This semester, Susama Agarwala and I have been sharing a lecture series for graduate students. (A caveat: there are lecture notes there, by student request, but they’re rough notes, and contain some mistakes, omissions, and represent a very selective view of the subject.) Being a “topics” course, it consists of a few different sections, loosely related, which revolve around the theme of categorical tools which are useful for geometry (and topology).

What this has amounted to is: I gave a half-semester worth of courses on toposes, sheaves, and the basics of derived categories. Susama is now giving the second half, which is about motives. This post will talk about the part of the course I gave. Though this was a whole series of lectures which introduced all these topics more or less carefully, I want to focus here on the part of the lecture which built up to a discussion of sheaves as spaces. Nothing here, or in the two posts to follow, is particularly new, but they do amount to a nice set of snapshots of some related ideas.

Coming up soon: John Huerta is currently visiting Hamburg, and on  July 8, he gave a guest-lecture which uses some of this machinery to talk about supermanifolds, which will be the subject of the next post in this series. In a later post, I’ll talk about Susama’s lectures about motives and how this relates to the discussion here (loosely).

### Grothendieck Toposes

The first half of our course was about various aspects of Grothendieck toposes. In the first lecture, I talked about “Elementary” (or Lawvere-Tierney) toposes. One way to look at these is to say that they are categories $\mathcal{E}$ which have all the properties of the category of Sets which make it useful for doing most of ordinary mathematics. Thus, a topos in this sense is a category with a bunch of properties – there are various equivalent definitions, but for example, toposes have all finite limits (in particular, products), and all colimits.

More particularly, they have “power objects”. That is, if $A$ and $B$ are objects of $\mathcal{E}$, then there is an object $B^A$, with an “evaluation map” $B^A \times A \rightarrow B$, which makes it possible to think of $B^A$ as the object of “morphisms from A to B”.

The other main thing a topos has is a “subobject classifier”. Now, a subobject of $A \in \mathcal{E}$ is an equivalence class of monomorphisms into $A$ – think of sets, where this amounts to specifying the image, and the monomorphisms are the various inclusions which pick out the same subset as their image. A classifier for subobjects should be thought of as something like the two-element set is $Sets$, whose elements we can tall “true” and “false”. Then every subset of $A$ corresponds to a characteristic function $A \rightarrow \mathbf{2}$. In general, a subobject classifies is an object $\Omega$ together with a map from the terminal object, $T : 1 \rightarrow \Omega$, such that every inclusion of subobject is a pullback of $T$ along a characteristic function.

Now, elementary toposes were invented chronologically later than Grothendieck toposes, which are a special class of example. These are categories of sheaves on (Grothendieck) sites. A site is a category $\mathcal{T}$ together with a “topology” $J$, which is a rule which, for each $U \in \mathcal{T}$, picks out $J(U)$, a set of collections of maps into $U$, called seives for $U$. They collections $J(U)$ have to satisfy certain conditions, but the idea can be understood in terms of the basic example, $\mathcal{T} = TOP(X)$. Given a topological space, $TOP(X)$ is the category whose objects are the open sets $U \subset X$, and the morphisms are all the inclusions. Then  that each collection in $J(U)$ is an open cover of $U$ – that is, a bunch of inclusions of open sets, which together cover all of $U$ in the usual sense.

(This is a little special to $TOP(X)$, where every map is an inclusion – in a general site, the $J(U)$ need to be closed under composition with any other morphism (like an ideal in a ring). So for instance, $\mathcal{T} = Top$, the category of topological spaces, the usual choice of $J(U)$ consists of all collections of maps which are jointly surjective.)

The point is that a presheaf on $\mathcal{T}$ is just a functor $\mathcal{T}^{op} \rightarrow Sets$. That is, it’s a way of assigning a set to each $U \in \mathcal{T}$. So, for instance, for either of the cases we just mentioned, one has $B : \mathcal{T}^{op} \rightarrow Sets$, which assigns to each open set $U$ the set of all bounded functions on $U$, and to every inclusion the restriction map. Or, again, one has $C : \mathcal{T}^{op} \rightarrow Sets$, which assigns the set of all continuous functions.

