A formal analogy between flat morphisms and open maps

By applying the functional approach to general topology proposed by Clementino, Giuli, and Tholen, we may make a formal analogy between flat morphisms of affine schemes and open maps of topological spaces. First, the set-up:

• We are given a category 𝒞 and a class ℱ of morphisms in 𝒞; for convenience, we assume 𝒞 has pullbacks.
• A proper morphism is a morphism f : XY such that, for any pullback f′ : X′ → Y′ of f : XY along any morphism Y′ → Y, f′ : X′ → Y′ is a member of ℱ.
• A closed embedding is a monomorphism in 𝒞 that is also a proper morphism.
• A dominant morphism is a morphism that is left orthogonal to every closed embedding.
• A open morphism is a morphism f : XY such that pullback along f preserves dominant morphisms in the following sense: for any pullback f′ : X′ → Y′ of f : XY along any morphism Y′ → Y, the pullback of any dominant morphism Y″ → Y′ along f′ : X′ → Y′ is a dominant morphism X″ → X′.

If we take 𝒞 to be the category of topological spaces and ℱ to be the class of closed maps (i.e. continuous maps such that the image of every closed subset is again a closed subset), then these definitions agree with the classical ones. On the other hand, we could take 𝒞 to be the category of affine schemes and ℱ to be the class of closed immersions, in which case the dominant morphisms correspond to injective homomorphisms of commutative rings.

Unfortunately, the analogy ends there: although open immersions of schemes are flat monomorphisms, they are also required to be locally of finite presentation, which is not automatic. There are also some subtleties when trying to apply these definitions to the category of all schemes: although a dominant morphism in the sense above is a morphism whose scheme-theoretic image is the whole codomain, the connection between open morphisms in the sense above and flat morphisms of schemes is much less obvious.

The standard model structure on Cat is canonical

When I previously posted about a model structure on Cat, I called the usual (categorical equivalence, injective-on-objects, isofibration) model structure the “standard model structure”. The nLab calls it the canonical model structure on Cat, but I dislike that name because it always seems to suggest that there is some mechanical procedure for constructing it. As it turns out – there is!

Recall that Rezk’s classifying diagram for a (small) category 𝒞 is the bisimplicial set (= simplicial simplicial set) defined by N(𝒞)n, m = Fun([n] × I[m], 𝒞), where [n] is the standard n-simplex in Cat and I is the groupoid completion functor (= left adjoint of the inclusion GrpdCat). Rezk [2001] has shown that N : CatssSet is fully faithful and is homotopically conservative in the sense of sending categorical equivalences to degreewise weak homotopy equivalences and reflecting degreewise weak homotopy equivalences as categorical equivalences. As is usual with presheaf categories, N has a left adjoint, namely the colimit-preserving functor τ1 : ssSetCat that sends the bisimplicial set representing [n, m] to the category [n] × I[m].

I claim that the standard model structure on Cat is the model structure obtained by transferring the projective model structure on ssSet (i.e. degreewise weak homotopy equivalences and degreewise Kan fibrations) along the left adjoint τ1 : ssSetCat. We have already remarked that N preserves and reflects weak equivalences, so it is enough to show that N also preserves and reflects fibrations. This is fairly straightforward.

The Morita model structure on Cat

There is a well-known model structure on Cat where the weak equivalences are the categorical equivalences, i.e. the functors that are fully faithful and essentially surjective on objects, the cofibrations are the functors that are injective on objects, and the fibrations are the isofibrations, i.e. the functors that lift isomorphisms. Let us say that a functor f : 𝒞 → 𝒟 is a Morita equivalence if the induced functor f* : [𝒟op, Set] → [𝒞op, Set] is a categorical equivalence. Clearly, every categorical equivalence is a Morita equivalence. Does the left Bousfield localisation of Cat with respect to Morita equivalences exist?

The standard model structure on Cat is combinatorial and simplicial, and all objects are cofibrant, so the model structure is also left proper. Thus, we may apply Smith’s theorem on the existence of left Bousfield localisations. It is a straightforward exercise to verify that left Bousfield localisation with respect to the inclusion of the free idempotent into the free split idempotent gives the desired model structure: the local objects are the Cauchy-complete categories, so the local equivalences are the Morita equivalences. (That Morita equivalences are local equivalences is easy; for the converse, consider a small category of sufficiently large sets.)

But one could (in principle) also establish the existence of the Morita model structure on Cat by hand: the fibrations are the isofibrations that also lift splittings of idempotents, i.e. the isofibrations with the right lifting property with respect to the inclusion mentioned above. Thus the Morita model structure on Cat is generated by the same cofibrations as the standard model structure on Cat and just one extra trivial cofibration.

Local lifting properties

For various reasons (one of which was adumbrated in my previous post), I have been thinking about generalising the classical notion of ‘weak factorisation system’ to work better in the intuitionistic context.

The difficulty, really, is in how we understand existence. Let us consider a pair of morphisms, say f : AB and g : CD, in an elementary topos 𝒮. What does it mean to say that f has the left lifting property with respect to g in the internal logic of 𝒮? That is, suppose when given morphisms h : AC, k : BD, if

• for all a in A, k(f(a)) = g(h(a)),

then there exists a morphism : BC such that

• for all a in A, (f(a)) = h(a); and
• for all b in B, g((b)) = k(b).

