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For precategories and , their product is a precategory with and
Identities are defined by and composition by
For precategories , the following types are equivalent?:
Proof. Given , for any we obviously have a functor . This gives a function . Next, for any , we have for any the morphisms .
These are the components of a natural transformation . Functoriality in is easy to check, so we have a functor .
Conversly, suppose given . Then for any and we have the object , giving a function . And for and , we have the morphism
in . Functoriality is again easy to check, so we have a functor Finally, it is also clear that these operations are inverses.
Category theory precategory functor functor precategory
Revision on September 6, 2018 at 18:44:41 by Ali Caglayan. See the history of this page for a list of all contributions to it.