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April 7, 2014

On a Topological Topos

Posted by Emily Riehl

Guest post by Sean Moss

In this post I shall discuss the paper “On a Topological Topos” by Peter Johnstone. The basic problem is that algebraic topology needs a “convenient category of spaces” in which to work: the category 𝒯\mathcal{T} of topological spaces has few good categorical properties beyond having all small limits and colimits. Ideally we would like a subcategory, containing most spaces of interest, which is at least cartesian closed, so that there is a useful notion of function space for any pair of objects. A popular choice for a “convenient category” is the full subcategory of 𝒯\mathcal{T} consisting of compactly-generated spaces. Another approach is to weaken the notion of topological space, i.e. to embed 𝒯\mathcal{T} into a larger category, hopefully with better categorical properties.

A topos is a category with enough good properties (including cartesian closedness) that it acts like the category of sets. Thus a topos acts like a mathematical universe with ‘sets’, ‘functions’ and its own internal logic for manipulating them. It is exciting to think that if a “convenient topos of spaces” could be found, then its logical aspects could be applied to the study of its objects. The set-like nature of toposes might make it seem unlikely that this can happen. For instance, every topos is balanced, but the category of topological spaces is famously not. However, sheaves (the objects in Grothendieck toposes) originate from geometry and already behave somewhat like generalized spaces.

I shall begin by elaborating on this observation about Grothendieck toposes, and briefly examine some previous attempts at a “topological topos”. I shall explore the idea of replacing open sets with convergent sequences and see how this leads us to Johnstone’s topos. Finally I shall describe how Johnstone’s topos is a good setting for homotopy theory.

I would to thank Emily Riehl for organizing the Kan Extension Seminar and for much useful feedback. Thanks go also to the other seminar participants for being a constant source of interesting perspectives, and to Peter Johnstone, for his advice and for writing this incredible paper.

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)(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 Set\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 nn-sphere into a space. Given a small category 𝒞\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 𝒞\mathcal{C}, which is testable by the ‘classical’ objects of 𝒞\mathcal{C}: it is described entirely by what the maps into it from objects of 𝒞\mathcal{C} ought to be. If we have in mind that 𝒞\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 𝒞\mathcal{C}.

(B) The category of presheaves on a small category 𝒞\mathcal{C} is its free cocompletion. Intuitively, it is the category of objects obtained by formally gluing together objects of 𝒞\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 𝒞\mathcal{C} into its presheaves typically does not preserve any colimits already in 𝒞\mathcal{C}. But if 𝒞\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 𝒞\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 𝒞\mathcal{C} in a way that respects these coverings. This is strongly connected with the preservation of colimits by the embedding of 𝒞\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 𝒞\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 𝒞\mathcal{C} of 𝒯\mathcal{T}, which is closed under open subspaces. The gros topos is the topos of sheaves for this site.

The Yoneda embedding exhibits 𝒞\mathcal{C} as a subcategory, and in fact we can ‘embed’ 𝒯\mathcal{T} via the functor Xhom 𝒯(,X)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 𝒞\mathcal{C}. This turns out not to be useful, since the site does not have enough covers for colimits to agree with those in 𝒯\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 MM to be the full subcategory of 𝒯\mathcal{T} whose only object is the closed unit interval II. So MM is the monoid of continuous endomorphisms of II. Lawvere’s topos \mathcal{L} is the topos of sheaves on MM with respect to the canonical Grothendieck coverage (the largest Grothendieck coverage on MM for which hom M(,I)\hom_M(-,I) is a sheaf).

Then an object XX of \mathcal{L} is a set X(I)X(I) of paths, together with, for any continuous γ:II\gamma\colon I \to I, a reparametrization map X(γ):X(I)X(I)X(\gamma) \colon X(I) \to X(I), where this assignment is functorial. The points of such a space are given by natural transformations 1X1 \to X, i.e. ‘constant paths’ or paths which are fixed by every reparametrization. We can see which point a path visits at time tt by reparametrizing that path by the constant map III \to I with value tt. 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:𝒯P \colon \mathcal{T} \to \mathcal{L} given by Xhom 𝒯(I,X)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 PP 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 II, so it is possible to construct the circle S 1S^1 out of copies of II, but we cannot do this in the usual way. The coequalizer of II by its endpoints in \mathcal{L} is not S 1S^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 𝒯(,I)\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 XX, a convergent sequence in XX is a function from a:{}Xa \colon \mathbb{N}\cup\{\infty\} \to X such that whenever UXU \subseteq X is an open set containing a a_\infty, then there exists an NN such that a nUa_n \in U for all n>Nn \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:XYf\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 YY is a sequential space. Given a topological space XX, A set UXU \subseteq X is sequentially open if for any convergent sequence (a n)(a_n), with a Ua_\infty \in U, (a n)(a_n) is eventually in UU. (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 XX (of points) and family of “convergent sequences”: a specified subset of the set of functions {}X\mathbb{N}\cup\{\infty\} \to X, such that:

