What is a Presheaf-category enriched pullback?

Question

I have a question about presheaf-enriched categories, like sSet for example that I think is pretty basic, but I don't know how to go about.

So I have a category $C$, like $\Delta^\text{op}$, that is monoidal. I also have a $\mathsf{Set}^C$ enriched category $B$, such as a simplicial set enriched category.

In the underlying category $B_0$ which has homsets $B(x,y)(I)$, I have a pullback square $P$, so $P$ is a pullback diagram in the normal sense. I abuse notation by writing $B_0(x,P)$ for the corresponding pullback diagram in $\mathsf{Set}$. Moreover, for every $n\in \mathsf{Ob}(C)$, I have a pullback diagram $B(x,P)(n)$ in $\mathsf{Set}$.

Is this the definition of a $\mathsf{Set}^C$ enriched pullback? If not what am I missing?

Answer 1

You just have an object $p$ together with morphisms $p\to a \to c$ and $p\to b\to c$ in $B_0$ such that the presheaf $B(x,p)$ is isomorphic to the pullback of $B(x,a)$ and $B(x,b)$ over $B(x,c)$, for every $x$ in $B$.

To understand this better, we can recall that pullbacks in $\mathbf{Set}^C$ are created by evaluation at every point, so the condition in the previous paragraph holds if and only if, for every $n$ in $C$, we have that $B(x,p)_n$ is a pullback of $B(x,a)_n\to B(x,c)_n\leftarrow B(x,b)_n$, which I think is what you were suggesting.

As a side comment, it's not $\Delta^{\mathrm{op}}$ that's monoidal, at least in any natural way, but its category of presheaves. There seems to be a related confusion here in that $B_0(x,y)$ is defined as $\mathbf{Set}^C(I, B(x,y))$. For simplicially enriched categories, this happens to be the same as $B(x,y)([0])$, but that is a coincidence coming from the fact that an initial object in $C$ corepresents the terminal object $I$ of $\mathbf{Set}^C$. Even then, $[0]$ is not quite the same thing as $I$, while if $C$ lacks an initial object, then the expression $B(x,y)(I)$ could be very misleading.