# nLab product-preserving functor

Let $F \colon C \to D$ be a functor, and suppose a collection of objects $\{c_i\}$ in $C$ admits a product, with projections

$\pi_i \colon \prod_i c_i \to c_i.$

We say $F$ preserves this product if the collection of maps

$F(\pi_i): F(\prod_i c_i) \to F(c_i)$

exhibits $F(\prod_i c_i)$ as a product of the collection of objects $F(c_i)$.

If $C$ has all (small) products, $F$ is product-preserving if it preserves every product in $C$.

If $C$ does not have all small products, then one wants a more subtle condition; compare flat functor (which is about finite limits instead of products).

Revised on August 17, 2012 01:51:07 by Toby Bartels (98.19.44.121)