    • (edited Mar 4th 2012)

    Hm, somehow I had missed that appendix A.1 “Locally constant sheaves” of Higher Algebra until now. I must have been being blind (likely; or was this added more recently?)

    I have added pointers to this now from locally n-connected (n+1,1)-topos and elsewhere.

    One comment (admittedly somewhat orthogonal to Marc’s comment):

    in the case that we have an ambient "gros" locally \infty-connected (,1)(\infty,1)-topos (= of locally constant shape) H\mathbf{H} given, all its objects XHX \in \mathbf{H} are also locally \infty-connected / of locally constant shape. By chasing through the definitions and theorems one finds that Jacob Lurie’s definition of locally constant objects in H/X\mathbf{H}/X is equivalent to that of locally constant \infty-stacks on XX as given in section 2.3.14 here.

    • CommentRowNumber48.
    • CommentAuthorMarc Hoyois
    • CommentTimeMar 4th 2012

    Urs, I’ve only looked at the appendix to Higher Algebra for the first time this week, so it may well be new for all I know. I think Higher Algebra was updated quite recently. I agree that the results there prove the equivalence of the two definitions in the locally \infty-connected case.

    Mike, I see that you’ve provided a counter-example (S 1× S^1\times \mathbb{N}_\infty) to the implication locally constant \Rightarrow local system? Is this still valid? I thought this implication would be the easier one, so I didn’t check it carefully. Just to be clear, so far I’m only trying to check that an object of H\mathbf H is locally constant iff it is the pullback along a morphism 1LConst(coreGrpd κ)1\to LConst(core\infty Grpd_\kappa) (for some cardinal κ\kappa). So I haven’t considered what the categories of such objects should be, which seemed to be an object of controversy on the blog (?). I think it’s true in general that local system \Rightarrow locally constant, and as Urs said, the whole equivalence is certainly true if H\mathbf H is locally \infty-connected.

    OK, so let XHX\in\mathbf H be locally constant, and let’s try to classsify XX by a map 1LConst(Grpd)1\to LConst(\infty Grpd) (let me drop the core and the cardinality bound from the notation). Choose a diagram {U α}\{U_\alpha\} with colimit 11 (a Cech nerve) such that X×U αX\times U_\alpha is constant. So X×U αLConst H/U αV αX\times U_\alpha\simeq LConst_{\mathbf H/U_\alpha} V_\alpha for some \infty-groupoid V αV_\alpha. We have cartesian squares

    V α Grpd * * Grpd \array{ V_\alpha & \to & \infty Grpd_\ast \\ \downarrow & & \downarrow \\ \ast & \to & \infty Grpd \\ }

    Now we apply the exact functor LConst H/U α:GrpdH/U αLConst_{\mathbf H/U_\alpha}:\infty Grpd\to \mathbf H/U_\alpha, which is the composition p α *LConstp_\alpha^\ast\circ LConst, where p α *:HH/U αp_\alpha^\ast:\mathbf H\to\mathbf H/U_\alpha is the pullback functor. We get cartesian squares

    p α *(X)=X×U α p α *LConst(Grpd *) p α *(*)=U α p α *LConst(Grpd) \array{ p_\alpha^\ast(X)=X\times U_\alpha & \to & p_\alpha^\ast LConst(\infty Grpd_\ast) \\ \downarrow & & \downarrow \\ p_\alpha^\ast(\ast)=U_\alpha & \to & p_\alpha^\ast LConst(\infty Grpd) \\ }

    All that remains to check is that the bottom map in these cartesian squares is natural for varying α\alpha (and it seems to me that this is true for formal reasons), and then in the colimit we will obtain XX as a pullback of a map 1LConst(Grpd)1\to LConst(\infty Grpd) as desired. Does any part of this “proof” sound particularly fishy?

    • CommentRowNumber49.
    • CommentAuthorMike Shulman
    • CommentTimeMar 5th 2012

    Mike, I see that you’ve provided a counter-example (S 1× S^1\times \mathbb{N}_\infty) to the implication locally constant \Rightarrow local system? Is this still valid?

    I still believe it. (Although I think I now prefer to use the name “locally constant” for what you are calling a “local system”.)