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

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 XYX \to Y is a function from the points of XX to the points of YY 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 XX, its sequentially open sets constitute a new (finer) topology; given a subsequential space YY, 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) seqX(X)_\text{seq} \to X, or throw in more convergent sequences - so there is a natural map Y(Y) seqY \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 Σ\Sigma be the full subcategory of 𝒯\mathcal{T} on the objects 11 (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 TT \subseteq \mathbb{N}, let f Tf_T denote the unique order-preserving injection + +\mathbb{N}^+ \to \mathbb{N}^+ whose image is T{}T \cup \{\infty\}. One can check that there is a Grothendieck coverage JJ on Σ\Sigma where 11 is covered by only the maximal sieve, and where +\mathbb{N}^+ is covered by any sieve RR such that:

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

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

The objects in our topos are a slight generalization of subsequential space. If XX \in \mathcal{E}, then X(1)X(1) is its set of points, and X( +)X(\mathbb{N}^+) is its set of convergent sequences. Each point n:1 +n \colon 1 \to \mathbb{N}^+ induces a ‘projection map’ X(n):X( +)X(1)X(n)\colon X(\mathbb{N}^+) \to X(1), giving you the point of the sequence at time nn. The unique map +1\mathbb{N}^+ \to 1 induces a map X(1)X( +)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( +)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 Σ\Sigma. The third axiom is encoded into the coverage. Conversely, any object XX of \mathcal{E} for which the projection maps X(n):X( +)X(1)X(n)\colon X(\mathbb{N}^+) \to X(1), n +n \in \mathbb{N}^+ are jointly injective is isomorphic to one coming from a subsequential space. There is a functor H:𝒯H \colon \mathcal{T} \to \mathcal{E} sending Xhom 𝒯(,X)X \mapsto \hom_\mathcal{T}(-,X), and it is indeed sheaf-valued since it is equal to the composite of the coreflection 𝒯\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 XX is the quotient in 𝒯\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 XX. Compare this with the canonical representation of a presheaf as a colimit of representables (one for each of its elements).


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 𝒯\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 𝒯\mathcal{T} because it is a coreflective subcategory. Surprisingly, the inclusion \mathcal{F} \to \mathcal{E} preserves many of these colimits.

Theorem Let XX be a sequential space, and {U ααA}\{U_\alpha \mid \alpha \in A\} an open cover of XX. Then the obvious colimit diagram in \mathcal{F}: U αU β U α U βU γ U β X U γ \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 JJ-dense, for then it will exhibit XX as the colimit upon reflecting into the topos \mathcal{E}. The colimit LL in presheaves is calculated “objectwise”, so LL has the same points of XX, but only those convergent sequences which are entirely within some U αU_\alpha (hence the comparison map LXL \to X is monic). To sheafify, we need to add in all those sequences xX( +)x \in X(\mathbb{N}^+) which are locally in LL, i.e. for which the sieve {f:? +X(f)(x)L(?)} \{f \colon ? \to \mathbb{N}^+ \mid X(f)(x) \in L(?) \} in Σ\Sigma is JJ-covering. For any xX( +)x \in X(\mathbb{N}^+), this sieve clearly contains all the points 1 +1 \to \mathbb{N}^+. But xx must also be eventually within one of the U αU_\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 Δ\Delta be the category of non-zero finite ordinals and order-preserving maps. Then objects of the presheaf category [Δ op,Set][\Delta^\mathrm{op},\mathrm{Set}] are known as simplicial sets.

