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…)

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.

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

]]>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 $\Shape(\mathbf H)$.

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

]]>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.~~

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 $Iso(V_\alpha,V_\beta)$.)

]]>[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.

]]>@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.

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’.

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.

]]>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.

]]>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.

]]>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?

]]>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.

]]>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?

]]>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?

]]>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. :)

]]>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).

]]>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.

]]>@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 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…

]]>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.

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?

]]>