It is known that for any small category $\mathcal{C}$ the presheaf category $[\mathcal{C}^{op}, \mathsf{Set}]$ forms a topos. What about the category of simplicial presheaves, i.e. $[\Delta^{op},[\mathcal{C}^{op}, \mathsf{Set}]]$? I would guess there is no difficulty to view such categories as ordinary presheaves due to the general isomorphism of functor categories $[\mathcal{A},[\mathcal{B}, \mathcal{C}]]\cong[\mathcal{A}\otimes \mathcal{B}, \mathcal{C}]$ (G.M.Kelly, Basic Concepts Of Enriched Category Theory, p.31), but I am not sure.
Is the category of simplicial presheaves an example of a topos
topos-theory
Best Answer
Your guess is good.
Actually, this is even true for sheaves: the category of functors to a Grothendieck topos $D^{op} \to Sh_J(C)$ for any Grothendieck topology $J$ on $C$ is again a Grothendieck topos - it is equivalent to $Sh_{J'}(C \times D)$ for certain topology $J'$. (Exercise III.9, Sheaves in geometry and logic.) A family $\{(C_i, D_i)\}$ should cover $(C,D)$ iff there is $(C_j, D)$ in it such that $C_j$ cover $C$. Thus, in our case both $J$ and $J'$ will be discrete.