The n-Category Café

Are there toposes of spaces?

We shall need to be flexible about what we mean by “space”. For the rest of this post I shall try to use the term “topological space” in a strict technical sense (a set of points plus specified open sets), whereas “space” will be a nebulous concept. The idea is that spaces have existence regardless of having been implemented as a topological space or not, and may naturally have more (or perhaps less) structure. Topology merely forms one setting for their rigorous study. In topology we can only detect the topological properties of spaces. For example, $\mathbb{R}$ and $(0,1)$ are isomorphic as topological spaces, but they are far from being the same space: consider how different their implementations as, say, metric spaces are. Some spaces are naturally considered as having algebraic or smooth structure. The type of question one wishes to ask about a space will bear upon the type of object as which it should be implemented.

An extremely important class of toposes consists of the Grothendieck toposes, which are categories of sheaves on a site. A site is a small category together with a Grothendieck coverage (also known as a Grothendieck topology). Informally, the Grothendieck coverage tells us how some objects can be “covered” by maps coming out of other objects. In the special case where the site is a topological space, the objects are open sets and the coverage tells us that an open set is covered by the inclusions of any family of open sets whose union is all of that open set. A sheaf on a site is then a contravariant $\mathrm{Set}$-valued functor on the underlying category (a presheaf) which satisfies a “unique patching” condition with respect to each covering sieve.

In the following two senses, a Grothendieck topos always behaves like a category of spaces:

(A) One way to describe the properties of a space is to consider the maps into that space. This is the idea behind the homotopy groups, where we consider (homotopy classes of) maps from the $n$-sphere into a space. Given a small category $\mathcal{C}$, each object is determined by knowing all the arrows into it and how these arrows “restrict” along other arrows, i.e. precisely the data of the representable presheaf. A non-representable presheaf can be viewed as a generalized object of $\mathcal{C}$, which is testable by the ‘classical’ objects of $\mathcal{C}$: it is described entirely by what the maps into it from objects of $\mathcal{C}$ ought to be. If we have in mind that $\mathcal{C}$ is some category of spaces, with some sense in which some spaces are covered by families of maps out of other spaces, (i.e. we have a Grothendieck coverage), then we should be able to patch maps into these generalized spaces together. So the topos of sheaves on this site is a setting in which we may be able to implement certain spaces, if we wish to study their properties testable by objects of $\mathcal{C}$.

(B) The category of presheaves on a small category $\mathcal{C}$ is its free cocompletion. Intuitively, it is the category of objects obtained by formally gluing together objects of $\mathcal{C}$. The use of the word “gluing” is itself a spatial metaphor. CW-complexes are built out of gluing together cells - simplicial sets are instructions for carrying out this gluing. Manifolds are built from gluing together open subsets of Euclidean space. Purely formal ‘gluing’ is not quite sufficient: the Yoneda embedding of $\mathcal{C}$ into its presheaves typically does not preserve any colimits already in $\mathcal{C}$. But if $\mathcal{C}$ is a category of spaces, its objects are not neutral with respect to each other: there may be a suitable Grothendieck coverage on $\mathcal{C}$ which tells us how some objects can cover others. The topos of sheaves is then the category of objects obtained by formally gluing objects of $\mathcal{C}$ in a way that respects these coverings. This is strongly connected with the preservation of colimits by the embedding of $\mathcal{C}$ into the sheaves. Colimits in the presheaf topos are constructed pointwise; to get the sheaf colimit one applies the reflection into the category of sheaves (“sheafification”) to the presheaf colimit. The more covers imposed on $\mathcal{C}$, the more work is done by the sheafification, so the closer we end up to the original colimit.

Are there toposes in topology?

It is far from clear that we can choose a site for which the space-like behaviour of sheaves accords with the usual topological intuition. If we want to use a topological topos for homotopy theory, then ideally it should contain objects that we can recognize as the CW-complexes, and we should be able to construct them via more or less the usual colimits.

Attempt 1: The “gros topos” of Giraud

The idea is to take sheaves on the ‘site’ of topological spaces, where covers are given by families of open inclusions of subspaces whose union is the whole space. We do not automatically get a topos unless the site is small, so instead take some small, full subcategory $\mathcal{C}$ of $\mathcal{T}$, which is closed under open subspaces. The gros topos is the topos of sheaves for this site.

The Yoneda embedding exhibits $\mathcal{C}$ as a subcategory, and in fact we can ‘embed’ $\mathcal{T}$ via the functor $X \mapsto \hom_\mathcal{T}(-,X)$, this will be full and faithful on a fairly large subcategory. By (B) one may like to consider the gros topos as the category of spaces glued together from objects of $\mathcal{C}$. This turns out not to be useful, since the site does not have enough covers for colimits to agree with those in $\mathcal{T}$. Moreover the site is so large that calculations are difficult.

