Category Archives: math.CT Category theory

The localic reflection of the petit étale topos

Recall that the petit étale topos of a scheme X is the topos of sheaves on the category of schemes étale over X (with respect to the étale topology). Since étale morphisms are open, any closed sieve on X must contain the union of the (open) images of all its members. Thus, the closed sieves on X are classified by the open subspaces of X, so the localic reflection of the petit étale topos of X is the petit Zariski topos of X.

Note also that the same argument works for the petit fppf topos of X – the point being that flat morphisms locally of finite presentation are open.

Local objects and projective objects

Let 𝒞 be a category and let J be a subcanonical Grothendieck topology on 𝒞. A J-local object in 𝒞 is an object X in 𝒞 such that the only J-covering sieve on X is the maximal sieve. For example, if 𝒞op is the category of commutative rings and J is the Zariski topology, then the J-local objects are precisely the local rings; or if 𝒞 is the category of T₁-spaces and J is the open cover topology, then J-local objects are precisely the points.

It is easy to see that representable J-sheaves on 𝒞 corresponding to J-local objects in 𝒞 are projective (with respect to sheaf epimorphisms). However, the converse is not true in general; after all, the class of projective objects is closed under coproduct, whereas the class of J-local objects in 𝒞 may fail to be closed under coproduct. For example, if 𝒞 is the category of (all) topological spaces and J is the open cover topology, then any representable sheaf corresponding to a discrete space is projective. In fact, a representable sheaf is projective if and only if every open cover of the corresponding topological space can be refined by a cover of clopen subsets.

So, whereas J-local objects in 𝒞 should be thought of as being point-like, it would seem that objects in 𝒞 such that the corresponding representable sheaf is projective should be thought of as being “highly atomised”, but not necessarily a coproduct of J-local objects. This is one heuristic explanation for why we should not expect sheaf toposes to have enough projectives.

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.

A pathological filtration on a bounded chain complex

Let A be any non-zero abelian group, let C0 be the group of (infinite) sequences of elements of A, and let C1 be the subgroup consisting of those sequences that are almost-everywhere zero. Then C is a bounded chain complex with non-zero homology, concentrated in degree 0; the homology group H0(C) is an example of a filterpower (of A), a generalisation of ultrapowers.

Now, consider the (ascending, integer-indexed) filtration FC where FmC0 is the subgroup of sequences whose first m entries are zero and FmC1 = FmC0C1. This is an exhaustive, separated filtration, albeit unbounded below. It is not hard to see that the inclusions F-(m + 1)CFmC are all quasi-isomorphisms, so the associated graded chain complex must be acyclic.

In particular, the E2 page of the homology spectral sequence associated with FC is zero! But this should not be surprising; after all, the induced filtration on H(C) is trivial. One can make various modifications to this example, but the essential point is that the filtration on H(C) can fail to be separable even if FC is separable.

The free strict monoidal category with k monoids

It is well known that the augmented simplex category is the free strict monoidal category with a monoid: the monoidal product is concatenation, and the monoid is the terminal object (called [0] or Δ0 by geometers). Perhaps less well known is the free strict monoidal category with k monoids (where k is any cardinal, possibly infinite):

  • The objects are pairs (n, P) where n is a natural number (possibly zero) and P is a partition of the set { 0, …, n – 1 } into k disjoint subsets Pj, 0 ≤ j < k.
  • The morphisms (n, P) → (m, Q) are monotone maps { 0, …, n – 1 } → { 0, …, m – 1 } that respect the partitions, i.e. send members of Pj to members of Qj.

In other words, it is the category of finite linear orders equipped k-colouring of its elements. From this point of view, the monoidal product is best interpreted as concatenation of (coloured) linear orders. The k monoids are the k different objects of the form (1, P). Amusingly, the monoidal product on the set of objects makes it the free monoid on k generators.

The opposite category, i.e. the free strict monoidal category with k comonoids, can be described concretely as follows:

  • The objects are pairs (n, P) as before.
  • The morphisms (m, Q) → (n, P) are monotone maps f : { 0, …, m } → { 0, …, n } that satisfy the following axioms:
    • f(0) = 0 and f(m) = n.
    • If i is a member of Pj, then for f(i) ≤ e < f(i + 1), e is a member of Qj .

The above should be interpreted as the category of finite intervals equipped with a k-colouring of its edges. From this point of view, the monoidal product is the wedge sum. The special case where k = 2 is especially useful in relation to the problem of inverting morphisms in a category: for this, see [Dwyer and Kan, Calculating simplicial localizations].

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.