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

The closed unit interval [0,1][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 [Δ op,Set]\mathcal{E} \to [\Delta^\mathrm{op},\mathrm{Set}] (an adjunction (f f )(f^\star \dashv f_\star) with f f^\star left-exact).

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

The usual geometric realization is not left-exact if considered to take values in 𝒯\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][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 Σ\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.

Posted at April 7, 2014 5:38 AM UTC

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Re: On a Topological Topos

In our discussions we could hardly ignore Lawvere’s notion of cohesion as a characteristic property of categories of spaces. So it was interesting to notice that this topological topos fails to be cohesive over Set, since the inclusion of discrete spaces does not have a left adjoint (as Tim pointed out, this functor does not preserve infinite products). This could be taken as a first indication that this topos fails to capture the homotopical properties of its objects.

Posted by: Alexander Campbell on April 7, 2014 9:36 AM | Permalink | Reply to this

Re: On a Topological Topos

I wonder if we can salvage some semblance of cohesion? The “discrete spaces” functor doesn’t have a left adjoint, but certain objects do have reflections along it – in particular, any sequential space has a reflection given by taking its connected components. As Sean discussed, Johnstone characterizes the subsequential spaces \mathcal{F}' \subset \mathcal{E} as the separated objects in the double-negation topology, but I think not all of these have a discrete reflection. I wonder if the sequential spaces \mathcal{F} admit some kind of topos-theoretic characterization within \mathcal{E}?

Posted by: Tim Campion on April 7, 2014 1:52 PM | Permalink | Reply to this

Re: On a Topological Topos

Is cohesion really characteristic, or is it an idealisation?

  1. It is not clear to me whether Top\mathbf{Top} has a π 0\pi_0 functor. I am sure we get π 0\pi_0 if we restrict to locally connected spaces, however.
  2. Haus\mathbf{Haus} does not have a “codiscrete” functor, of course. Nor does Mfd\mathbf{Mfd}.
  3. sSet\mathbf{sSet} has all the desired adjoints, but π 0\pi_0 does not preserve powers.

The only category of spaces that I know satisfies all of Lawvere’s axioms is the category of Kan complexes – but then we don’t have finite limits!

Posted by: Zhen Lin on April 7, 2014 2:41 PM | Permalink | Reply to this

Re: On a Topological Topos

I attended a talk today where the speaker cited a topos of topological spaces. For a fun second I thought he might mean Johnstone’s consequential spaces, but he explained that he wanted Kan complexes (if I heard correctly). I was surprised to hear the claim that Kan complexes are a topos. If such an obvious example is present, why the need for Johnstone’s elaborate construction?

But Zhen, you point out that Kan fibrations do not contain finite limits. Is that obvious? Maybe if you take the equalizer of the inclusion of two (n1)(n-1)-hemispheres of a sphere, you get a simplicial set with no nn-simplex filler? But surely that’s only if you take the equalizer in simplicial sets instead of Kan complexes. Can you clarify?

Posted by: Joe Hannon on April 8, 2014 2:57 AM | Permalink | Reply to this

Re: On a Topological Topos

Perhaps the speaker was using the implicit infinity-category convention.

The (full sub)category of Kan complexes is not closed under equalisers, as you say. But I am sure that it simply doesn’t have equalisers. Indeed, Δ 0\Delta^0 is a Kan complex, so equalisers must have the expected vertex set, and the full subcategory of Kan complexes is an exponential ideal of the category of simplicial sets, so exponentiating by Δ n\Delta^n lets us deduce the same about the sets of nn-simplices.

Posted by: Zhen Lin on April 8, 2014 8:28 AM | Permalink | Reply to this

Re: On a Topological Topos

The existence of π 0\pi_0 means that a cohesive category should be thought of not as an arbitrary category of spaces, but as a category of locally connected spaces. In particular, according to observation (B), a cohesive topos should be a category of spaces obtained by gluing together (or “locally modeled on”) locally connected spaces. From this perspective, it is unsurprising that Johnstone’s topos is not cohesive, since +\mathbb{N}^+ is not locally connected.