    The part of your “proof” that I think is fishy is the part that you think is true for “formal reasons”. It’s true that there is a canonical isomorphism over U α×U βU_\alpha \times U_\beta between the pullbacks of XX via U αU_\alpha and via U βU_\beta, hence an isomorphism between LConst H/U α×U βV αLConst_{\mathbf{H}/U_\alpha\times U_\beta} V_\alpha and LConst H/U α×U βV βLConst_{\mathbf{H}/U_\alpha\times U_\beta} V_\beta. The question is whether this isomorphism induces an isomorphism between the corresponding maps into p α,β *LConst(Gpd)p_{\alpha,\beta}^* LConst(\infty Gpd), which I believe is what fails in the non-locally-connected case.

    • CommentRowNumber50.
    • CommentAuthorMarc Hoyois
    • CommentTimeMar 5th 2012

    Ah great, I’ll try to figure out how it goes in your counter-example. The only example I have of a non-locally contractible topos in which this implication does work is the case of discrete sheaves in the étale \infty-topos of a simplicial scheme: any locally constant sheaf of sets (in the naive sense) induces a local system on the étale homotopy type (see Friedlander’s book Etale Homotopy of Simplicial Schemes). But this topos is locally 11-connected, so I guess that’s why it works for discrete sheaves.

    • CommentRowNumber51.
    • CommentAuthorMarc Hoyois
    • CommentTimeMar 5th 2012

    I think I understand. Let me know if I’ve gotten this right. So a locally constant sheaf should really come with the data of an isomorphism ϕ αβ\phi_{\alpha\beta} between V αV_\alpha and V βV_\beta, and then isomorphisms ϕ αβγ\phi_{\alpha\beta\gamma} between ϕ αβϕ βγ\phi_{\alpha\beta}\phi_{\beta\gamma} and ϕ αγ\phi_{\alpha\gamma}, and so on. And if the topos is locally contractible then such data is essentially unique on a contractible cover, so it’s not needed. More generally, no additional data is needed for nn-truncated sheaves on a locally nn-connected topos.

    • CommentRowNumber52.
    • CommentAuthorDavidRoberts
    • CommentTimeMar 5th 2012

    And if the topos is locally contractible then such data is essentially unique on a contractible cover, so it’s not needed.

    I’m not sure this is true. Even if we have a good cover, then the transition functions for a local system of discrete abelian groups are not automatic, and we still need the cocycle equation to hold - this isn’t automatic. If we have a local system of (connected, say) chain complexes, then we still need cocycle equations to hold ’all the way up’.

    • CommentRowNumber53.
    • CommentAuthorMike Shulman
    • CommentTimeMar 6th 2012

    @David, I think Marc is thinking of a locally constant sheaf as being given as a sheaf already, which implies that it has transition functions with cocycle equations etc. The point is that in addition to giving this sheaf and isomorphisms between X×U αX\times U_\alpha and the constant sheaf at V αV_\alpha over U αU_\alpha (making it “locally constant” in the traditional sense), which implies that the constant sheaves at V αV_\alpha and V βV_\beta over U αU βU_\alpha\cap U_\beta are isomorphic, we need to give a lifting of that isomorphism to a section over U αU βU_\alpha\cap U_\beta of the constant sheaf at Iso(V α,V β)Iso(V_\alpha,V_\beta). And so on.

    • CommentRowNumber54.
    • CommentAuthorMarc Hoyois
    • CommentTimeMar 6th 2012
    • (edited Mar 6th 2012)

    [Cross-posted with Mike]

    Right, I got it backwards. If you fix equivalences LConst U αV αX×U αLConst_{U_\alpha}V_\alpha\simeq X\times U_\alpha, then you already have an equivalence LConst U α×U βV αLConst U α×U βV βLConst_{U_\alpha\times U_\beta}V_\alpha\simeq LConst_{U_\alpha\times U_\beta} V_\beta, and the question is, does it come from an equivalence V αV βV_\alpha\simeq V_\beta. If the topos is locally contractible (or nn-connected and the VV’s are nn-truncated), there is an essentially unique such equivalence.

    So is there any characterization of those locally constant sheaves (in the naive sense) that are classified by a map 1LConst(CoreGrpd κ)1\to LConst(Core\infty Grpd_\kappa) that doesn’t stray too far from the naive definition? One could simply require the existence of those ϕ αβ\phi_{\alpha\beta}, ϕ αβγ\phi_{\alpha\beta\gamma}, and so on, so as to make the descent argument work, but that’s not very practical.

    • CommentRowNumber55.
    • CommentAuthorMike Shulman
    • CommentTimeMar 6th 2012

    I can’t think of any other way to state it.

    (By the way, an equivalence V αV βV_\alpha \simeq V_\beta is not quite the same as a section over U αU βU_\alpha\cap U_\beta of the constant sheaf at Iso(V α,V β)Iso(V_\alpha,V_\beta).)

