Vipul: New page: {{functor property}} ==Definition== ===Symbol-free definition=== A functor between two categories is termed '''product-preserving''' if it preserves [[categorical produc...

2008-12-10T01:07:50Z

New page: {{functor property}} ==Definition== ===Symbol-free definition=== A functor between two categories is termed '''product-preserving''' if it preserves [[categorical produc...

New page

{{functor property}}
==Definition==

===Symbol-free definition===

A [[functor]] between two [[category|categories]] is termed '''product-preserving''' if it preserves [[categorical product]]s.

===Definition with symbols===

Suppose <math>\mathcal{C},\mathcal{D}</math> are [[category|categories]]. A functor <math>\mathcal{F}:\mathcal{C} \to \mathcal{D}</math> is termed '''product-preserving''' if the following holds. Suppose <math>A_1,A_2 \in \operatorname{Ob}\mathcal{C}</math> and <math>C \in \operatorname{Ob}\mathcal{C}</math> is a product of <math>A</math> and <math>B</math> with projection maps <math>\pi_i:C \to A_i</math>.

Then, <math>\mathcal{F}C</math>, along with the projection maps <math>\mathcal{F}\pi_1,\mathcal{F}\pi_2</math>, is a product of <math>\mathcal{F}A_1</math> and <math>\mathcal{F}A_2</math>.

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

Product-preserving functor - Revision history

Vipul: New page: {{functor property}} ==Definition== ===Symbol-free definition=== A functor between two categories is termed '''product-preserving''' if it preserves [[categorical produc...