Posted by: Mike Shulman on April 7, 2014 7:10 PM | Permalink | Reply to this

Re: On a Topological Topos

Oh – it only just struck me that even the discrete space functor fails to preserve infinite products even when taking values in Top\mathbf{Top}. So it doesn’t have a left adjoint of cohesion even in that case (even though it’s not a topos anyway). This despite the fact that the “connected components” of an aribtrary space are well-defined in point-set topology. The map from a totally disconnected space, for instance, to its connected components is not continuous, so it doesn’t form a reflection.

But the discrete space functor from Set\mathbf{Set} to locally connected spaces does have a left adjoint, given by taking connected components. (This implies that products of locally connected spaces are not taken as in Top\mathbf{Top}.) If we take an arbitrary small, full subcategory of the category of locally connected spaces, and look at the category of sheaves in the canonical topology, do we expect to get a cohesive topos?

Is cohesion something that only really applies to gros toposes? Or is it also meaningful for petit toposes? Is the category of sheaves over a locally connected space cohesive?

Posted by: Tim Campion on April 10, 2014 1:46 AM | Permalink | Reply to this

Re: On a Topological Topos

These questions are addressed in Lawvere’s short article Categories of spaces may not be generalized spaces as exemplified by directed graphs. Proposition 1 shows that localic toposes (such as sheaves on a topological space) cannot be “sufficiently cohesive” over Set - meaning they are not cohesive over Set with connected subobject classifier (see Axiomatic cohesion for this definition).

Posted by: Alexander Campbell on April 10, 2014 4:30 AM | Permalink | Reply to this

Re: On a Topological Topos

Is the category of sheaves over a locally connected space cohesive?

Cohesivity is (strong) local connectedness plus locality. Locality says that the topos, considered “petitely”, must be a sort of “fat point”.

products of locally connected spaces are not taken as in Top

Does the category of locally connected spaces even have all products?

the “connected components” of an aribtrary space are well-defined in point-set topology. The map from a totally disconnected space, for instance, to its connected components is not continuous

Indeed: so in general, one should consider the connected components of a non-locally-connected space to be not a set but a space, with the induced quotient topology. Another way to say this is that we can generalize from a left adjoint to a left pro-adjoint, leading to shape theory.

Posted by: Mike Shulman on April 10, 2014 7:01 PM | Permalink | Reply to this

Re: On a Topological Topos

Does the category of locally connected spaces even have all products?

Yes, the category of locally connected spaces is coreflective in TopTop. The coreflector retopologizes a space XX by letting the connected components of open sets of XX generate a new topology. (So, for anyone reading this: to take an infinite product of locally connected spaces, one takes the product in TopTop and then retopologizes that according to this recipe.)

Posted by: Todd Trimble on April 10, 2014 8:18 PM | Permalink | Reply to this

Re: On a Topological Topos

Ah, very nice, thanks. So the category of locally connected spaces is cohesive over Set (though it is not of course a topos).

Posted by: Mike Shulman on April 10, 2014 11:25 PM | Permalink | Reply to this

Re: On a Topological Topos

Is there a “pure logic” way to see that [Δ op,Set][\Delta^{\mathrm{op}}, \mathbf{Set}] is the classifying topos for intervals (= total orders with distinct top and bottom elements, as Sean says)? Does it somehow follow directly from the fact that Δ op\Delta^{\mathrm{op}} is equivalent to the category of finitely presentable intervals in Set\mathbf{Set}? And how is this related to the fact that Δ\Delta is the category of finite inhabited total orders (NOT intervals!) in Set\mathbf{Set}?

On the flip side, what kind of geometric insight is there to be gained by viewing the singular complex / geometric realization adjunction as being generated by an “interval object”?

Posted by: Tim Campion on April 7, 2014 5:01 PM | Permalink | Reply to this

Re: On a Topological Topos

Is there a “pure logic” way to see that [Δ op,Set][\Delta^{\text{op}}, Set] is the classifying topos for intervals (= total orders with distinct top and bottom elements, as Sean says)?

You mean something other than the proof in Moerdijk-Mac Lane p. 457 (Chapter VIII, Section 8)? I’m not sure it’s “pure logic” but it’s not really “pure geometric” either.

Does it somehow follow directly from the fact that Δ op\Delta^{\text{op}} is equivalent to the category of finitely presentable intervals in SetSet?

