[[!redirects extensional categories]] #Contents# * table of contents {:toc} ## Idea ## An [[concrete category]] where morphisms satisfy the axiom of extensionality. ## Definition ## An extensional category $C$ is a [[concrete category]] such that for morphisms $f:Hom(A,B)$ and $g:Hom(A,B)$, if $f(x) = g(x)$ for all elements $x:El(A)$, then $f = g$. ## Examples ## * The category $Set$ of sets and functions is an extensional category. * The category $Mon$ of monoids and monoid homomorphisms is an extensional category. * The category $\mathbb{Z}Mod$ of [[Z-module|$\mathbb{Z}$-modules]] and $\mathbb{Z}$-module homomorphisms is an extensional category. * The category $\mathbb{Z}Alg$ of [[Z-algebra|$\mathbb{Z}$-algebras]] and $\mathbb{Z}$-algebra homomorphisms is an extensional category. * The category $CRing$ of [[commutative ring|commutative rings]] and commutative ring homomorphisms is an extensional category. * The category $Field$ of [[field (ring theory)|fields]] and field homomorphisms is an extensional category. * The category $HeytAlg$ of [[Heyting algebra|Heyting algebras]] and Heyting algebra homomorphisms is an extensional category. * The category $Frm$ of [[frame|frames]] and frame homomorphisms is an extensional category. * The category $Conv$ of set-truncated convergence spaces and continuous functions is an extensional category. * The category $Top$ of set-truncated topological spaces and continuous functions is an extensional category. * The category of empty sets and functions is an extensional category. ## See also ## * [[concrete category]] * [[Set]]