### Petit (∞,1)-Toposes

#### Posted by Mike Shulman

I think I’m gradually coming to understand why some people think $(\infty,1)$-toposes are the best thing since sliced bread. For me, I think a brief way to describe what’s so amazing about them is that they still have all the wonderful things I love about “good old” 1-toposes, but they also bring in homotopy theory and higher category theory in the “right” way, which fixes a number of problems (or inelegancies) that happen with good old 1-toposes.

In this post I want to describe one aspect of that, at an intuitive level that should hopefully make sense even if you have no idea what an $(\infty,1)$-topos is (yet). It’s closely related to stuff that Urs has been talking a lot about recently, but I prefer to think about it in terms of “petit toposes” instead of “gros” ones. From this perspective, the claim is that $(\infty,1)$-toposes elucidate the relationship between topological spaces and $\infty$-groupoids, by providing a context in which both can be embedded (almost) *disjointly*.

Imagine that you’ve never heard of the homotopy hypothesis, but you know what a topological space is, and you know what a groupoid is, and you have at least some vague idea of what an $\infty$-groupoid is. And you want to know how they’re related. You might notice first that there’s a particular class of topological spaces that can naturally be identified with a particular class of groupoids: *discrete* topological spaces are just sets, as are discrete (higher) groupoids (those with no nonidentity $k$-morphisms for $k\gt 0$). Thus, we have two categories which share a common subcategory, so one might naturally ask whether they are in fact both subcategories of a common *supercategory*. In other words, is there a commutative square:
$\array{\text{sets} & \to & \text{(higher) groupoids}\\
\downarrow & & \downarrow \\
\text{spaces} & \to & ??}$

Before we go any further, let’s replace topological spaces with their open-set lattices. (For suitably nice topological spaces, this doesn’t lose any information.) This is a good idea since lattices are special posets, which are special categories, and groupoids are also special categories. But we can’t just put “(higher) categories” in the ?? box, since continuous maps of spaces correspond, not to arbitrary poset-maps of their open-set lattices, but only to those which preserve finite intersections and arbitrary unions. On the other hand, we can’t put “categories with finite limits and arbitrary colimits” either, since groupoids don’t have those.

But actually, before we jump to that, we should think about *how* sets sit inside groupoids and spaces. They sit inside groupoids essentially as themselves (with identity morphisms added), but the *open-set lattice* of the discrete space on a set is not the set itself but its powerset. The powerset of a set is equivalently its free cocompletion *qua* poset. Thus, we could put something like “categories with finite limits and arbitrary colimits” in our ?? box, as long as we interpret the right-hand vertical arrow not as “inclusion” but as “free cocompletion.”

Of course, it doesn’t make much sense to start with a (higher) groupoid and take its free cocompletion as a poset, since it isn’t a poset. So instead let’s call the right-hand arrow the free cocompletion of a (higher) groupoid as a (higher) category, which of course is just its presheaf category $[G^{op},\infty Grpd]$. Now, however, we need to adjust the bottom horizontal arrow, since while open-set lattices are very well-behaved cocomplete posets, they are not that well-behaved when regarded directly as cocomplete categories. We don’t want to exactly take their free cocompletion, though, since we want to preserve the colimits they already have; thus the bottom arrow is something like “boosting up a cocomplete poset (= (0,1)-category) to a well-behaved cocomplete (higher) category.”

This is essentially the correct picture. One has to be a little more careful with the definition than “categories with finite limits and arbitrary colimits”—the limits have to distribute over the colimits in a suitable way—but once we do that, and explain how that bottom arrow is to be constructed, we’ve arrived at the right notion of (higher) topos. The point of taking this route is that *the category of $(\infty,1)$-toposes contains topological spaces and $\infty$-groupoids as full subcategories, whose intersection consists of the discrete spaces.* If we think of an $(\infty,1)$-topos as a generalized topological space, then we might evocatively say that the right-hand vertical arrow in our square “equips an $\infty$-groupoid with the discrete topology.”

Next, now that we’ve got spaces and $\infty$-groupoids in a common framework, we can ask how they’re related. For instance, what is a map from a space $X$ to an $\infty$-groupoid $Y$ (both considered as (∞,1)-toposes)? Well, if $Y$ is a set, then we know the answer: it’s a decomposition of $X$ into connected components labeled by the points of $Y$. That is, we have to write $X = \bigcup_i U_i$ where each $U_i$ is open and labeled by some $f(U_i)\in Y$, and moreover if $U_i \cap U_j \neq \emptyset$, then $U_i$ and $U_j$ are labeled by the same point of $Y$. In yet other words, for every nonempty intersection $U_i \cap U_j$, we have to *specify an equality* $f(U_i) = f(U_j)$.