Unless I’m confused, I think it does in the sense that if for some theory TT, Mod(T)=Hom(Set,[𝒞,Set])Mod(T)=Hom(Set,[\mathcal{C},Set]) (i.e. if TT is of presheaf type) then we can take 𝒞\mathcal{C} to be the full subcategory of finitely presentable objects in Mod(T)Mod(T). So in this case since Mod(T)Mod(T) is the category of intervals, the fact that [Δ op,Set][\Delta^{\text{op}},Set] is the classifying topos for intervals would follow directly from the fact that Δ op\Delta^{\text{op}} is equivalent to the category of finitely presentable intervals. But this assumes a lot of stuff on accessible/presentable categories, so I’m not sure you’d like to say that it follows “directly”. Or perhaps you had something different in mind.

Posted by: Dimitris on April 7, 2014 10:54 PM | Permalink | Reply to this

Re: On a Topological Topos

The proof that in Mac Lane and Moerdijk that Delat op,Set]\Delat^{\mathrm{op}}, \mathbf{Set}] classifies intervals is similar to Johnstone’s proof: both are rather “fiddly”, I’d say: they require really unpacking the terms of Diaconescu’s theorem and proving them bit-by-bit.

The nlab entry at Theories of Presheaf Type claims that the classifying topos B[𝕋]\mathbf{B}[\mathbb{T}] of a geometric theory 𝕋\mathbb{T} is a presheaf category (the theory is “of presheaf type”) if and only if the category Mod(𝕋)\mathrm{Mod}(\mathbb{T}) of models in Set\mathbf{Set} is finitely accessible. The linked paper by Beke refers to Mac Lane and Moerdijk pp. 381-386 for a proof, which is confusing because Mac Lane and Moerdijk don’t discuss accessible categories at all… Beke does prove, following Joyal and Wraith, under slightly stronger hypotheses, that the classifying topos is the presheaf topos on the finitely presentable objects of Mod(𝕋)\mathrm{Mod}(\mathbb{T}) (apparently in general it might be presheaves on some other category, and this can happen for the theory of fields, at least under certain axiomatizations). The proof is short, but I don’t really follow it. It takes the locally finitely presentable case for granted, without citation.

This is exactly the sort of thing I had in mind: some systematic framework into which the fiddly calculations fall. I wish I could follow the details.

Posted by: Tim Campion on April 8, 2014 3:57 AM | Permalink | Reply to this

Re: On a Topological Topos

You won’t find the proof stated in topos-theoretic language, but it is in e.g. [Adámek and Rosický, Proposition 2.26].

But there are subtleties. The category of fields is 0\aleph_0-accessible, hence is the category of models of a theory of presheaf type. This theory is not the coherent theory of fields.

Posted by: Zhen Lin on April 8, 2014 8:35 AM | Permalink | Reply to this

Re: On a Topological Topos

Yes, it was the Beke paper that I had in mind in the argument above.

It takes the locally finitely presentable case for granted, without citation.

The fact that Mod(T)Mod(T) is locally finitely presentable can be seen to follow from e.g. Theorem D.2.3.6 in the Elephant, since TT in this case is a finite limit theory. So you are asking about the fact that Set 𝒞 TSet^{\mathcal{C}_T} is the classifying topos for TT where 𝒞 T\mathcal{C}_T are the fp objects in Mod(T)Mod(T)? I don’t think I’ve ever seen that worked out explicitly but I think it follows from the discussion leading up to Corollary 1.2.5 in Makkai-Pare’s Accessible Categories, if that’s at all helpful.

Also you’re right that there is no mention whatsoever of accessible categories in Moerdijk Mac-Lane. I guess you need both Adamek-Rosicky and Mac Lane-Moerdijk to get Beke’s proposition 0.1. (ii)(iii)(iv)(ii)(ii) \Rightarrow (iii) \Rightarrow (iv) \Rightarrow (ii) is provided by Mac Lane-Moerdijk Theorem 2 [bis] on p. 382 since Flat(𝒞)Flat(\mathcal{C}) is the free filtered cocompletion of C\C and also obviously a category of models for the geometric theory of flat functors CSet\C \rightarrow Set. And then (i)(ii)(i) \Leftrightarrow (ii) follows from the Representation Theorem 2.26 quoted by Zhen.

