Many naturally arising functors in mathematics have adjoints. This makes them a useful thing to study.
A functor is a left adjoint if there exists
If s a category and is a precategory then the type “ is a left adjoint” is a mere proposition?.
Proof. Suppose we are given with the triangle identities and also . Define to be . Then
using Lemma 9.2.8 (see natural transformation) and the triangle identities. Similarly, we show , so is a natural isomorphism . By Theorem 9.2.5 (see functor precategory), we have an identity .
Now we need to know that when and are [transported]] along this identity, they become equal to and . By Lemma 9.1.9,
Lemma 9.1.9 needs to be included. For now as transports are not yet written up I didn’t bother including a reference to the page category. -Ali
this transport is given by composing with or as appropriate. For , this yields
using Lemma 9.2.8 (see natural transformation) and the traingle identity. The case of is similar. FInally, the triangle identities transport correctly automatically, since hom-sets are sets.
Category theory functor natural transformation
Revision on September 6, 2018 at 18:49:58 by Ali Caglayan. See the history of this page for a list of all contributions to it.