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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
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
$[X, \mathcal{B}C] \cong [\Pi X_* ,C]$where $X_*$ is the skeletal filtration of the CW-complex $X$, $C$ is a crossed complex, and $\mathcal{B}C$ is the classifying space of $C$, 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 $X$ is a covariant functor from the fundamental groupoid of $X$ 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)
(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)
(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)
Went through this cluster of links, polished slightly here and there and added links back and forth:
A locally constant function is a section of a constant sheaf;
a locally constant sheaf is a section of a constant stack;
a locally constant stack is a section of (… and so on…)
a locally constant ∞-stack is a section of a constant ∞-stack.
A locally constant sheaf / $\infty$-stack is also called a local system.
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 $\mathcal{S} := Fin \infty Grpd \in \infty Grpd$.
A locally constant infinity-stack on an object $X$ in an $\infty$-topos $\mathbf{H}$ is equivalently
a morphism $\tilde \nabla : X \to LConst \mathcal{S}$;
the object in the over-topos $\mathbf{H}/X$ obtained by the $(\infty,1)$-Grothendieck construction, i.e. the pullback
$\array{ P &\to& LConst \mathcal{Z} \\ \downarrow && \downarrow \\ X &\stackrel{\tilde \nabla}{\to}& LConst \mathcal{S} }$of the universal fibration.
This $(P \to X) \in \mathbf{H}/X$ is the genuine “$\infty$-sheaf on $X$” in that $\mathbf{H}/X$ is the little $\infty$-topos over $X$.
Now by general abstraction, cohomology with coefficients in the localy system of coefficients deserves to be nothing but
$H(X,\tilde \nabla) := \pi_0 \mathbf{H}_{/X} (X, P_{\tilde \nabla})$which is to be read as nothing but the nonabelian sheaf cohomology on $X$ with coefficients in the locally constant $\infty$-sheaf $P_{\tilde \nabla} \to X$.
And indeed, if you unwind the definitions and use the universality of the comma pullback, you find that a cocycle in here is in $\mathbf{H}$ a diagram
$\array{ && * \\ & \nearrow &\Downarrow& \searrow \\ X &\underset{\tilde \nabla}{\to}& LConst \mathcal{S} }$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.
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.
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.
Sure. I added a brief remark to local system.
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 $\Pi_2(X) \to Gpd$ are equivalent to sections of the constant 2-stack with fibre $Gpd$). I don’t think it should be the definition.
sections of the constant 2-stack with fibre $Gpd$
It’s just that taken at face value there is no 2-stack with values in $Gpd$. Since $Gpd$ is not an object in $2 Grpd$ but in $2 GRPD$. But of course one can deal with this.
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 $n$-stack, and since the notion of sections of constant $n$-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.
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.
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.
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 $\mathbf{H}$ and
$X \to LConst \mathcal{S}$a morphism. i still think the corresponding locally constant stack regarded as an object of $\mathbf{H}/X$ should be the pullback
$\array{ P &\to& LConst \mathcal{Z} \\ \downarrow && \downarrow \\ X &\to& LConst \mathcal{S} }$for $\mathcal{Z} \to \mathcal{S}$ the universal fibration.
So how would we recognize this as being “locally constant”? Like this: say a cover of $X$ “by contractible patches” is a morphism $U \to X$ such that
$U \to X$ is an effective epimorphism
$\Pi(U)$ is 0-truncated (a set).
The second conditon encodes that each connected component of $U$ is contractible. We are to think of this is $U = \coprod_i U_i$ of contractible patches $U_i$.