Attempt 2: Lawvere’s topos

We use observation (A). Motivated by the use of paths in homotopy theory, we take $M$ to be the full subcategory of $\mathcal{T}$ whose only object is the closed unit interval $I$. So $M$ is the monoid of continuous endomorphisms of $I$. Lawvere’s topos $\mathcal{L}$ is the topos of sheaves on $M$ with respect to the canonical Grothendieck coverage (the largest Grothendieck coverage on $M$ for which $\hom_M(-,I)$ is a sheaf).

Then an object $X$ of $\mathcal{L}$ is a set $X(I)$ of paths, together with, for any continuous $\gamma\colon I \to I$, a reparametrization map $X(\gamma) \colon X(I) \to X(I)$, where this assignment is functorial. The points of such a space are given by natural transformations $1 \to X$, i.e. ‘constant paths’ or paths which are fixed by every reparametrization. We can see which point a path visits at time $t$ by reparametrizing that path by the constant map $I \to I$ with value $t$. A word of caution: a given object in $\mathcal{L}$ may have distinct paths which agree on points for all time.

This site is much easier to calculate with than the gros site (once we have a handle on the canonical coverage). Again there is a functor $P \colon \mathcal{T} \to \mathcal{L}$ given by $X \mapsto \hom_\mathcal{T}(I,X)$, which is full and faithful on a fairly large subcategory (including CW-complexes). However, it is still the case that the site could do with more covers: the functor $P$ does not preserve all the colimits used to build up CW-complexes. By observation (B), an object of $\mathcal{L}$ is obtained by gluing together copies of the unit interval $I$, so it is possible to construct the circle $S^1$ out of copies of $I$, but we cannot do this in the usual way. The coequalizer of $I$ by its endpoints in $\mathcal{L}$ is not $S^1$, but a “signet-ring”: it is a circle with a ‘lump’, through which a path can cross only if waits there for non-zero time. We cannot solve this problem by adding in more covers, because the coverage is already canonical (adding in more covers will evict the representable $\hom_\mathcal{T}(-,I)$ from the topos).

The key idea in Johnstone’s topos is to replace paths with convergent sequences. Given a topological space $X$, a convergent sequence in $X$ is a function from $a \colon \mathbb{N}\cup\{\infty\} \to X$ such that whenever $U \subseteq X$ is an open set containing $a_\infty$, then there exists an $N$ such that $a_n \in U$ for all $n \gt N$. The convergent sequences are precisely the continuous maps out of $\mathbb{N}\cup\{\infty\}$ when we give it the topology that makes it the one-point compactification of the discrete space $\mathbb{N}$ - we denote this topological space by $\mathbb{N}^+$.

Convergent sequences as primitive

It is a basic theorem in general topology that, given a function $f\colon X \to Y$ between topological spaces, if it is continuous then it preserves convergent sequences. The converse is not true for general topological spaces, but it is true whenever $Y$ is a sequential space. Given a topological space $X$, A set $U \subseteq X$ is sequentially open if for any convergent sequence $(a_n)$, with $a_\infty \in U$, $(a_n)$ is eventually in $U$. (Clearly any open subset is sequentially open.) A topological space is then said to be sequential if all of its sequentially open sets are open. The sequential spaces include all first-countable spaces and in fact they can be characterized as the topological quotients of metrizable spaces, so they certainly include all CW-complexes.

The notion of convergent sequence is arguably more intuitive than that of open set. For example, each convergent sequence gives you concrete data about the nearness of some family of points to another point, whereas open sets only give you such data when the topology (or at least a neighbourhood basis) is considered as a whole. It would be compelling to define a continuous function as one that preserves convergent sequences. This motivates the study of subsequential spaces.

A subsequential space consists of a set $X$ (of points) and family of “convergent sequences”: a specified subset of the set of functions $\mathbb{N}\cup\{\infty\} \to X$, such that:

1. for every point $x \in X$, the constant sequence $(x)$ converges to $x$;
2. if $(x_n)$ converges to $x$, then so does every subsequence of $(x_n)$;
3. if $(x_n)$ is a sequence and $x$ is a point such that every subsequence of $(x_n)$ contains a (sub)subsequence converging to $x$, then $(x_n)$ converges to $x$.

