## Idea ## ## Definition ## A [[functor]] $F : A \to B$ is an **equivalence of precategories** if it is a [[left adjoint]] for which $\eta$ and $\epsilon$ are [[isomorphisms]]. We write $A \equiv B$ for the type of equivalences of [[precategories]] from $A$ to $B$. ## Properties ## ### Lemma 9.4.2 ### If for $F : A \to B$ there exists $G : B \to A$ and [[isomorphisms]] $G F \cong 1_A$ and $F G \cong 1_B$, then $F$ is an equivalence of precategories. **Proof.** +--{.query} The proof from the HoTT book analogues the proof of Theorem 4.2.3 for equivalence of types. This has not been written up at time of writing. =-- ### Lemma 9.4.5 ### For any [[precategories]] $A$ and $B$ and functor $F : A \to B$, the following types are [[equivalent]]. * $F$ is an [[equivalence of precategories]] * $F$ is [[fully faithful]] and [[split essentially surjective]] **Proof.** Suppose $F$ is an [[equivalence of precategories]] with $G,\eta,\epsilon$ specified. Then we have the function $$ \begin{aligned} hom_B(F a, F b) &\to hom_A(a,b), \\ g &\mapsto \eta_b^{-1}\circ G(g) \circ \eta_a. \end{aligned} $$ For $f:hom_A(a,b)$, we have $$\eta_b^{-1} \circ G(F(f)) \circ \eta_a = \eta_b^{-1} \circ \eta_b \circ f = f$$ while for $g: hom_B(F a, F b)$ we have $$ \begin{aligned} F(\eta_b^{-1} \circ G(g) \circ \eta_a) &= F(\eta_b^{-1}) \circ F(G(g)) \circ F(\eta_a) \\ &= \epsilon_{F b} \circ F(G(g)) \circ F(\eta_a) \\ &= g \circ \epsilon_{F a} \circ F(\eta_a) \\ &= g \end{aligned} $$ using [[natural transformation|naturality]] of $\epsilon$, and the triangle identities twice. Thus, $F_{a,b}$ is an [[equivalence of precategories|equivalence]], so $F$ is [[fully faithful]]. Finally, for any $b:B$, we have $G b : A$ and $\epsilon_b : F G b \cong b$. On the other hand, suppose $F$ is [[fully faithful]] and [[split essentially surjective]]. Define $G_0:B_0\to A_0$ by sending $b:B$ to the $a:A$ given by the spcified [[split essentially surjective|essential splitting]], and write $\epsilon_b$ for the likewise specified [[isomorphism]] $F G b \cong b$. ## See also ## [[Category theory]] [[functor]] [[fully faithful functor]] [[split essentially surjective]] ## References ## [[HoTT Book]] category: category theory