A monoidal category that preserves subobjects

Question

Let $X$, $Y$ be objects in a monoidal category $\mathcal{C}$, s.t. the functors $X \otimes \_$ and $\_\otimes Y$ preserve monomorphisms. Moreover, let $A \hookrightarrow X$, $B \hookrightarrow Y$ be subobjects and $P$ be a pullback of the cospan $$X \otimes B \hookrightarrow X \otimes Y \hookleftarrow A \otimes Y.$$ Do we have $P \cong A \otimes B$? If not, what do we have to assume on $\mathcal{C}$? On the functors $X \otimes \_$ and $\_\otimes Y$? It clearly follows when $\_ \otimes \_$ preserves pullbacks of monos, but this seems a bit too much.

Answer 1

sorry for necroposting; I hope this might be useful to someone.

One does not, in general, have $P\cong A\otimes B$; in fact, one doesn't even have that the arrow $A\otimes Y\to X\otimes Y$ is a monomorphism as you seem to have indicated using $\hookrightarrow$ – even if $\mathcal C$ has the pullback you are considering.

Example (of failure of pullback): Consider pointed topological spaces, with continuous functions between them that preserve their base point. In this category, a continuous injection preserving the base point is monomorphic. Take the smash product as a monoidal operation ($\otimes = \wedge$), with the space with the discrete space with two points $I=\{0,1\}$ and base point $0$ as its monoidal unit.

By definition $X\otimes I\xrightarrow\cong X\xleftarrow\cong I\otimes X$, and so the pullback of those isomorphisms is also isomorphic to $X$ (and is – in fact – as a set, the unique isomorphism $X\otimes I\to I\otimes X$ making the resulting triangle commute, since functions are subsets of pairs!)

Now, if $X\not\cong I$ then clearly the pullback $P\not\cong I\otimes I \cong I$.