The third axiom is the general form of intertwining two or more sequences with the same limit or changing a finite initial segment of a sequence. Note that there is no ‘Hausdorff‘-style condition on the convergent sequences: a sequence may converge to more than one limit. A continuous map between subsequential spaces $X \to Y$ is a function from the points of $X$ to the points of $Y$ that preserves convergence of sequences.

The axioms above are all true of the set of convergent sequences which arise from a topology on a set. In fact, this process gives a full and faithful embedding of sequential spaces into subsequential spaces. Thus sequential spaces live inside both topological and subsequential spaces. They are coreflective in the former and reflective in the latter: given a topological space $X$, its sequentially open sets constitute a new (finer) topology; given a subsequential space $Y$, we can consider the “sequentially open” sets with respect to its convergent sequences, and then take all convergent sequences in the resulting (sequential) topological space. Observe that the sense of the adjunction in each case comes from the fact that we either throw in more open sets - so there is a natural map $(X)_\text{seq} \to X$, or throw in more convergent sequences - so there is a natural map $Y \to (Y)_\text{seq}$.

In the following I shall denote the category of sequential spaces by $\mathcal{F}$ and that of subsequential spaces by $\mathcal{F}'$.

Johnstone’s topos

Let $\Sigma$ be the full subcategory of $\mathcal{T}$ on the objects $1$ (the singleton space) and $\mathbb{N}^+$ (the one-point compactification of the discrete space of natural numbers). The arrows in this category can be described without topology as well: as functions, the maps $\mathbb{N}^+ \to \mathbb{N}^+$ are the eventually constant ones and the ones that “tend to infinity”.

Given an infinite subset $T \subseteq \mathbb{N}$, let $f_T$ denote the unique order-preserving injection $\mathbb{N}^+ \to \mathbb{N}^+$ whose image is $T \cup \{\infty\}$. One can check that there is a Grothendieck coverage $J$ on $\Sigma$ where $1$ is covered by only the maximal sieve, and where $\mathbb{N}^+$ is covered by any sieve $R$ such that:

1. $R$ contains all of the points $n \colon 1 \to \mathbb{N}^+$, $n \in \mathbb{N}^+$.
2. For any infinite subset $T \subseteq \mathbb{N}$ there exists an infinite subset $T' \subseteq T$ such that $f_{T'} \in R$.

The topos $\mathcal{E}$ is then defined to be $\mathrm{Sh}(\Sigma,J)$.

The objects in our topos are a slight generalization of subsequential space. If $X \in \mathcal{E}$, then $X(1)$ is its set of points, and $X(\mathbb{N}^+)$ is its set of convergent sequences. Each point $n \colon 1 \to \mathbb{N}^+$ induces a ‘projection map’ $X(n)\colon X(\mathbb{N}^+) \to X(1)$, giving you the point of the sequence at time $n$. The unique map $\mathbb{N}^+ \to 1$ induces a map $X(1) \to X(\mathbb{N}^+)$, which sends each point to a canonical choice of constant sequence. Note that there may be more than one convergent sequence with the same points, thus it may be helpful to think of $X(\mathbb{N}^+)$ as the set of proofs of convergence for sequences.

Clearly we can embed $\mathcal{F}'$, the subsequential spaces, into $\mathcal{E}$: the points are the same, and the convergence proofs are just the convergent sequences. The first two axioms are satisfied because of the equations that hold in $\Sigma$. The third axiom is encoded into the coverage. Conversely, any object $X$ of $\mathcal{E}$ for which the projection maps $X(n)\colon X(\mathbb{N}^+) \to X(1)$, $n \in \mathbb{N}^+$ are jointly injective is isomorphic to one coming from a subsequential space. There is a functor $H \colon \mathcal{T} \to \mathcal{E}$ sending $X \mapsto \hom_\mathcal{T}(-,X)$, and it is indeed sheaf-valued since it is equal to the composite of the coreflection $\mathcal{T} \to \mathcal{F}$ with the inclusions $\mathcal{F} \to \mathcal{F}' \to \mathcal{E}$. In fact, the Grothendieck coverage defining $\mathcal{E}$ is canonical, so it is the largest for which this functor is well-defined.

We can use observation (B) to think of $\mathcal{E}$ as all spaces constructed from gluing sequences together. It is just about possible that we could have motivated the construction of $\mathcal{E}$ this way: classically, any sequential space $X$ is the quotient in $\mathcal{T}$ of a metrizable space, which may be taken to be a disjoint union of copies of $\mathbb{N}^+$ - one for every convergent sequence in $X$. Compare this with the canonical representation of a presheaf as a colimit of representables (one for each of its elements).