By the way in their remark after Theorem 2.26, Adamek-Rosicky write that the fact that 𝒞\mathcal{C} can be chosen to be the finitely presentable objects of Mod(T)Mod(T) “generalizes [well not really in this case since we are talking about finite presentability] the well-known fact that each Scott domain is a free completion of its finite elements.” I don’t really understand this, but it sounds like a “pure logic” fact.

Posted by: Dimitris on April 8, 2014 4:40 PM | Permalink | Reply to this

Re: On a Topological Topos

It’s very simple: if 𝒞\mathcal{C} is a κ\kappa-accessible category, then 𝒞\mathcal{C} is the free completion of the full subcategory of κ\kappa-presentable objects under κ\kappa-filtered colimits. Conversely, the free completion of any small category under κ\kappa-filtered colimits is κ\kappa-accessible. The order-theoretic analogue of 0\aleph_0-accessible categories are the so-called “algebraic posets”.

I suppose you can phrase it in logic as a downward Löwenheim–Skolem theorem of some kind. It is quite telling that the proof that the category of models for a small geometric sketch is accessible requires such a theorem.

Posted by: Zhen Lin on April 8, 2014 8:30 PM | Permalink | Reply to this

Re: On a Topological Topos

I understand that finitely accessible categories are the Ind-completions of small categories, which is (i)(iii)(i) \Leftrightarrow (iii), so in theory I should be satisfied with Mac Lane and Moerdijk for establishing Beke’s Proposition 0.1. Having digested this a bit more, I think I’d be inclined to say that there’s very little “topos-theoretic content” here – it’s mostly just playing with accessible categories.

Anyway, as Zhen hinted at, we can conclude from the finite accessibility of the category of Set\mathbf{Set}-intervals that [Δ op,Set][\Delta^{\mathrm{op}}, \mathbf{Set}] classifies a geometric theory whose Set\mathbf{Set}-models are intervals. But in order to identify which geometric theory, we need to do more work, like Johnstone does, or using a more technical theorem like Beke’s Theorem 1.1 and Example 3.3. The way Beke’s theorem works is we let TT be the theory of a partial order with bottom and top elements. This is a Cartesian theory. We let T +T^+ be TT augmented by the axiom x,y.xyyx\forall x, y . x \leq y \vee y \leq x, a theory of intervals. Then because in Set\mathbf{Set}, every interval is a filtered colimit of intervals which are finitely presentable when considered as models of TT, and also there are “enough” models of T +T^+, it follows that T +T^+ is classified by copresheaves on its objects which are finitely presentable when considered as models of TT.

I’m a little concerned about the condition that T +T^+ has “enough Set\mathbf{Set}-models”. Does this just mean that the inverse image functors of the Set\mathbf{Set}-points of its classifying topos B[T +]\mathbf{B}[T^+] are jointly conservative? If so, how do we verify this without constructing B[T +]\mathbf{B}[T^+] (which is, after all, what the the theorem is supposed to do for us)?

Posted by: Tim Campion on April 9, 2014 3:26 AM | Permalink | Reply to this

Re: On a Topological Topos

There are “classical completeness theorems” that assert that the classifying topos of, say, theories in coherent logic automatically have enough points. (This particular one is also known as Deligne’s theorem on coherent toposes.) The theory of intervals is a coherent theory, so its classifying topos, whatever it may be, must have enough points.

Posted by: Zhen Lin on April 9, 2014 8:49 AM | Permalink | Reply to this

Re: On a Topological Topos

As Sean pointed out, an interesting consequence of \mathcal{E}’s being a topos is that it is balanced: any monic epic is an isomorphism. This is kind of interesting. In Top\mathbf{Top}, a map which is the identity on points but goes from a finer topology to a coarser one violates balancedness. How are things different in \mathcal{E}?

The monics XYX \to Y in \mathcal{E}, like in any sheaf topos, are the maps that are pointwise injective as maps of presheaves. So they are the functions which are injective on points (X(1)Y(1)X(1) \to Y(1) is injective) and injective on “proofs of convergence” (X(N +)Y(N +)X(N^+) \to Y(N^+) is injective). In particular, if XX is subsequential, so that a “proof of convergence” is determined by its underlying sequence, then it’s necessary and sufficient that XYX \to Y be injective on points.

