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

- degenerate simplex $s(1)$
- a loop on $1$ (let us conditionally write: $[11]$)
- 1-simplex $[10]$
- 1-simplex $[01]$
- 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?