* table of contents {:toc} ## Definition ## For [[precategories]] $A,B$, there is a [[precategory]] $B^A$, called the **functor precategory**, defined by * $(B^A)_0$ is the type of [[functors]] from $A$ to $B$. * $hom_{B^A}(F,G)$ is the type of [[natural transformations]] from $F$ to $G$. **Proof.** We define $(1_F)_a \equiv 1_{F a}$. Naturality follows by the unit axioms of a [[precategory]]. For $\gamma : F \to G$ and $\delta : G \to H$, we define $(\delta \circ \gamma)_a \equiv \delta_a \circ \gamma_a$. Naturality follows by associativity. Similarly, the unit and associativity laws for $B^A$ follow from those for $B$. ## Properties ## ### Lemma 9.2.4 ### A natural transformation $\gamma : F \to G$ is an isomorphism in $B^A$ if and only if each $\gamma_a$ is an isomorphism in $B$. **Proof.** If $\gamma$ is an [[isomorphism]], then we have $\delta : G \to F$ that is its inverse. By definition of composition in $B^A$, $(\delta \gamma)_a \equiv \delta_a \gamma_a$. Thus, $\delta \gamma = 1_F$ and $\gamma \delta=1_G$ imply that $\delta_a \gamma_a = 1_{F a}$ and $\gamma_a \delta_a = 1_{G a}$, so $\gamma_a$ is an isomorphism. Conversely, suppose each $\gamma_a$ is an [[isomorphism]], with inverse called $\delta_a$. We define a [[natural transformation]] $\delta : G \to F$ with components $\delta_a$; for the naturality axiom we have $$F f \circ \delta_a=\delta_b \circ \gamma_b \circ F f \circ \delta_a = \delta_b \circ G f \circ \gamma_a \circ \delta_a = \delta_b \circ G f$$ Now since composition and identity of [[natural transformations]] is determined on their components, we have $\gamma \delta=1_G$ and $\delta \gamma 1_F.\ \square$ ### Theorem 9.2.5 ### If $A$ is a [[precategory]] and $B$ is a [[category]], then $B^A$ is a [[category]]. ## See also ## [[Category theory]] [[functor]] [[category]] [[Rezk completion]] ## References ## [[HoTT Book]] category: category theory