Adjunctions

Written by: Gabriel Field.

Hom-set definition

The hom-set definition of an adjunction is how adjunctions are most typically introduced. Let $\mathsf{C}$ and $\mathsf{\widetilde{C}}$ be categories. A hom-set adjunction consists of a pair of opposing functors $$ F : \mathsf{C} \rightleftarrows \mathsf{\widetilde{C}} : U $$ and a family of isomorphisms^[1] [1] This family is often suppressed in practice. $$ \varphi_{x, \tilde{y}} : \widetilde{\mathsf{C}}(F x, \tilde{y}) \simeq \mathsf{C}(x, U \tilde{y}) $$ natural in $x \in \mathsf{C}^{\mathrm{op}}$ and in $\tilde{y} \in \widetilde{\mathsf{C}}$.

Unit-counit definition

The unit-counit definition is useful in 2-category theory, and for practically working with adjunctions. Let $\mathsf{C}$ and $\widetilde{\mathsf{C}}$ be categories. A unit-counit adjunction consists of a pair of opposing functors $$ F : \mathsf{C} \rightleftarrows \mathsf{\widetilde{C}} : U $$ and two natural transformations^[2] [2] These are known as the unit and the counit respectively. $$\begin{align*} \eta : 1_{\mathsf{C}} &\to U F &\varepsilon : F U &\to 1_{\widetilde{\mathsf{C}}} \end{align*}$$ such that the triangle identities shown below commute:

Equivalence of definitions

The two definitions provided above are equivalent in the following sense.

Lemma.

Fix a pair of opposing functors $F : \mathsf{C} \rightleftarrows \mathsf{\widetilde{C}} : U$. Then, there is a family of isomorphisms $$ \varphi_{x, \tilde{y}} : \widetilde{\mathsf{C}}(F x, \tilde{y}) \simeq \mathsf{C}(x, U \tilde{y}) $$ specifying a hom-set adjunction if and only if there is a pair of natural transformations $$\begin{align*} \eta : 1_{\mathsf{C}} &\to U F &\varepsilon : F U &\to 1_{\widetilde{\mathsf{C}}} \end{align*}$$ specifying a unit-counit adjunction.

Result: RAPL

Perhaps the most frequently used result in category theory is the following.

Theorem. (RAPL)

Let $U : \mathsf{\widetilde{C}} \to \mathsf{C}$ be a right adjoint functor and let $\widetilde{D} : \mathsf{J} \to \mathsf{\widetilde{C}}$ be a diagram with a limit in $\mathsf{\widetilde{C}}$. Then, the composite diagram $U \widetilde{D} : \mathsf{J} \to \mathsf{C}$ has a limit in $\mathsf{C}$ and $$ \lim_{\mathsf{J}} \left( U \widetilde{D} \right) \simeq U \left( \lim_{\mathsf{J}} \widetilde{D} \right) $$ Consequently, if $\mathsf{C}$ and $\mathsf{\widetilde{C}}$ have all $\mathsf{J}$-shaped limits, then the following diagram commutes up to isomorphism.

Bibliography

Emily Riehl. Category Theory in Context. URL: https://emilyriehl.github.io/files/context.pdf.