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

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?

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 $$\forall_i A'$$.