[[!redirects evaluational categories]] #Contents# * table of contents {:toc} ## Idea ## A [[concrete category]] with a good notion of evaluation of morphisms and elements. ## Definition ## An evaluational category $C$ is a [[concrete category]] with a function $(-)((-)): Hom(A,B) \times El(A) \to El(B)$ for objects $A:Ob(C)$ and $B:Ob(C)$ such that for morphisms $f:Hom(A,B)$ and $g:Hom(B,C)$ and elements $x:El(A)$, $(g \circ f)(x) = g(f(x))$. ## Examples ## * The category $Set$ of sets and functions is an evaluational category. * The category $Mon$ of monoids and monoid homomorphisms is an evaluational category. * The category $\mathbb{Z}Mod$ of [[Z-module|$\mathbb{Z}$-modules]] and $\mathbb{Z}$-module homomorphisms is an evaluational category. * The category $\mathbb{Z}Alg$ of [[Z-algebra|$\mathbb{Z}$-algebras]] and $\mathbb{Z}$-algebra homomorphisms is an evaluational category. * The category $CRing$ of [[commutative ring|commutative rings]] and commutative ring homomorphisms is an evaluational category. * The category $Field$ of [[field (ring theory)|fields]] and field homomorphisms is an evaluational category. * The category $HeytAlg$ of [[Heyting algebra|Heyting algebras]] and Heyting algebra homomorphisms is an evaluational category. * The category $Frm$ of [[frame|frames]] and frame homomorphisms is an evaluational category. * The category $Conv$ of set-truncated convergence spaces and continuous functions is an evaluational category. * The category $Top$ of set-truncated topological spaces and continuous functions is an evaluational category. * The category of empty sets and functions is an evaluatioinal category. ## Non-examples ## * The category $Rel$ of sets and relations is not an evaluational category. ## See also ## * [[concrete category]] * [[concrete dagger 2-poset]] * [[concrete precategory]] * [[extensional category]] * [[evaluational dagger 2-poset]] * [[Set]]