Tag Archives: homotopy theory

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 : EB 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.

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 pα 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 pα 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 pα 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, pβ is the trivial loop based at b, so pαβ is path-homotopic to γ. Since γ is arbitrary, this shows that π1(E, e) → π1(B, b) is surjective.

Cisinski trivial fibrations and the subobject classifier

Recall that a Cisinski model structure is a cofibrantly generated model structure on a Grothendieck topos whose cofibrations are the monomorphisms. Accordingly, a Cisinski trivial fibration is a morphism that has the right lifting property with respect to all monomorphisms; note that this makes sense in any category, but we will mainly focus on the case of elementary toposes.

Also recall that a subobject classifier in a category with pullbacks is an object Ω equipped with a natural bijection between morphisms XΩ and isomorphism classes of subobjects of X. It is easy to see that any subobject classifier Ω is an injective object with respect to the class of monomorphisms, and so in a category with finite limits, the projection Ω × XX is a Cisinski trivial fibration (by the stability of right lifting properties under pullback). If, in addition, the category has a strict initial object, then Ω has two disjoint global points and is therefore an interval object. Cisinski calls this the Lawvere cylinder.

Putting these observations together, we may deduce the following:

Proposition. For an elementary topos, inverting Cisinski trivial fibrations is the same thing as quotienting out the congruence of homotopy with respect to the Lawvere cylinder.

In particular, every homotopy equivalence with respect to the Lawvere cylinder is a weak equivalence in any Cisinski model structure. Joyal’s determination principle for model structures implies that a Cisinski model structure is entirely determined by its fibrant objects, and the above proposition implies that the homotopy category of a Cisinski model category is equivalent to a reflective subcategory of the homotopy category with respect to the Lawvere cylinder, where the reflector is induced by fibrant replacement.