A Grothendieck topology on $\Delta$

Question

Is there a choice for a Grothendieck topology on $\Delta$ for which most interesting simplicial sets are sheaves (like representables, horns and boundaries, and more generally all categories)? I suspect I can look at the Segal condition as a sheaf condition, but I'm not able to go further, nor I find information googling something similar to my question (which is, I admit it, somewhat vague).

Answer 1

No, there is no such a topology. In fact, the simplex category $\mathbf{\Delta}$ admits only two Grothendieck topologies, namely the indiscrete topology (which is the canonical topology in this case), and the discrete topology; and sheaves, in either case, are not interesting.

The indiscrete topology on $\mathbf{\Delta}$ is given by the maximal sieves, i.e. it is the topology with covering sieves $\{\Delta^n\mid n\in \mathbb{N}\}$. All simplicial sets are indiscrete-sheaves. More generally, all presheaves on any category are indiscrete-sheaves.

Let $\tau$ be any topology on $\mathbf{\Delta}$ that is finer than the indiscrete topology, then there exists $[n]\in \mathbf{\Delta}$ and a $\tau$-covering sieves $S\subsetneq \Delta^n$, and hence $\mathrm{id}_{[n]}\notin S_n$. For every $i\in [n]$, one has $$ {(\sigma^i_{n})}^\ast(S)\subset \Lambda^{n+1}_i \bigcap \Lambda^{n+1}_{i+1} $$ that $\sigma^i_{n} \partial_{n+1}^i=\sigma^i_{n} \partial_{n+1}^{i+1}=\mathrm{id}_{[n]}\notin S_n$. Thus, in particular, $\Lambda^{n+1}_i$ is a $\tau$-covering sieve, for every $i\in [n+1]$. Let $\underline{j}:[0]\to [n+1]$ denote the unique morphism in $\mathbf{\Delta}$ whose image is $\{j\}$, for some $j\in[n+1]$. Then, $$ \underline{j}^\ast (\bigcap_{\substack{i\in [n+1]\\i\neq j}} \Lambda^{n+1}_i)=\underline{j}^\ast (<\partial_{n+1}^{j}>)=\emptyset_{[0]}. $$ Thus, the empty sieve $\emptyset_{[0]}$ is a $\tau$-covering sieve. For every $[m]\in \mathbf{\Delta}$, one has $0_m^\ast \emptyset_{[0]}=\emptyset_{[m]}$, for the terminal morphism $0_m:[m]\to [0]$. Therefore, $\tau$ is the discrete topology on $\mathbf{\Delta}$, i.e. all sieves in $\mathbf{\Delta}$ (including the empty sieves) are $\tau$-covering sieves. The discrete-sheaves are precisely the terminal simplicial sets. Hence, the indiscrete topology is the canonical topology on $\mathbf{\Delta}$.