These two examples illustrate the condition which distinguishes those presheaves $S$ which are sheaves – namely, those which satisfy some “gluing” conditions. Thus, suppose we’re, given an open cover $\{ f_i : U_i \rightarrow U \}$, and a choice of one element $x_i$ from each $S(U_i)$, which form a “matching family” in the sense that they agree when restricted to any overlaps. Then the sheaf condition says that there’s a unique “amalgamation” of this family – that is, one element $x \in S(U)$ which restricts to all the $x_i$ under the maps $S(f_i) : S(U) \rightarrow S(U_i)$.

### Sheaves as Generalized Spaces

There are various ways of looking at sheaves, but for the purposes of the course on categorical methods in geometry, I decided to emphasize the point of view that they are a sort of generalized spaces.

The intuition here is that all the objects and morphisms in a site $\mathcal{T}$ have corresponding objects and morphisms in $Psh(\mathcal{T})$. Namely, the objects appear as the representable presheaves, $U \mapsto Hom(-,U)$, and the morphisms $U \rightarrow V$ show up as the induced natural transformations between these functors. This map $y : \mathcal{T} \rightarrow Psh(\mathcal{T})$ is called the Yoneda embedding. If $\mathcal{T}$ is at all well-behaved (as it is in all the examples we’re interested in here), these presheaves will always be sheaves: the image of $y$ lands in $Sh(\mathcal{T})$.

In this case, the Yoneda embedding embeds $\mathcal{T}$ as a sub-category of $Sh(\mathcal{T})$. What’s more, it’s a full subcategory: all the natural transformations between representable presheaves come from the morphisms of $\mathcal{T}$-objects in a unique way. So  $Sh(\mathcal{T})$ is, in this sense, a generalization of $\mathcal{T}$ itself.

More precisely, it’s the Yoneda lemma which makes sense of all this. The idea is to start with the way ordinary $\mathcal{T}$-objects (from now on, just call them “spaces”) $S$ become presheaves: they become functors which assign to each $U$ the set of all maps into $S$. So the idea is to turn this around, and declare that even non-representable sheaves should have the same interpretation. The Yoneda Lemma makes this a sensible interpretation: it says that, for any presheaf $F \in Psh(\mathcal{T})$, and any $U \in \mathcal{T}$, the set $F(U)$ is naturally isomorphic to $Hom(y(U),F)$: that is, $F(U)$ literally is the collection of morphisms from $U$ (or rather, its image under the Yoneda embedding) and a “generalized space” $F$. (See also Tom Leinster’s nice discussion of the Yoneda Lemma if this isn’t familiar.) We describe $U$ as a “probe” object: one probes the space $F$ by mapping $U$ into it in various ways. Knowing the results for all $U \in \mathcal{T}$ tells you all about the “space” $F$. (Thus, for instance, one can get all the information about the homotopy type of a space if you know all the maps into it from spheres of all dimensions up to homotopy. So spheres are acting as “probes” to reveal things about the space.)

Furthermore, since $Sh(\mathcal{T})$ is a topos, it is often a nicer category than the one you start with. It has limits and colimits, for instance, which the original category might not have. For example, if the kind of spaces you want to generalize are manifolds, one doesn’t have colimits, such as the space you get by gluing together two lines at a point. The sheaf category does. Likewise, the sheaf category has exponentials, and manifolds don’t (at least not without the more involved definitions needed to allow infinite-dimensional manifolds).

These last remarks about manifolds suggest the motivation for the first example…

### Diffeological Spaces

The lecture I gave about sheaves as spaces used this paper by John Baez and Alex Hoffnung about “smooth spaces” (they treat Souriau’s diffeological spaces, and the different but related Chen spaces in the same framework) to illustrate the point. They describe In that case, the objects of the sites are open (or, for Chen spaces, convex) subsets of $\mathbb{R}^n$, for all choices of $n$, the maps are the smooth maps in the usual sense (i.e. the sense to be generalized), and the covers are jointly surjective collections of maps.

