Let be a functor, and suppose a collection of objects in admits a product, with projections
\pi_i \colon \prod_i c_i \to c_i.
We say preserves this product if the collection of maps
F(\pi_i): F(\prod_i c_i) \to F(c_i)
exhibits as a product of the collection of objects .
If has all (small) products, is product-preserving if it preserves every product in .
If 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