The epics are a little more subtle. In a topos, a map f:XYf: X \to Y is epic iff the usual logical formula yx.f(x)=y\forall y \exists x . f(x)=y is valid in the internal logic. I’m a little hazy on how this interpretation works exactly, but I think it should mean that ff is surjective on points, and that any convergent sequence in YY has a subsequence which is image of a convergent sequence in XX. Or rather, this should be said of “proofs of convergence”, but it comes to the same thing if YY is subsequential.

So being an epimorphism in \mathcal{E} is a stronger condition than being an epimorphism in Top\mathbf{Top}, or in \mathcal{F} or \mathcal{F}'. This is because there are more maps out of a given object to exhibit a map as a non-epimorphism. Any surjection is an epimorphism in the category \mathcal{F}' of subsequential spaces, so we need to use all the flexibility that \mathcal{E} provides to get the strengthened epimorphism condition: exhibiting a surjection XYX \to Y as a non-epimorphism in \mathcal{E} by two maps YZY \overset{\to}{\to} Z requires sending some convergence proof in Y(N +)Y(N^+) to two different proofs in Z(N +)Z(N^+); in particular ZZ can’t be subsequential.

As a cogenerator, this means that the subobject classifier Ω\Omega must be very far from subsequential. This is borne out in Johnstone’s description of it: Ω\Omega has two points {t,f}\{t, f\}, and every sequence converges, but for sequences converging to tt there are many, many different proofs.

Does this make it easier or harder to do topology? I’m not sure.

Posted by: Tim Campion on April 9, 2014 3:36 AM | Permalink | Reply to this

Re: On a Topological Topos

Something which hasn’t been mentioned yet (at least, not on this blog post) is that the category of subsequential spaces is, while not a topos, at least a quasitopos. Intuitively, a quasitopos is “a topos that may not be balanced”. Moreover, subsequential spaces are:

  • a Grothendieck quasitopos: the category of presheaves on a small category (here Johnstone’s category Σ\Sigma) that are sheaves for one topology (here, Johnstone’s topology JJ described above) and separated for another topology (here, the topology where +\mathbb{N}^+ is covered by the collection of all points 1 +1\to \mathbb{N}^+).

  • the category of concrete sheaves on a concrete site. This is a special sort of Grothendieck quasitopos, where the second topology consists of the “surjections on global points” in the site.

  • the category of separated objects for the double negation Lawvere-Tierney topology on Johnstone’s topos \mathcal{E}.

A surprising amount of topos theory can also be done for quasitoposes, so if one is bothered by the non-balancedness of Johnstone’s topos, one might consider using subsequential spaces instead.

Posted by: Mike Shulman on April 9, 2014 6:03 AM | Permalink | Reply to this

Re: On a Topological Topos

One answer to the question of why you’d want a topos of spaces is so that you get a cartesian closed category. One thing I was surprised to learn a few years ago is that there is in fact a (unique) closed monoidal structure on the category of all spaces.

This is all worked out very clearly in the obvious chapter of Borceux’s Handbook of categorical algebra volume II. I don’t know whether the results are due to him.

The first theorem is that any closed monoidal structure on all spaces must have the underlying set of the monoidal product the cartesian product of the underlying sets, and it must have the underlying set of the internal hom equal to the set of all continuous functions between the given spaces.

The next result is that there is a unique such structure, with the function spaces given the topology of pointwise convergence. So the monoidal product is not the cartesian product: “asterisks” are open sets.

The “pointwise convergence” appearing here suggests a connection to the Johnstone topos. Can anyone illuminate this?

Posted by: Emily Riehl on April 11, 2014 10:45 PM | Permalink | Reply to this

Re: On a Topological Topos

Pardon me for being dense, but what’s an asterisk, in this context?

Posted by: David Roberts on April 12, 2014 1:54 AM | Permalink | Reply to this

Re: On a Topological Topos

Yeah, I should have spelled this out.

The product is topologized so that f:X×YZf \colon X \times Y \to Z is continuous just when ff is separately continuous in each variable, i.e., when f(x,):YZf(x,-) \colon Y \to Z and f(,y):XZf(-,y) \colon X \to Z is continuous for each point xXx \in X and yYy \in Y. So this is the largest topology so that the maps (x,):YX×Y(x,-) \colon Y \to X\times Y and (,y):XX×Y(-,y) \colon X \to X \times Y are continuous.