Then pick such a cover $p$ and pull back to it
$\array{ p^* P &\to & P &\to& LConst \mathcal{Z} \\ \downarrow && \downarrow && \downarrow \\ U &\stackrel{p}{\to}& X &\to& LConst \mathcal{S} } \,.$The total rectangle here is still a pullback. Now use that $(\Pi \dashv LConst)$ and that $LConst$ is full and faithful to deduce that the adjunct
$\array{ \Pi (p^* P) &\to& \mathcal{Z} \\ \downarrow && \downarrow \\ \Pi(U) &\to& \mathcal{S} }$is still a pullback. Since by assumption $\Pi(U)$ is a set, the bottom morphism picks a bunch of finite $\infty$-groupoids $\mathcal{F}_i$ – which I think we can deduce must all be equivalent – and by the property of $\mathcal{Z}$ is follows that
$\Pi(p^* P) \simeq \Pi(U) \times \mathcal{F} \simeq \Pi(U \times LConst \mathcal{F})$(Hm, maybe I am assuming now also that $\Pi$ preserves finite products.) So we conclude that
$p^*P \simeq U \times LConst \mathcal{F} \,.$so this says that $P$ looks locally like a contractible times a constant $\infty$-stack. Which should be the defining property of locally constant objects in $\mathbf{H}/X$. I guess.
And now conversely, suppose that $P \to X$ is an object in $\mathbf{H}/X$ such that there is an effective epimorphism $U \to X$ and a pullback diagram
$\array{ U \times LConst \mathcal{F} &\to & P \\ \downarrow &&\downarrow \\ U &\stackrel{p}{\to}& X } \,.$We want to conclude that then there is a morphism $X \to LConst \mathcal{S}$ such that we have a pasting of pullbacks
$\array{ p^* P &\to & P &\to& LConst \mathcal{Z} \\ \downarrow && \downarrow && \downarrow \\ U &\stackrel{p}{\to}& X &\to& LConst \mathcal{S} } \,.$For this now use the assumption that $U \to X$ is an effective epi, hence that
$\cdots U \times_X U \times_X U \stackrel{\to}{\stackrel{\to}{\to}} U \times_X U \stackrel{\to}{\to} \to U \to X$is a colimiting cone under the Cech nerve. This implies that the morphism $U \to LConst \mathcal{S}$ that classifies $U \times LConst \mathcal{F}$ factors through $X$ and we get
$\array{ p^* P &\to & P & & LConst \mathcal{Z} \\ \downarrow && \downarrow && \downarrow \\ U &\stackrel{p}{\to}& X &\to& LConst \mathcal{S} } \,.$Now we need to show that the right square fills. For that use that colimits in $\mathbf{H}$ are universal.
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 $X$ such that there exists a well-supported object $U$ (i.e. $U\to 1$ is an (effective) epi) such that $U\times X \cong U \times_{\Delta I} \Delta S$ over $U$, for some map $S\to I$ in Set and $U\to \Delta I$ in the topos $\mathcal{E}$, where I am writing $\Delta= L Const$ for brevity. Of course if $\mathcal{E}$ is locally connected, then $U\to \Delta I$ is equivalently $\Pi_0(U)\to I$, 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 $\mathcal{E}/\Delta I \simeq \mathcal{E}^I$, by extensivity of $\mathcal{E}$, so that $U\to \Delta I$ is equivalently an $I$-indexed family $(U_i)$ of objects of $\mathcal{E}$. Saying that $U = \sum_i U_i$ is well-supported then means that the $U_i$ jointly cover $1\in\mathcal{E}$, and the isomorprhism $U\times X \cong U \times_{\Delta I} \Delta S$ over $U$ is equivalently an isomorphism $U_i\times X \cong U_i \times \Delta S_i$ over each $U_i$—so that $X$ actually becomes literally constant over each $U_i$.
Now suppose we translate that straightforwardly into an $(\infty,1)$-topos, and define a locally constant object to be an $X$ such that there exists a well-supported object $U$ (i.e. $U\to 1$ is an effective epi) such that $U\times X \simeq U \times_{\Delta I} \Delta S$ over $U$, for some maps $S\to I$ in $\infty Gpd$ and $U\to \Delta I$ 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 $U=1$, this says that $X$ is the fiber of $\Delta S \to \Delta I$ along a global section of $\Delta I$. Clearly a special case of this is when $I = \infty Gpd_\kappa$ is the $\infty$-groupoid of $\infty$-groupoids bounded in size by some cardinal $\kappa$ and $S$ 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 $U$, $S$, and $I$ as defined above. Then there exists a cardinal $\kappa$ such that $S\to I$ is the pullback of the universal fibration over $\infty Gpd_\kappa$ along some map $I\to \infty Gpd_\kappa$. Since $\Delta$ preserves pullbacks, we may as well assume that $I= \infty Gpd_\kappa$ and $S$ is the universal fibration. Now, since $U\times X$ over $U$ is a pullback of the “global object” $X$, it has descent data over the kernel of $U\to 1$, and hence (via the equivalence) so does $U \times_{\Delta I} \Delta S$.
