I am new to the theory of simplicial sets. I want to show that geometric realization of a small Category C is naturally homeomorphic to the geometric realization of its dual.
Initially, I tried to show that Nerve of small Category C is naturally isomorphic to the Nerve of its dual. I thought this to be the case, since we can send each n-simplex of the Nerve of C to the corresponding n-simplex of the Nerve of the dual via:
$\phi: (c_0 \to c_1 \to ... \to c_n) \longrightarrow (c_n \to c_{n-1} \to ... \to c_0)$
However, this mapping is not natural since it does not commute with face maps, as one can see from:
$\phi(d_0(c_0 \to c_1 \to ... \to c_n))= \phi (c_1 \to ... \to c_n)= (c_n \to ... \to c_1)$
But:
$d_0 (\phi (c_n \to c_{n-1} \to ... \to c_0))= (c_{n-1} \to ... \to c_0)$
So is there another natural isomorphism? If not, how can one show that the geometric realization of a category is homeomorphic to the realization of its dual?
The nerve functor $N:\mathsf{Cat} \to \mathsf{sSet}$ is fully faithful. In particular, it reflects isomorphisms: hence if there is an isomorphism $\phi:NC\to ND$, then necessarily $C$ and $D$ are isomorphic. So you can't expect your property to hold unless $C$ is isomorphic to its dual already in $\mathsf{Cat}$.
But indeed the geometric realizations will be homeomorphic, because morally the geometric realization "forgets" about the direction of the simplexes. You can refer to this page for more details.