    • CommentRowNumber56.
    • CommentAuthorMarc Hoyois
    • CommentTimeMar 20th 2012
    • (edited Mar 21st 2012)

    One property of local systems that locally \infty-connected topoi have and that would be pleasant to have in general is that they themselves form an \infty-topos. Jacob Lurie has a definition of the \infty-topos Grpd /X\infty Grpd_{/X} of local systems on a pro-space XX in HTT 7.1.6, which as far as I can see differs from the usual definition. By the usual definition I mean that the category of local systems on X=``limX iX=``lim'' X_i is the corresponding colimit of categories of local systems, where the transition maps are the pullback functors. The problem with this definition is that the result doesn’t look like an \infty-topos anymore (for example, if L iL_i is a local system on X iX_i for each ii, there need not be a coproduct of all L iL_i’s). To fix this, Lurie takes this colimit within the category of \infty-topoi, which gives you the limit of the underlying diagram of categories, where the transition maps are the pushforwards. This looks like some kind of completion of the previous category. The nice thing is that XGrpd /XX\mapsto \infty Grpd_{/X} is right adjoint to the functor HShape(H)\mathbf H\mapsto Shape(\mathbf H) (at least Lurie claims so in HTT).

    If H\mathbf H is locally \infty-connected, then the full subcategory of locally constant objects is equivalent to Grpd /Shape(H)\infty Grpd_{/Shape(\mathbf H)} (of course Shape(H)Shape(\mathbf H) is just a constant pro-space in this case). In general, what the nLab defines is local systems on the shape of H\mathbf H in the “usual” sense, so they do not form an \infty-topos. I would say that we should take instead Grpd /Shape(H)\infty Grpd_{/Shape(\mathbf H)} as the definition of the \infty-topos of local systems in H\mathbf H. There is a functor Grpd /Shape(H)H\infty Grpd_{/Shape(\mathbf H)}\to\mathbf H sending a local system to its “underlying sheaf”, which is the left adjoint of the unit of the above adjunction, and which is fully faithful in the locally \infty-connected case. I’m not sure what kind of “stuff” Grpd /Shape(H)H\infty Grpd_{/Shape(\mathbf H)}\to\mathbf H forgets in general. From the above discussion I guess it could forget structure?

    I’m still quite confused by all this because it seems that the image of Grpd /Shape(H)H\infty Grpd_{/Shape(\mathbf H)}\to\mathbf H is the same as what the nLab definition gives you. If so then either the two definitions of local systems on a pro-space that I think are different are in fact the same, or this functor definitely forgets structure.

    • CommentRowNumber57.
    • CommentAuthorMike Shulman
    • CommentTimeMar 20th 2012

    Why do you say it seems the image of this functor is the same as the nLab definition?

    • CommentRowNumber58.
    • CommentAuthorMarc Hoyois
    • CommentTimeMar 20th 2012

    Hi Mike,

    I was wrong, of course. I realized soon after I posted but it was time for bed! The image does seem bigger: the pullback of any map 1LConst(CoreGrpd κ)1\to LConst(Core\infty Grpd_\kappa) is in the image, and any object in the image is a filtered colimit of such. Maybe the image even contains objects that cannot be made constant over a cover. At least the proof that any local system is trivializable that I sketched in #45 does not work for local systems that do not live on a single space of the pro-space Shape(H)\Shape(\mathbf H).

    • CommentRowNumber59.
    • CommentAuthorMike Shulman
    • CommentTimeMar 20th 2012

    Okay. I guess it depends on what we want “local system” to mean, and therefore what we want to use it for.

    • CommentRowNumber60.
    • CommentAuthorMarc Hoyois
    • CommentTimeMar 21st 2012
    • (edited Mar 21st 2012)

    By the way, this is unrelated to my recent posts but I found this paper addressing in details the case of the 1-topos of a topological space, which I didn’t see mentioned in the blog discussion: http://matwbn.icm.edu.pl/ksiazki/fm/fm156/fm15611.pdf. It defines a category of locally constant sheaves with some extra structure and proves that it is equivalent to Set-representations of the fundamental Cech pro-groupoid.

    • CommentRowNumber61.
    • CommentAuthorUrs
    • CommentTimeMar 21st 2012

    Thanks. Let’s archive this refrence somewhere where we can later find it. I added it to fundamental groupoid and to some other entries. (I see that the related entries all deserve to be expanded…)