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 $U\to \Delta I$ has descent data over the kernel and hence factors through $1$. That seems to require that knowing the pullback functor $Hom_{\mathcal{E}}(U,\Delta (\infty Gpd_\kappa)) \to Core(\mathcal{E}/U)$ is fully faithful, which seems plausible, but I don’t know how to prove it. Is this a known fact?
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 $\mathcal{T}/B$ in order to be able to connect back to a situation in $\mathcal{T}$ in cases where that is gros and suitably connected. (And I’ll write $B$ for the base object now, not to collide with your use of $X$.)
For given that $\Delta_{\mathcal{T}/B}$ is given by $S \mapsto (\Delta_{\mathcal{T}} S) \times B$ your formula
$U \times X \simeq U \times_{\Delta_{\mathcal{E}} I} \Delta_{\mathcal{E}} S$in $\mathcal{E}$ identifies with my formula
$U \times_B X \simeq U \times_{\Delta_{\mathcal{T}} I} \Delta_{\mathcal{T}} S$in $\mathcal{T}$. Notably we may restrict to the case that $B = *$ and $\mathcal{E} = \mathcal{T}$. 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 $\Delta_{\mathcal{T}} \mathcal{S}$ in $\mathcal{T}$.
Now I need to think more about the other points.
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 $\Delta F \to X \to B$ is a fiber sequence that is locally trivial (in our sense here) then it is classified by a morphism $B \to \mathbf{B}AUT(\Delta F)$.
For our case $F$ is a $\kappa$-bounded $\infty$-groupoid and since $\Delta$ is full and faithful (in the $\infty$-connected $\mathcal{T}$, compare my remarks above) we have that this is a morphism $B \to \Delta ( \mathbf{B} AUT(F))$. But here $\mathbf{B} AUT(F) \hookrightarrow Core \infty Grpd_\kappa$ is precisely the sub-$\infty$-groupoid of $Core \infty Grpd_\kappa$ on $F$, so we conclude that we have a pullback
$\array{ X &\to& \Delta \mathcal{Z} \\ \downarrow && \downarrow \\ B &\to& \Delta Core \infty Grpd_\kappa }$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.
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 $U\times \Delta S$ rather than a pullback $U\times_{\Delta I} \Delta S$.
(Sorry about mixing up the meaning of $X$; 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 $\Delta_{\mathcal{T}} \mathcal{S}$ in $\mathcal{T}$.
I think that should follow automatically if we just do everything for a general topos $\mathcal{E}$, and then at the end specialize to $\mathcal{T}/B$, 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 $\Delta_{\mathcal{E}}(Core\infty Gpd_\omega)$, then those in $\mathcal{T}/B$ are classified by global sections in $\mathcal{T}/B$ of $\Delta_{\mathcal{T}/B}(Core\infty Gpd_\omega) = B \times \Delta_{\mathcal{T}}(Core\infty Gpd_\omega)$, which are equivalent to maps $B\to \Delta_{\mathcal{T}}(Core\infty Gpd_\omega)$ in $\mathcal{T}$ by adjunction.
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…
But we will then also not have the alternative slick definition.
What do you mean? Why not?
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 $Hom(1,\Delta(Core \infty Gpd)) \to Core(\mathcal{E})$ 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 $U$.)
