The following statements are true:\
- if $A,B$ are small categories, then: $\underline{Hom}(N(A),N(B))= N(Fun(A,B))$
- if $A\in sSet$ and $B$ a small category, then $\underline{Hom}(A,N(B))\cong N(Fun(\tau(A),B))$
Notation: $\tau:sSet\to Cat$ is the left adjoint of the nerve $N:Cat\to sSet$.
My progress: Honestly I am not sure what to do with them. I am trying to solve the first one and I think two simplicial sets are equal if the sets we get after applying them to each $[n]\in \Delta$ are the same. So if w elook at the $n=0$ case first, what we need to prove is $$Hom_{sSet}(\Delta^0\times N(A),N(B))=Hom_{sSet}(N(A),N(B))=Hom_{Cat}([0],Fun(A,B))$$
Then if we choose $f\in Hom_{sSet}(N(A),N(B))$ which is a natural transformation between functors $N(A),N(B)$. So when we consider the situation when it acts on $[0],[1]$, we see that there should be maps on objects and morphisms (so maybe higher cases implies the composition? I am not sure!). Then this might give us a functor between $A$ and $B$.
Somehow I am confused with the converse. Also, I feel like the higher cases in general is not that intuitive to understand either. Can anyone help a bit with this?
Best Answer
Here are instructions for the first item. Evaluate each simplicial set at $n$, and use the definition of the internal hom and of the nerve in order to express each side as a hom-set; the left-hand side is a hom-set in $\mathbf{sSet}$ and the right-hand side is a hom-set in $\mathbf Cat$. Compute these hom-sets utilising the left adjoint $\tau$, and its properties. Similar principles work for the second item.