product-preserving functor

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

π i: ic ic i.\pi_i \colon \prod_i c_i \to c_i.

We say FF preserves this product if the collection of maps

F(π i):F( ic i)F(c i)F(\pi_i): F(\prod_i c_i) \to F(c_i)

exhibits F( ic i)F(\prod_i c_i) as a product of the collection of objects F(c i)F(c_i).

If CC has all (small) products, FF is product-preserving if it preserves every product in CC.

If CC 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 (