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 𝒞^op 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 : X → Y such that, for any pullback f′ : X′ → Y′ of f : X → Y 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 : X → Y such that pullback along f preserves dominant morphisms in the following sense: for any pullback f′ : X′ → Y′ of f : X → Y 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.

PSA: Effective epimorphisms in higher category theory may not be epimorphisms

In ordinary category theory, coequalisers are always epimorphisms, so effective epimorphisms are indeed epimorphisms. But this is not so in higher category theory.

Indeed, consider the unique morphism from the circle S¹ to the point Δ⁰. This is surjective on connected components, so it is an effective epimorphism in the (∞, 1)-category of spaces. But it is not an epimorphism: indeed, a morphism X → Δ⁰ is an epimorphism in the (∞, 1)-category of spaces if and only if the (unreduced) suspension of X is (weakly) contractible, and the suspension of S¹ is the sphere S², which is certainly not contractible.

On the other hand, there are non-trivial epimorphisms X → Δ⁰ is the (∞, 1)-category of spaces, i.e. there is a space X whose (unreduced) suspension is (weakly) contractible: see here for details.

[Thanks to Aaron Mazel-Gee and Saul Glasman for helping me figure this out.]

When are fibres path-connected?

Question. Let p : E → B be a fibration, let b be a point in B, and let F be the fibre of p over b. Suppose E is path-connected. When is F path-connected?

Answer. Choose any point e in F and consider the long exact sequence of homotopy groups induced by p. Exactness of

π₁(E, e) → π₁(B, b) → π₀(F, e) → π₀(E, e)

implies F is path-connected if and only if the homomorphism π₁(E, e) → π₁(B, b) is surjective, e.g. when B is simply connected.

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As far as answers go, this one is quite neat. But it leaves something to be desired: after all, a geometric question deserves a geometric answer. So let’s unfold the proof to see what’s going on “under the hood” (so to speak).

Choose a point e′ in F. Since E is path-connected, there is a path α from e′ to e. Then p ∘ α is a loop in B based at b. Suppose π₁(E, e) → π₁(B, b) is surjective, i.e. each loop in B based at b is path-homotopic to the image of some loop in E based at e. By replacing α if necessary, we may assume p ∘ α is path-homotopic to the trivial loop at b. Since p is a fibration, the homotopy lifting property implies α is homotopic to a path in F from e′ to e. Since e′ is arbitrary, this shows that F is path-connected.

Conversely, suppose F is path-connected. Let γ be a loop in B based at b. By the homotopy lifting property, there is a path α in E from e′ to e such that p ∘ α is path-homotopic to γ. Since F is path-connected, there is also a path β in F from e to e′, so the composite path αβ is a loop in E based at e. By definition, p ∘ β is the trivial loop based at b, so p ∘ αβ is path-homotopic to γ. Since γ is arbitrary, this shows that π₁(E, e) → π₁(B, b) is surjective.

Are sheaf theorists fascists?

A friend observes: Italian fascismo “fascism” ← Italian fascio ← Latin fascis → French faisceau, “sheaf”, hence sheaf theorists are fascists.

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 F_•C_• where F_–mC₀ is the subgroup of sequences whose first m entries are zero and F_–mC₁ = F_–mC₀ ∩ C₁. This is an exhaustive, separated filtration, albeit unbounded below. It is not hard to see that the inclusions F_{-(m + 1)}C_• → F_–mC_• 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 F_•C_• 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 F_•C_• is separable.