Colimits

It turns out that $\mathcal{F}'$ is the subcategory of $\neg\neg$-separated objects in $\mathcal{E}$, hence it is a reflective subcategory. $\mathcal{F}$ is reflective in $\mathcal{F}'$, hence it is also reflective in $\mathcal{E}$. In particular, all limits in $\mathcal{F}$ are preserved by the inclusion into $\mathcal{E}$. Take some caution, however, since products do not agree with those in $\mathcal{T}$: one has to take the sequential coreflection of the topological product. This is only a minor issue; having to modify the product arises in other “convenient categories” such as compactly-generated spaces.

The colimits in $\mathcal{F}$ do agree with those in $\mathcal{T}$ because it is a coreflective subcategory. Surprisingly, the inclusion $\mathcal{F} \to \mathcal{E}$ preserves many of these colimits.

Theorem Let $X$ be a sequential space, and $\{U_\alpha \mid \alpha \in A\}$ an open cover of $X$. Then the obvious colimit diagram in $\mathcal{F}$: $$\begin{matrix} U_\alpha\cap U_\beta & \rightarrow & U_\alpha & & \\ & \searrow & & \searrow & \\ U_\beta \cap U_\gamma & \rightarrow & U_\beta & \rightarrow & X \\ \vdots & \searrow & & \nearrow & \\ & & U_\gamma & & \\ & & \vdots & & \end{matrix}$$ is preserved by the embedding $\mathcal{F} \to \mathcal{E}$.

Proof The recipe for this sort of theorem is: take the colimit in presheaves, show that the comparison map is monic, then show that it is $J$-dense, for then it will exhibit $X$ as the colimit upon reflecting into the topos $\mathcal{E}$. The colimit $L$ in presheaves is calculated “objectwise”, so $L$ has the same points of $X$, but only those convergent sequences which are entirely within some $U_\alpha$ (hence the comparison map $L \to X$ is monic). To sheafify, we need to add in all those sequences $x \in X(\mathbb{N}^+)$ which are locally in $L$, i.e. for which the sieve $$\{f \colon ? \to \mathbb{N}^+ \mid X(f)(x) \in L(?) \}$$ in $\Sigma$ is $J$-covering. For any $x \in X(\mathbb{N}^+)$, this sieve clearly contains all the points $1 \to \mathbb{N}^+$. But $x$ must also be eventually within one of the $U_\alpha$, so the second condition for the covering sieves is also satisfied. $\square$

There are several other colimit preservation results one can talk about (with similar proofs to the above). The amazing consequence of these is that the colimits used to construct CW-complexes are all preserved by the embedding $\mathcal{F} \to \mathcal{E}$. Thus classical homotopy theory embeds into $\mathcal{E}$ and we have successfully found a topos of spaces which agrees with the classical theory.

Geometric realization

Let $\Delta$ be the category of non-zero finite ordinals and order-preserving maps. Then objects of the presheaf category $[\Delta^\mathrm{op},\mathrm{Set}]$ are known as simplicial sets.

Theorem $[\Delta^\mathrm{op},\mathrm{Set}]$ is the classifying topos for intervals in $\mathrm{Set}$-toposes.

The closed unit interval $[0,1]$ is sequential and is in fact an interval (a totally ordered object with distinct top and bottom elements). Thus it corresponds to a geometric morphism $\mathcal{E} \to [\Delta^\mathrm{op},\mathrm{Set}]$ (an adjunction $(f^\star \dashv f_\star)$ with $f^\star$ left-exact).

Theorem If $S \in [\Delta^\mathrm{op},\mathrm{Set}]$ is a simplicial set, then $f^\star(S)$ is its geometric realization, considered as a sequential space and hence as an object of $\mathcal{E}$. If $X \in \mathcal{E}$ is a sequential space, then $f_\star(E)$ is its singular complex.

The usual geometric realization is not left-exact if considered to take values in $\mathcal{T}$, one must choose a “convenient subcategory” first, and then there is some work to do in proving it. Here the left-exactness just arises out of the general theory of geometric morphisms. Should we wish to do so, the above method allows us to replace $[0,1]$ with any other object that the internal logic of $\mathcal{E}$ sees as an interval to get a different realization of simplicial sets.

The above is far from a complete survey of “On a Topological Topos”, which contains several more results of interest relating to $\mathcal{E}$ and captures the elegance of using the site $\Sigma$ for calculation - I thoroughly recommend taking a look if you know some topos theory. We have seen enough though to understand that for many spaces the sequential properties align with the topological properties. Unfortunately, $\mathcal{E}$ is yet to receive the attention it deserves.