Category Archives: Uncategorized

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.

The number of abstract simplicial complexes

On MathOverflow, François G. Dorais pointed out that the partially ordered set of abstract simplicial complexes with vertices drawn from the set { 0, …, n } is a free distributive lattice (with free top element but without free bottom element). This is not hard to see in hindsight: the generators are the (n – 1)-simplices. This poset is of interest to topos theorists because it is precisely the poset of simplicial subsets of the standard n-simplex; hence, we obtain a nice description of the subobject classifier of the topos of simplicial sets.

Amusingly, this observation implies it is very difficult to count the number of abstract simplicial complexes: apparently, the number of elements in a free distributive lattice is not known in general. This is Dedekind’s problem, and the known numbers are enumerated as OEIS sequence A000372.

When is the realisation functor conservative?

Let 𝒞 be a cocomplete category and let U : ℐ → 𝒞 be a small diagram. Then there is an induced realisation–nerve adjunction,

RN : 𝒞 → [ℐop, Set]

where N(C) is the presheaf 𝒞(U–, C) and R is defined by left Kan extension. The comonad induced by the realisation–nerve adjunction is precisely the density comonad induced by U, so it is natural to ask, when is R comonadic?

Beck’s monadicity theorem implies R is comonadic if and only if R is conservative and preserves certain equaliser diagrams in [ℐop, Set], and Thomas Athorne described some sufficient conditions in his thesis. We discuss here his condition for conservativity.

Let us say that U satisfies the repeated element condition if the components of the adjunction unit are monomorphisms in [ℐop, Set]. Unravelling the definitions, this amounts to saying R reflects equality between presheaf morphisms with representable domain.

First, suppose R is conservative. Since [ℐop, Set] has coequalisers, R must also be faithful. Thus, R reflects monomorphisms, and in particular, U satisfies the repeated element condition.

Conversely, suppose U satisfies the repeated element condition. By general nonsense about adjunctions, this happens if and only if R is faithful. Since [ℐop, Set] is a balanced category, to verify that R reflects isomorphisms, it is enough to show that R reflects epimorphisms. This is easily done by considering cokernel pairs.

We may therefore conclude that the following are equivalent:

  1. U satisfies the repeated element condition.
  2. R reflects monomorphisms.
  3. R is faithful.
  4. R is conservative.