nLab
pushout-product

Definition

Let $\otimes :{ℰ}_1\times {ℰ}_2\to {ℰ}_3$ be a functor (e.g. a tensor product, tensoring). Let ${ℰ}_3$ have pushouts.

Definition

For $f:A\to B$ in ${ℰ}_1$ and $g:X\to Y$ in ${ℰ}_2$, the pushout product morphism is the morphism

$$A\otimes Y\coprod _{A\otimes X}B\otimes Y\to B\otimes Y$$A \otimes Y \coprod_{A \otimes X} B \otimes Y \to B \otimes Y

out of the coproduct, induced from the commuting diagram

$$\begin{array}{ccc}A\otimes X& \to & B\otimes X\\ \downarrow & & \downarrow \\ B\otimes X& \to & B\otimes Y\end{array}.$$\array{ A \otimes X &\to& B \otimes X \\ \downarrow && \downarrow \\ B \otimes X &\to& B \otimes Y } \,.

Related concepts

• pushout-product axiom

• Joyal-Tierney calculus