For if $\mathcal{E} = Psh(C)$ for some small (∞,1)-category $C$, then $\mathcal{E}$ is locally ∞-connected, and $\Pi(1)$ is the ∞-groupoid reflection of $C^{op}$, call it $gpd(C^{op})$. Thus we have
$Hom_{\mathcal{E}}(1,\Delta(Core\infty Gpd)) \simeq Hom_{\infty Gpd}(gpd(C^{op}),Core\infty Gpd) \simeq Hom_{(\infty,1)Cat}(C^{op}, Core \infty Gpd)$using the adjunctions $\Pi \dashv \Delta$ and that $gpd$ is a reflection of (∞,1)-categories into ∞-groupoids. But this latter ∞-groupoid is the same as (the core of) the full subcategory of $Hom_{(\infty,1)Cat}(C^{op},\infty Gpd)$ spanned by those presheaves $C^{op}\to \infty Gpd$ which take all maps in $C$ to equivalences. Since $\mathcal{E} = Psh(C)$ is the category of all presheaves $C^{op}\to \infty Gpd$, 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…
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
$Hom_{Sh(C)}(1, \Delta Core \infty Grpd) = Hom_{PSh(C)}(1, \Delta Core \infty Grpd) \to PSh(C) \to Sh(C)$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.
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 $\Delta AUT(F)$-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 $\mathbf{B} \Delta AUT(F)$. And since $\Delta$ preserves looping this is $\Delta \mathbf{B}AUT(F)$ and this in turn is the image under $\Delta$ of the full subcategory of $Core \infty Grpd$ on the given $F$. So that finally we find this way that $AUT(F)$-principal $\infty$-bundles for arbitrary $F$ are represented by $\Delta Core \infty Grpd$.
Notice that this does reproduce our previous notion of total spaces of these beasts:
if
$\array{ P &\to& * \\ \downarrow && \downarrow \\ X &\to& \Delta \mathbf{B}G }$is a $\Delta G$-principal $\infty$-bundle and $\rho : \mathbf{B}G \to \infty Grpd$ an $\infty$-permutation representation, then the corresponding associated $\infty$-bundle is the pullback of the universal associated $\infty$-bundle which is
$\array{ P &\to& \Delta \rho(*)//G \\ \downarrow && \downarrow \\ X &\to& \Delta \mathbf{B}G }$Here the action $\infty$-ggroupoid $\rho(*)//G$ is the colimit $\cdots \simeq \lim_\to \rho$ 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
$\array{ P &\to& \Delta \rho(*)//G &\to& \Delta \mathcal{Z} \\ \downarrow && \downarrow && \downarrow \\ X &\to& \Delta \mathbf{B}AUT(F) &\hookrightarrow& Core \infty Grpd }$Gee, I really have to run now. I’ll be late for the seminar after all…
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?
$Hom_{Sh(C)}(1, \Delta Core \infty Grpd) = Hom_{PSh(C)}(1, \Delta Core \infty Grpd)$
Wait… those two $\Delta$s are different! One of them is the constant presheaf, the other is the constant sheaf.
Concerning $\Delta$s:
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 $G$-action” which talks about total objects $P$ and actions by group objects $G$.
I am proposing to make use of the fact that every locally constant $\infty$-sheaf $X$ with typical fiber $F$ is (or should be) a $\Delta AUT(F)$-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 $\Delta AUT(F)$-principal $\infty$-bundle is given locally trivial, with transition functions in $\Delta AUT(F)_0$, and so on.
The corresponding $(\rho : \mathbf{B}AUT(F) \to Core \infty Grpd)$-associated bundle hece is locally constant on $F$, 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-$\infy$-category of the ambient topos on the objects of the core.
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 $AUT(\Delta F)$-bundle, not a $\Delta(AUT(F))$-bundle. And identifying $AUT(\Delta F)$ with $\Delta(AUT(F))$ seems like the same sort of problem as showing that $Hom(1,\Delta(Core \infty Gpd)) \to Core(\mathcal{E})$ is fully faithful.
I only see why it would be an $AUT(\Delta F)$-bundle, not a $\Delta(AUT(F))$-bundle.
But this is our design criterion for locally constant $\infty$-sheaves: we want them to be classified by morphisms into $\Delta Core \infty Grpd_\kappa$. That is $\cdots \simeq \Delta \coprod_{[F]} \mathbf{B}AUT(F)$.