Borceux points out that an “asterisk” is then an open neighborhood of the origin in ×\mathbb{R} \times \mathbb{R}; the figure is at the top of page 359. What it looks like is a four-leaf clover whose leaves align with the axes and meet at the origin. The boundary points are not in the open set, with the exception of the origin.

If you look at horizontal or vertical cross sections of the asterisk, you see open intervals in \mathbb{R}, so this is open in the asterisk topology. But of course it’s not open in the usual product topology because this set contains no open ball around the origin.

Posted by: Emily Riehl on April 12, 2014 6:21 PM | Permalink | Reply to this

Re: On a Topological Topos

That’s very interesting! I had never run across that result.

For what it’s worth, the function spaces in Johnstone’s topos are not pointwise convergence; they’re “continuous convergence”. A sequence of functions f nf_n converges continuously to a limit ff if for any sequence x nx_n in the domain converging to a point xx, the images f n(x n)f_n(x_n) converge to f(x)f(x).

Posted by: Mike Shulman on April 12, 2014 6:29 AM | Permalink | Reply to this

Re: On a Topological Topos

Oh, hmm, nevermind then.

Posted by: Emily Riehl on April 12, 2014 6:22 PM | Permalink | Reply to this

Re: On a Topological Topos

This is an interesting notion of convergence: I wonder if it has a standard name?

Sean mentioned in the seminar that the subsequential spaces form an exponential ideal in Johnstone’s topos: if YY is subsequential and XX is arbitrary, then [X,Y][X,Y] is subsequential. I’ve just managed to convince myself of this by unwinding the definitions; apparently it follows directly from the fact that the subsequential spaces are the separated objects for some topology on Johnstone’s topos, i.e. the fact that they form a Grothendieck quasitopos, as Mike mentioned above.

In particular, this means that subsequential spaces form a cartesian closed subcategory of Johnstone’s topos. I’m curious now: do the sequential spaces have similar nice properties? Are they cartesian closed? If not, then are the subsequential spaces somehow a “minimal” cartesian closed supercategory of the sequential spaces?

Posted by: Tim Campion on April 13, 2014 3:23 PM | Permalink | Reply to this

Re: On a Topological Topos

It is the smallest topology with respect to which the evaluation map is (sequentially) continuous, and is closely related to the topology of uniform convergence on compacts. I would hypothesise it is equivalent to uniform convergence on (sequential) compacts when the range is a uniform space.

Posted by: Tom Ellis on April 13, 2014 4:35 PM | Permalink | Reply to this

Re: On a Topological Topos

I think “continuous convergence” is the standard name.

do the sequential spaces have similar nice properties? Are they cartesian closed?

According to the nlab, yes.

Posted by: Mike Shulman on April 14, 2014 3:37 AM | Permalink | Reply to this

Re: On a Topological Topos

As for the attribution: Borceux refers to a paper of Pedicchio and Solomini. In his book, Borceux only proves the existence of a symmetric monoidal closed structure on Top\mathbf{Top}, but not its uniqueness. Pedicchio and Solomini do prove the uniqueness, and in fact they show that Top\mathbf{Top} admits a unique monoidal biclosed structure, symmetric or not. In contrast, they mention that there is a proper class of monoidal closed structures on Top\mathbf{Top} which are not biclosed.

Skimming their introduction, they apparently use a notion of Isbell called a topological topology, which is a topology on the lattice of open sets of a topological space XX which makes arbitrary union and finite intersection into continuous maps. Weird!

Posted by: Tim Campion on April 13, 2014 1:49 AM | Permalink | Reply to this

Re: On a Topological Topos

It seems like Peter Scholze’s work on condensed mathematics is much like top. Compactly generated spaces embed fully faithfully into condensed sets (sheaves over the etale site on a point, covers are jointly surjective collections of maps), not minding certain size constraints they put on the category. I think this is equivalent to sheaves on the category of compact hausdorff spaces.

Posted by: Dean Young on August 9, 2019 4:17 PM | Permalink | Reply to this

Re: On a Topological Topos

Yes, condensed sets and the related pyknotic sets are certainly another approach to this sort of thing. Do they have good behavior relative to colimits like Johnstone’s topos?

Posted by: Mike Shulman on August 10, 2019 10:10 AM | Permalink | Reply to this

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