Product-preserving functor

This article defines a functor property: a property that can be evaluated to true/false given a functor between two categories.
View a complete list of functor properties|Get functor property lookup help |Get exploration suggestions

Definition

Symbol-free definition

A functor between two categories is termed product-preserving if it preserves categorical products.

Definition with symbols

Suppose ${\mathcal {C}},{\mathcal {D}}$ are categories. A functor ${\mathcal {F}}:{\mathcal {C}}\to {\mathcal {D}}$ is termed product-preserving if the following holds. Suppose $A_{1},A_{2}\in \operatorname {Ob} {\mathcal {C}}$ and $C\in \operatorname {Ob} {\mathcal {C}}$ is a product of $A$ and $B$ with projection maps $\pi _{i}:C\to A_{i}$.

Then, ${\mathcal {F}}C$, along with the projection maps ${\mathcal {F}}\pi _{1},{\mathcal {F}}\pi _{2}$, is a product of ${\mathcal {F}}A_{1}$ and ${\mathcal {F}}A_{2}$.

Note that this term is typically used for categories that both admit finite products, though it makes sense for arbitrary categories.