What is known about the homotopy type of the classifier of subobjects of simplicial sets?

Question

For the presheaf topos $\mathrm{PSh}(C)$, the subobject classifier is the presheaf $\Omega$ such that

• For $c \in C$, $\Omega(c)$ is the set of all subobjects of the functor $\mathrm{Hom}(-, c)$
• For $f: c \to c'$, $\Omega(f)(F) = \{g: a \to c | gf \in F\}$, where a subfunctor $F < \mathrm{Hom}(-, c')$, and the definite subfunctor $\mathrm{Hom}(-, c)$ we have written as a set of morphisms with arbitrary $\mathrm{dom}$ and fixed $\mathrm{cod}$.

So in $\mathrm{sSet}$ we have:

• 2 points $0$ and $1$ corresponding to the empty and full subsets of $\Delta^0$
• 5 segments corresponding to subsets of $\Delta^1$ ($\Delta^1, \partial \Delta^1, \{0\}, \{1\}, \varnothing$) which are glued respectively as

1. degenerate simplex $s(1)$
2. a loop on $1$ (let us conditionally write: $[11]$)
3. 1-simplex $[10]$
4. 1-simplex $[01]$
5. degenerate simplex $s(0)$

• 19 triangles corresponding to subsets of $\Delta^2$ ..

The sequence of numbers of simplices is called dedekind numbers.

What is known about the homotopy type of this space? Maybe it is contractible? Or is it homotopically equivalent to a well-known space? If not, maybe we can say something about its homotopy groups, cohomology rings, etc?

Answer 1

It’s not hard to check that the subobject classifier $\Omega$ is trivially fibrant, and so its homotopy type is trivial. Orthogonality of $\Omega \to 1$ against a map $i : A \to B$ corresponds to the property that any subobject $A' \to A$ extends to a subobject $B' \to B$ such that $i^*B' = A'$; when $i$ is mono, this is always possible, with canonical solutions given by the direct image $\exists_i A'$ and the dual image $\forall_i A'$.

Nothing here is special to simplicial sets: this argument anpplies equally in any model structure on a topos in which all cofibrations are monomorphisms.

This argument appears (tersely!) in the proof of Theorem 1.4.3 of Cisinski 2006, Les préfaisceaux comme modèles des types d’homotopie (Presheaves as models for homotopy types), 2006 (thanks to Tim Campion in comments for the precise reference). A couple of closely related arguments — firstly the fibrancy of a universe classifying certain fibrations, corresponding to showing that those fibrations extend along trivial cofibrations, and secondly the univalence of this universe — appear in my 2012 paper with Chris Kapulkin, The simplicial model of univalent foundations (after Voevodsky), in Sections 2.1, 2.2, and 3.2.