left adjoint (Rev #4)

Many naturally arising functors in mathematics have adjoints. This makes them a useful thing to study.

A functor $F: A \to B$ is a **left adjoint** if there exists

- A functor $G : B \to A$
- A natural transformation $\eta:1_A \to G F$ (the
**unit**). - A natural transformation $\epsilon G \to 1_B$ (the
**counit**). - Which satisfy the
**triangle identities**(a.k.a zig-zag identities)- $(\epsilon F)(F \eta) = 1_F$
- $(G \epsilon)(\eta G) = 1_G$

If $A$ s a category and $B$ is a precategory then the type “$F$ is a left adjoint” is a mere proposition?.

**Proof.** Suppose we are given $(G, \eta, \epsilon)$ with the triangle identities and also $(G', \eta', \epsilon')$. Define $\gamma: G \to G'$ to be $(G' \epsilon )(\eta G')$. Then

$\begin{aligned}
\delta \gamma &= (G \epsilon')(\eta G')(G' \epsilon) (\eta' G)\\
&= (G \epsilon')(G F G' \epsilon_))\eta G' F G)(\eta' G)\\
&= (G \epsilon ')(G \epsilon' F G)(G F \eta' G)(\eta G)\\
&= (G \epsilon)(\eta G)\\
&= 1_G
\end{aligned}$

using Lemma 9.2.8 (see natural transformation) and the triangle identities. Similarly, we show $\gamma \delta=1_{G'}$, so $\gamma$ is a natural isomorphism $G \cong G'$. By Theorem 9.2.5 (see functor precategory), we have an identity $G=G'$.

Now we need to know that when $\eta$ and $\epsilon$ are [transported]] along this identity, they become equal to $\eta'$ and $\epsilon '$. 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 $\gamma$ or $\delta$ as appropriate. For $\eta$, this yields

$(G' \epsilon F)(\eta' G F)\eta = (G' \epsilon F)(G' F \eta)\eta'=\eta'$

using Lemma 9.2.8 (see natural transformation) and the traingle identity. The case of $\epsilon$ is similar. FInally, the triangle identities transport correctly automatically, since hom-sets are sets. $\square$

Category theory functor natural transformation

Revision on September 6, 2018 at 14:49:59 by Ali Caglayan. See the history of this page for a list of all contributions to it.