Now, that example is a somewhat special situation: they talk about concrete sheaves, on concrete sites, and the resulting categories are only quasitoposes – a slightly weaker condition than being a topos, but one still gets a useful collection of spaces, which among other things include all manifolds. The “concreteness” condition – that $\mathcal{T}$ has a terminal object to play the role of “the point”. Being a concrete sheaf then means that all the “generalized spaces” have an underlying set of points (namely, the set of maps from the point object), and that all morphisms between the spaces are completely determined by what they do to the underlying set of points. This means that the “spaces” really are just sets with some structure.

Now, if the site happens to be $TOP(X)$, then we have a slightly intuition: the “generalized” spaces are something like generalized bundles over $X$, and the “probes” are now sections of such a bundle. A simple example would be an actual sheaf of functions: these are sections of a trivial bundle, since, say, $\mathbb{C}$-valued functions are sections of the bundle $\pi: X \times \mathbb{C} \rightarrow X$. Given a nontrivial bundle $\pi : M \rightarrow X$, there is a sheaf of sections – on each $U$, one gets $F_M(U)$ to be all the one-sided inverses $s : U \rightarrow M$ which are one-sided inverses of $\pi$. For a generic sheaf, we can imagine a sort of “generalized bundle” over $X$.

### Schemes

Another example of the fact that sheaves can be seen as spaces is the category of schemes: these are often described as topological spaces which are themselves equipped with a sheaf of rings. “Scheme” is to algebraic geometry what “manifold” is to differential geometry: a kind of space which looks locally like something classical and familiar. Schemes, in some neighborhood of each point, must resemble varieties – i.e. the locus of zeroes of some algebraic function on $\mathbb{k}^n$. For varieties, the rings attached to neighborhoods are rings of algebraic functions on this locus, which will be a quotient of the ring of polynomials.

But another way to think of schemes is as concrete sheaves on a site whose objects are varieties and whose morphisms are algebraic maps. This is dual to the other point of view, just as thinking of diffeological spaces as sheaves is dual to a viewpoint in which they’re seen as topological spaces equipped with a notion of “smooth function”.

(Some general discussion of this in a talk by Victor Piercey)

### Generalities

These two viewpoints (defining the structure of a space by a class of maps into it, or by a class of maps out of it) in principle give different definitions. To move between them, you really need everything to be concrete: the space has an underlying set, the set of probes is a collection of real set-functions. Likewise, for something like a scheme, you’d need the ring for any open set to be a ring of actual set-functions. In this case, one can move between the two descriptions of the space as long as there is a pre-existing concept of the right kind of function  on the “probe” spaces. Given a smooth space, say, one can define a sheaf of smooth functions on each open set by taking those whose composites with every probe are smooth. Conversely, given something like a scheme, where the structure sheaf is of function rings on each open subspace (i.e. the sheaf is representable), one can define the probes from varieties to be those which give algebraic functions when composed with every function in these rings. Neither of these will work in general: the two approaches define different categories of spaces (in the smooth context, see Andrew Stacey’s comparison of various categories of smooth spaces, defined either by specifying the smooth maps in, or out, or both). But for very concrete situations, they fit together neatly.

The concrete case is therefore nice for getting an intuition for what it means to think of sheaves as spaces. For sheaves which aren’t concrete, morphisms aren’t determined by what they do to the underlying points i.e. the forgetful “underlying set” functor isn’t faithful. Here, we might think of a “generalized space” which looks like two copies of the same topological space: the sheaf gives two different elements of $F(U)$ for each map of underlying sets. We could think of such generalized space as built from sets equipped with extra “stuff” (say, a set consisting of pairs $(x,i) \in X \times \{ blue , green \}$ – so it consists of a “blue” copy of X and a “green” copy of X, but the underlying set functor ignores the colouring.

Still, useful as they may be to get a first handle on this concept of sheaf as generalized space, one shouldn’t rely on these intuitions too much: if $\mathcal{T}$ doesn’t even have a “point” object, there is no underlying set functor at all. Eventually, one simply has to get used to the idea of defining a space by the information revealed by probes.

In the next post, I’ll talk more about this in the context of John Huerta’s guest lecture, applying this idea to the category of supermanifolds, which can be seen as manifolds built internal to the topos of (pre)sheaves on a site whose objects are called “super-points”.