---- > [!definition] Definition. ([[category equivalence]]) > Let $\mathsf{C}$ and $\mathsf{D}$ be [[category|categories]] and $[\mathsf{C}, \mathsf{D}]$ the [[natural transformation|category of functors]] between them. A **category equivalence** between $\mathsf{C}$ and $\mathsf{D}$ consists of a [[covariant functor|functor]] $\mathscr{F}:\mathsf{C} \to \mathsf{D}$ and a [[covariant functor|functor]] $\mathscr{G}:\mathsf{D} \to \mathsf{C}$ such that $\mathscr{G} \circ \mathscr{F}$ is [[natural transformation|naturally isomorphic]] to the [[identity functor]] $1_{\mathsf{C}}$ and $\mathscr{F} \circ \mathscr{G}$ is [[natural transformation|naturally isomorphic]] to the [[identity functor]] $1_{\mathsf{D}}$. ^definition > [!intuition] > This notion relaxes that of [[category isomorphism]], a bit like how the notion of [[homotopy equivalent|homotopy equivalence]] relaxes that of [[homeomorphism]]. ^intuition > [!equivalence] > A [[covariant functor|functor]] $\mathscr{F}:\mathsf{C} \to \mathsf{D}$ yields an equivalence of categories if and only if it is [[fully faithful functor|fully faithful]] and [[essentially surjective functor|essentially surjective]]. ^equivalence ---- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```