Interpreted naïvely, this says that f has the left lifting property with respect to g in the ordinary sense. Nonetheless, in the internal logic of 𝒮, the above is true when f is the unique morphism 0 → 1 and g is any epimorphism CD – exactly as in the archetypical topos Set. On the other hand, it is simply not true that epimorphisms induce surjections of global sections! What the above really says is the following: a lifting exists – after base change along an epimorphism!

In light of the preceding discussion, and also bearing in mind the work of Brown, Jardine, Joyal et al. on the homotopy theory of sheaves of simplicial sets, it would appear that the right definition of left/right lifting property in the relative context must be local, in the sense of the following definition.

Definition. Let 𝒮 be a regular category, let ℂ be an 𝒮-indexed category, let J be an object in 𝒮, and let 𝒞J be the fibre of ℂ over J. Given morphisms f and g in 𝒞J, we say f has the local left lifting property with respect to g if, for all morphisms x : IJ in 𝒮 and all morphisms h and k in 𝒞I such that kx*f = x*gh, there exist a regular epimorphism e : EI and a morphism in 𝒞E such that e*x*f = e*h and e*x*g = e*k.

It is easy to verify that the class of morphisms in 𝒞J that have the local left lifting property with respect to g is closed under composition and retracts. It is also closed under finite coproducts and pushouts, provided we restrict attention to those colimits that are stable under reindexing. The distinguishing feature is the descent property: if x : IJ is any regular epimorphism in 𝒮 and f is a morphism in 𝒞J, then f has the local left lifting property with respect to g if and only if x*f has the local left lifting property with respect to x*g.

Whether or not the class is closed under transfinite composition and 𝒮-indexed coproducts seems rather more tricky. If our metatheory has the axiom of choice, then the class is indeed closed under reindexing-stable transfinite composition; the same can be said for reindexing-stable infinite coproducts. It should therefore be unsurprising that there is a connection between the internal axiom of choice in 𝒮 and the question of whether or not the class is closed under 𝒮-indexed coproducts.

Indeed, suppose 𝒮 is an elementary topos, and consider the self-indexing of 𝒮. We earlier observed that 0 → 1 has the local left lifting property with respect to epimorphisms. Thus, if the class of morphisms with the local left lifting property with respect to epimorphisms is closed under 𝒮-indexed coproducts, then an epimorphism g : CD must have the local right lifting property with respect to the unique morphism 0 → D; but this happens if and only if the morphism [D, g] : [D, C] → [D, D] is an epimorphism, i.e. if and only if g : CD has a section in the internal logic of 𝒮.

So, should we accept the definition of ‘local lifting property’ proposed above, and with it, the failure of the basic saturation properties of classes of morphisms with a left/right lifting property? Or should we continue searching for a definition that allows us to carry over more of the classical theory?

Weak factorisation systems and the axiom of choice

Recall that a trivial Kan fibration is a morphism in sSet that has the right lifting property with respect to the boundary inclusions ∂Δn ↪ Δn for all n ≥ 0. The following is well-known:

Proposition. A morphism in sSet has the left lifting property with respect to all trivial Kan fibrations if and only if it is a monomorphism.

The standard proof uses the fact that monomorphisms in sSet admit a cellular decomposition, and the fact that the class of morphisms that have the left lifting property with respect to another class is closed under transfinite composition, pushouts, and retracts.

Yet, the proposition is equivalent to the axiom of choice! Indeed, let ∇ : SetsSet be the right adjoint of the functor XX0. (∇ is also known as cosk0.) It is clear that, if f : XY is a surjection, then ∇f : ∇X → ∇Y is a trivial Kan fibration: indeed, f : XY is a surjection if and only if ∇f : ∇X → ∇Y has the right lifting property with respect to ∂Δ0 ↪ Δ0, and for any map f : XY, ∇f : ∇X → ∇Y has the unique right lifting property with respect to ∂Δn ↪ Δn for all n ≥ 1. But 0 → ∇Y is a monomorphism, so if the proposition holds, then ∇f : ∇X → ∇Y must have a section; and ∇ : SetsSet is fully faithful, so f : XY has a section if and only if ∇f : ∇X → ∇Y has a section.

A few minutes thinking should reveal where we use the axiom of choice in the proof of the proposition. While it is true that every monomorphism in sSet is a relative cell complex (provided you allow arbitrarily many cells to be attached at each stage), it would appear that we need the axiom of choice to deduce that every relative cell complex is in the class of morphisms generated by the boundary inclusions under transfinite compositions and pushouts: after all, we have to choose the order in which we attach the cells!

But that is not the end of the story: if that were the only problem, then we could deduce that trivial Kan fibrations have the right lifting property with respect to monomorphisms in sSet whose codomain is well-orderable. This is not obviously true either – in fact, it seems that we need the axiom of choice to show that the class of morphisms with the left lifting property with respect to another class is closed under coproducts and transfinite composition!

So perhaps the moral of the story is that the theory of cofibrantly-generated model categories is non-constructive in subtle, easily missed ways.