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 $\Delta Core \infty Gpd$ 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 $U\times X \simeq U\times_{\Delta I} \Delta S$, 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 $1\to \Delta Core \infty Gpd$.
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”?
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 $F \in \infty Grpd_\kappa$.
Claim Principal $G$-torsors are represented by $\mathbf{B}G$ in that $G Tor_{\mathcal{E}} \simeq \mathcal{E}(*, \mathbf{B}G)$.
Definition
A principal $G$-torsor $P$ is locally trivial if there exists an effective epi $U \to *$ such that $U \times P \simeq U \times G$.
Write $F//AUF(F) := \lim_{\to} ( \mathbf{B}AUT(F) \hookrightarrow \infty Grpd_\kappa)$.
The $\Delta F$-bundle $P \times_{\Delta AUT(F)} \Delta F$ associated to a $\Delta AUT(F)$-torsor $P$ is $* \times_{\Delta \mathbf{B}AUT(F)} \Delta (F// AUT(F))$.
Proposition If $P \in \Delta AUT(F) Tor$ is locally trivial with trivializing cover $U$ then the associated bundle is a locally trivial $\Delta F$-bundle in that
$U \times (P \times_{\Delta AUT(F)} \Delta F) \simeq U \times \Delta F \,.$Proof : Consider the diagram
$\array{ U \times \Delta F &\to& P \times_{\Delta AUT(F)} \Delta F//AUT(F) &\to& \Delta (F//AUF(F)) &\to& \Delta \mathcal{Z}_\kappa \\ \downarrow && \downarrow && \downarrow && \downarrow \\ U &\to& * &\to& \mathbf{B}\Delta AUT(F) &\to& \Delta Core \infty Grpd_\kappa } \,,$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 $\Delta$ 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 $U \times_{\Delta \infty Grpd_\kappa} \Delta \mathcal{Z}$. By the assumption that on $U$ the torsor $P$ and hence its cocycle $* \to \mathbf{B} \Delta AUT(F)$ are trivial and again using that $\Delta$ preserves finite limits this is
$\array{ U \times \Delta F &\to& \Delta F &\to & \Delta \mathcal{Z} \\ \downarrow && \downarrow && \downarrow \\ U &\to& \Delta * &\stackrel{}{\to}& \Delta Core \infty Grpd_\kappa } \,,$QED.
Remark If $*$ has several direct summands it is straightforward to generalize this discussion to one where $F$ may be different on each summand.
Propososal Call an object $X$ locally constant if it is equivalent to a $\Delta F$-bundle associated to an $\Delta AUT(F)$-torsor.
Remark In the case that all $AUT(F)$-torsors are locally trivial, this means that every locally constant object locally looks like $U \times \Delta F$. But the fact that it is associated to a $\Delta AUT(F)$-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 $\Delta AUT(F)$.
Replies to your questions
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 $U$s at all.
The associtated-bundle definition makes sense without any assumption on existence of good $U$s. But if good $U$s exist, it reproduces objects that look locally like $U \times \Delta F$ plus locally constant descent data. So that seems to be good to me.
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.
Just a quick thought:
keep in mind the example of $G$-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 $U$-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.
@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?
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).
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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?
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.
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…
@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 $X$ is a map $Sh(X) \to G-set$ (geometric morphism?) If that is the case the Gr. topology one uses on $X$ determines what sort of cover the bundle is trivialised by.
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.
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 $B \Pi_\infty(G)$—the delooping of the ∞-group $\Pi_\infty(G)$—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 $[G,\infty Gpd]$ has the nice property that for any other topos E, geometric morphisms $E\to [G,\infty Gpd]$ are equivalent to global sections of the constant stack $\Delta G$ in E. So that’s why I expect $\Delta G$ to also classify all G-bundles (defined in the appropriate way relative to the topology we used to construct the topos E itself).
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. :)
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 $T$, as defined on the nLab, is an element of $Sh(T)(core\infty Grpd_\kappa)$, where $Sh(T)$ is the shape of $T$. If $(X_i)$ is a cofiltered system corepresenting $Sh(T)$, then any local system on $T$ is represented by a local system on some $X_i$ (i.e. an object of $\infty Grpd/X_i$).