That last immediately suggests what the right notion of “map from a space $X$ into a (higher) groupoid $Y$ is going to be; we should give a decomposition of $X = \bigcup_i U_i$ where each $U_i$ is labeled by an object $f(U_i)\in Y$, together with, for every nonempty intersection $U_i \cap U_j$, an *isomorphism* (or equivalence) $f(U_i) \cong f(U_j)$, and for every triple intersection, … etc. In other words, a map from a space $X$ to a groupoid $Y$ is the same as a “$Y$-bundle over $X$” (specified in terms of transition functions). In yet other words, if “space” means “generalized space” in the sense of “$(\infty,1)$-topos,” then the “classifying space” of an $\infty$-groupoid is just itself, regarded as a generalized space “with the discrete topology.”

The really lovely observation, however, is that the full subcategory $\infty Grpd\hookrightarrow (\infty,1)Topos$ is (almost) *reflective*. In other words, for a space $X$, there is an $\infty$-groupoid $\Pi(X)$ such that for any $\infty$-groupoid $Y$, maps $X\to Y$ are equivalent to maps $\Pi(X)\to Y$. This $\infty$-groupoid $\Pi(X)$ is just the usual *fundamental $\infty$-groupoid* of $X$, whose objects are the points of $X$, whose 1-morphisms are paths in $X$, whose 2-morphisms are homotopies of paths, and so on. In other words, the fundamental $\infty$-groupoid of a topological space is not an *ad hoc* algebraic gadget we invent in order to give us information about the space—it has a universal property.

As was just pointed out in another thread, the fact that $\Pi$ is a left adjoint can be regarded as *the ultimate van Kampen theorem*, since any left adjoint preserves all colimits. (The classical van Kampen theorem, of course, says that $\Pi_1$ preserves a particular class of pushouts.) Of course, the work is then moved into verifying that a given pushout of topological spaces induces a pushout of (∞,1)-toposes, but I think the conceptual basis of the theorem, at least, is much clearer this way.

Now this does come with a couple of caveats. One is that in order to identify $\Pi(X)$ with the fundamental $\infty$-groupoid as usually defined, we need there to exist lots of maps into $X$ from the interval. Some spaces have issues with that; for instance, they could be connected but not path-connected. One of my favorite spaces to think about is the “long circle:” take the extended long line and identify its endpoints.
The Warsaw circle is another. Both of these clearly have a “loop,” but classical $\pi_1$ can’t detect it, since no map out of $[0,1]$ can “make it all the way around.” However, the topos-theoretic $\Pi(X)$ does detect this loop. This is generally the way things go: for not-so-nice spaces where the notions disagree, the topos-theoretic version is usually better-behaved. This sort of “covering fundamental groupoid” has been studied in the 1-topos-theoretic literature, using hypercoverings and other model-categorey sort of things, but I think it’s much more intuitively clear what’s going on when we at least *state* it in the language of higher topoi.

The second caveat is that some spaces may be so badly behaved that $\Pi(X)$ doesn’t exist (hence why I said that $\infty Grpd$ is only “almost” reflective). For $\Pi(X)$ to exist, we essentially need $X$ to be locally contractible. What we do in this case is the same thing that we do when constructing “moduli stacks”—we look at the functor that $\Pi(X)$ would represent if it did exist, and study that functor as a stand-in for the nonexistent object. Here that functor is just $Hom_{(\infty,1)Topos}(X,-)\colon\infty Grpd \to \infty Grpd$. This functor is called the shape of $X$. And it’s still true that the map from (∞,1)-toposes to their shapes preserves colimits.

## Re: Petit (∞,1)-Toposes

Thanks, Mike.

One could tell this story of forming “$Top \coprod_{Set} \infty Grpd$” also nicely from the gros perspective, leading to the cohesive $\infty$ TopGrpd.

But of course you are right, this does miss objects of non-representable shape. I hope we can see how to put it all together to a big picture that includes the nice aspects of both points of view.

This here might be a way to go: remember the story of

Structured Spacesas summarized at Notions of space:for a given site $C$, we want to be looking at the sequence of inclusions of $\infty$-categories of spaces

$C \hookrightarrow Schemes(C) \hookrightarrow (\infty,1)Topos^{C} \hookrightarrow (\infty,1)\hat Sh(Pro(C))$

where on the far right we have the very large $(\infty,1)$-topos on pro-objects in $C$, and where $(\infty,1)Topos^C$ is supposed to denote here the $C$-structured $(\infty,1)$-toposes – supposed to be the

petitones! – but equipped with “$C$-structure” that makes them be objects in the very large $(\infty,1)$-topos themselves.If we take $C = OpenBalls$ or similar then every toplogical space $X$ naturally gives rise to a petit $(\infty,1)Topos^C$.

Maybe passing to that

very largeandgros$(\infty,1)\hat Sh(Pro(C))$ helps to put all aspects into the same perspective.