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 $U=1$, 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: $X$ is locally constant if there exists an effective epimorphism $\coprod_\alpha U_\alpha\to 1$ such that $X$ becomes constant over each $U_\alpha$. 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 $\coprod_\alpha U_\alpha\to 1$ that will work comes from a contractible cover of $\Pi(1)$ (= the shape). I’m hoping that this argument can be adapted to the non-locally $\infty$-connected case (but of course the cover $\coprod_\alpha U_\alpha$ will now depend on the local system).
Here’s how it would go, very roughly. Let $X:I\to \infty Grpd$ be a cofiltered system corepresenting the shape of $T$. Morally $I\to \infty Grpd$ is the pullback of $\infty Grpd_\ast\to\infty Grpd$ along the shape (really $I$ is a small final subcategory of what you get this way). Start with a local system $L$ on $T$. This is given by a local system on $X_i$ for some $i\in I$. Choose a contractible cover $\coprod_\alpha K_\alpha\to X_i$. Going around the pullback square defining $X:I\to \infty Grpd$ gives us a canonical point of $Sh(T)(X_i)$, i.e., a morphism $1\to LConst(X_i)$ in $T$. The local system $L$ should then be the image of the local system over $X_i$ under the composition
$f\colon \infty Grpd/X_i\to T/LConst(X_i)\to T$,
where the last functor is pullback along $1\to LConst(X_i)$. Set $U_\alpha=f(K_\alpha)$. Then $\coprod_\alpha U_\alpha\to 1$ is a cover of $1$. The functor $f$ followed by pullback to $U_\alpha$ factors through $\infty Grpd/K_\alpha\simeq \infty Grpd$, so $L\times U_\alpha$ is constant, as desired. Sorry this is still very sketchy.
Thoughts?
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?
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 $(\infty,1)$-topos (= of locally constant shape) $\mathbf{H}$ given, all its objects $X \in \mathbf{H}$ 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 $\mathbf{H}/X$ is equivalent to that of locally constant $\infty$-stacks on $X$ as given in section 2.3.14 here.
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 ($S^1\times \mathbb{N}_\infty$) 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 $\mathbf H$ is locally constant iff it is the pullback along a morphism $1\to LConst(core\infty Grpd_\kappa)$ (for some cardinal $\kappa$). 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 $\mathbf H$ is locally $\infty$-connected.
OK, so let $X\in\mathbf H$ be locally constant, and let’s try to classsify $X$ by a map $1\to LConst(\infty Grpd)$ (let me drop the core and the cardinality bound from the notation). Choose a diagram $\{U_\alpha\}$ with colimit $1$ (a Cech nerve) such that $X\times U_\alpha$ is constant. So $X\times U_\alpha\simeq LConst_{\mathbf H/U_\alpha} V_\alpha$ for some $\infty$-groupoid $V_\alpha$. We have cartesian squares
$\array{ V_\alpha & \to & \infty Grpd_\ast \\ \downarrow & & \downarrow \\ \ast & \to & \infty Grpd \\ }$Now we apply the exact functor $LConst_{\mathbf H/U_\alpha}:\infty Grpd\to \mathbf H/U_\alpha$, which is the composition $p_\alpha^\ast\circ LConst$, where $p_\alpha^\ast:\mathbf H\to\mathbf H/U_\alpha$ is the pullback functor. We get cartesian squares
$\array{ p_\alpha^\ast(X)=X\times U_\alpha & \to & p_\alpha^\ast LConst(\infty Grpd_\ast) \\ \downarrow & & \downarrow \\ p_\alpha^\ast(\ast)=U_\alpha & \to & p_\alpha^\ast LConst(\infty Grpd) \\ }$All that remains to check is that the bottom map in these cartesian squares is natural for varying $\alpha$ (and it seems to me that this is true for formal reasons), and then in the colimit we will obtain $X$ as a pullback of a map $1\to LConst(\infty Grpd)$ as desired. Does any part of this “proof” sound particularly fishy?
Mike, I see that you’ve provided a counter-example ($S^1\times \mathbb{N}_\infty$) 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 $U_\alpha \times U_\beta$ between the pullbacks of $X$ via $U_\alpha$ and via $U_\beta$, hence an isomorphism between $LConst_{\mathbf{H}/U_\alpha\times U_\beta} V_\alpha$ and $LConst_{\mathbf{H}/U_\alpha\times U_\beta} V_\beta$. The question is whether this isomorphism induces an isomorphism between the corresponding maps into $p_{\alpha,\beta}^* LConst(\infty Gpd)$, which I believe is what fails in the non-locally-connected case.
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 $1$-connected, so I guess that’s why it works for discrete sheaves.
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 $\phi_{\alpha\beta}$ between $V_\alpha$ and $V_\beta$, and then isomorphisms $\phi_{\alpha\beta\gamma}$ between $\phi_{\alpha\beta}\phi_{\beta\gamma}$ and $\phi_{\alpha\gamma}$, 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 $n$-truncated sheaves on a locally $n$-connected topos.
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’.
@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 $X\times U_\alpha$ and the constant sheaf at $V_\alpha$ over $U_\alpha$ (making it “locally constant” in the traditional sense), which implies that the constant sheaves at $V_\alpha$ and $V_\beta$ over $U_\alpha\cap U_\beta$ are isomorphic, we need to give a lifting of that isomorphism to a section over $U_\alpha\cap U_\beta$ of the constant sheaf at $Iso(V_\alpha,V_\beta)$. And so on.
[Cross-posted with Mike]
Right, I got it backwards. If you fix equivalences $LConst_{U_\alpha}V_\alpha\simeq X\times U_\alpha$, then you already have an equivalence $LConst_{U_\alpha\times U_\beta}V_\alpha\simeq LConst_{U_\alpha\times U_\beta} V_\beta$, and the question is, does it come from an equivalence $V_\alpha\simeq V_\beta$. If the topos is locally contractible (or $n$-connected and the $V$’s are $n$-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 $1\to LConst(Core\infty Grpd_\kappa)$ that doesn’t stray too far from the naive definition? One could simply require the existence of those $\phi_{\alpha\beta}$, $\phi_{\alpha\beta\gamma}$, and so on, so as to make the descent argument work, but that’s not very practical.
I can’t think of any other way to state it.
(By the way, an equivalence $V_\alpha \simeq V_\beta$ is not quite the same as a section over $U_\alpha\cap U_\beta$ of the constant sheaf at $Iso(V_\alpha,V_\beta)$.)
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 $\infty Grpd_{/X}$ of local systems on a pro-space $X$ 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 $X=``lim'' X_i$ 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 $L_i$ is a local system on $X_i$ for each $i$, there need not be a coproduct of all $L_i$’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 $X\mapsto \infty Grpd_{/X}$ is right adjoint to the functor $\mathbf H\mapsto Shape(\mathbf H)$ (at least Lurie claims so in HTT).
If $\mathbf H$ is locally $\infty$-connected, then the full subcategory of locally constant objects is equivalent to $\infty Grpd_{/Shape(\mathbf H)}$ (of course $Shape(\mathbf H)$ is just a constant pro-space in this case). In general, what the nLab defines is local systems on the shape of $\mathbf H$ in the “usual” sense, so they do not form an $\infty$-topos. I would say that we should take instead $\infty Grpd_{/Shape(\mathbf H)}$ as the definition of the $\infty$-topos of local systems in $\mathbf H$. There is a functor $\infty Grpd_{/Shape(\mathbf H)}\to\mathbf H$ 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” $\infty Grpd_{/Shape(\mathbf H)}\to\mathbf H$ 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 $\infty Grpd_{/Shape(\mathbf H)}\to\mathbf H$ 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.
Why do you say it seems the image of this functor is the same as the nLab definition?
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 $1\to LConst(Core\infty Grpd_\kappa)$ 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 $\Shape(\mathbf H)$.
Okay. I guess it depends on what we want “local system” to mean, and therefore what we want to use it for.
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.
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…)
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