\[ \require{mathtools} \let\DeclarePairedDelimiter\DeclarePairedDelimiters % MathJax typo \DeclarePairedDelimiter{\floor}{\lfloor}{\rfloor} \DeclarePairedDelimiter{\ceil}{\lceil}{\rceil} % sup \newcommand{\op}{^\mathrm{op}} \newcommand{\inv}{^{-1}} % rm \newcommand{\compL}{\mathbin{\circ}} \newcommand{\Kl}{\mathrm{Kl}} % arrow \newcommand{\xto}{\xrightarrow} \newcommand{\xot}{\xleftarrow} \newcommand{\xnat}{\xRightarrow} \newcommand{\toot}{\rightleftarrows} \newcommand{\klto}{\rightsquigarrow} \newcommand{\comma}{\downarrow} \newcommand{\incl}{\hookrightarrow} \newcommand{\mono}{\rightarrowtail} \newcommand{\epi}{\twoheadrightarrow} \newcommand{\iso}{\cong} \newcommand{\nat}{\Rightarrow} \newcommand{\adj}{\dashv} \newcommand{\adjto}[2]{\overset{{}\xto[]{#1}{}}{\underset{{}\xot[#2]{}{}}{\bot}}} % delimiter \DeclarePairedDelimiter{\parens}{(}{)} % parentheses ( ) \DeclarePairedDelimiter{\bracks}{[}{]} % brackets [ ] \DeclarePairedDelimiter{\braces}{\{}{\}} % braces { } \DeclarePairedDelimiter{\angles}{\langle}{\rangle} % angles \newcommand{\set}{\braces} \newcommand{\singleton}{\set{*}} % operator \DeclareMathOperator{\Obj}{Obj} \DeclareMathOperator{\Hom}{Hom} \DeclareMathOperator{\Eq}{Eq} \DeclareMathOperator{\Coeq}{Coeq} \DeclareMathOperator{\Pull}{Pull} \DeclareMathOperator{\Push}{Push} \DeclareMathOperator{\Lan}{Lan} \DeclareMathOperator{\Ran}{Ran} \DeclareMathOperator{\Lift}{Lift} \DeclareMathOperator{\Rift}{Rift} % \DeclareMathOperator{\id}{id} \DeclareMathOperator{\act}{act} \DeclareMathOperator{\colim}{colim} % exponential \DeclareMathOperator{\partf}{partial} \DeclareMathOperator{\curry}{curry} % natural number \DeclareMathOperator{\zero}{zero} \DeclareMathOperator{\successor}{succ} % list \DeclareMathOperator{\nil}{nil} \DeclareMathOperator{\cons}{cons} \newcommand{\concat}{\mathbin{{+}\mspace{-8mu}{+}}} % fold \DeclareMathOperator{\fold}{fold} \DeclareMathOperator{\map}{map} \DeclareMathOperator{\filter}{filter} % bool \DeclareMathOperator{\cond}{cond} % cd \DeclareMathOperator{\copy}{copy} \DeclareMathOperator{\del}{del} % sets \newcommand{\N}{\mathbb{N}} % natural numbers \newcommand{\Z}{\mathbb{Z}} % integers \newcommand{\Q}{\mathbb{Q}} % rational numbers \newcommand{\R}{\mathbb{R}} % real numbers % category \newcommand{\catf}[1]{{\mathbf{#1}}} \newcommand{\cA}{\catf{A}} \newcommand{\cB}{\catf{B}} \newcommand{\cC}{\catf{C}} \newcommand{\cD}{\catf{D}} \newcommand{\cE}{\catf{E}} \newcommand{\cF}{\catf{F}} \newcommand{\cG}{\catf{G}} \newcommand{\cH}{\catf{H}} \newcommand{\cI}{\catf{I}} \newcommand{\cJ}{\catf{J}} \newcommand{\cK}{\catf{K}} \newcommand{\cL}{\catf{L}} \newcommand{\cM}{\catf{M}} \newcommand{\cN}{\catf{N}} \newcommand{\cO}{\catf{O}} \newcommand{\cP}{\catf{P}} \newcommand{\cQ}{\catf{Q}} \newcommand{\cR}{\catf{R}} \newcommand{\cS}{\catf{S}} \newcommand{\cT}{\catf{T}} \newcommand{\cU}{\catf{U}} \newcommand{\cV}{\catf{V}} \newcommand{\cW}{\catf{W}} \newcommand{\cX}{\catf{X}} \newcommand{\cY}{\catf{Y}} \newcommand{\cZ}{\catf{Z}} \newcommand{\cZero}{\catf{0}} \newcommand{\cOne}{\catf{1}} \newcommand{\cTwo}{\catf{2}} % \newcommand{\cArr}{\catf{Arr}} % arrow \newcommand{\cPSh}{\catf{PSh}} % presheaf \newcommand{\cCone}{\catf{Cone}} % cone \newcommand{\cCocone}{\catf{Cocone}} % cocone % \newcommand{\cFin}{\catf{Fin}} % finite prefix \newcommand{\cSet}{\catf{Set}} % functions \newcommand{\cRel}{\catf{Rel}} % relations \newcommand{\cPos}{\catf{Pos}} % posets \newcommand{\cMon}{\catf{Mon}} % monoids \newcommand{\cGrp}{\catf{Grp}} % groups \newcommand{\cAb}{\catf{Ab}} % abelian groups \newcommand{\cCat}{\catf{Cat}} % categories \newcommand{\cGrpd}{\catf{Grpd}} % groupoids \newcommand{\cVect}{\catf{Vect}} % vector spaces \newcommand{\cMeas}{\catf{Meas}} % measurable spaces and measurable functions \newcommand{\cStoch}{\catf{Stoch}} %measurable spaces and stochastic maps \newcommand{\cProb}{\catf{Prob}} % probability measures and measure-preserving functions % logic \newcommand{\lfal}{\bot} \newcommand{\ltru}{\top} \newcommand{\lcon}{\wedge} \newcommand{\lncon}{\uparrow} \newcommand{\ldis}{\vee} \newcommand{\lndis}{\downarrow} \newcommand{\limp}{\rightarrow} \newcommand{\lnimp}{\nrightarrow} \newcommand{\lcim}{\leftarrow} \newcommand{\lncim}{\nleftarrow} \newcommand{\leqv}{\leftrightarrow} \newcommand{\lneqv}{\nleftrightarrow} \]
An enriched category is a category with generalized morphisms.
1 Enriched category
A category \(\cC = (\Obj, \Hom, \compL, \id)\) enriched in a monoidal category \((\cV, \otimes, I, \alpha, \lambda, \rho)\) (\(\cV\)-enriched category) consists of
- a collection of objects \(\Obj\)
- a hom-object between objects \(\Hom(A, B) \in \Obj_\cV\)
- a composition morphism \(\compL: \Hom(B, C) \otimes \Hom(A, B) \to \Hom(A, C)\)
- an identity element for each object \(\id_A: I \to \Hom(A, A)\)
subject to
- associativity: \(\compL (\id \otimes \compL) \alpha = \compL (\compL \otimes \id)\)
- identity: \(\compL (\id_B \otimes \id) = \lambda\) and \(\compL (\id \otimes \id_A) = \rho\)
2 Enriched functor
For \(\cV\)-enriched categories \(\cC\) and \(\cD\), a \(\cV\)-enriched functor \(F: \cC \to \cD\) consists of
- a function sending \(\cC\)-objects to \(\cD\)-objects \(F: \Obj_\cC \to \Obj_\cD:= A \mapsto FA\)
- a collection of \(\cV\)-morphisms indexed by pairs of \(\cC\)-objects \(F_{A, B}: \Hom_\cC(A, B) \to \Hom_\cD(FA, FB)\)
in such a way that \(F\) preserves
- composition: \(F_{A, C} \compL_\cC = \compL_\cD (F_{B, C} \otimes F_{A, B})\)
- identity: \(F_{A, A} \id_{\cC:A} = \id_{\cD:FA}\)
3 Enriched natural transformation
For \(\cV\)-enriched functors \(F, G: \cC \to \cD\), a \(\cV\)-enriched natural transformation \(\alpha: F \nat G\) is a collection of \(\cV\)-morphisms indexed by \(\cC\)-objects \(\alpha_A: I \to \Hom_\cD(FA, GA)\) that satisfies
- naturality: \(\compL_\cD (\alpha_B \otimes F_{A, B}) \lambda\inv = \compL_\cD (G_{A, B} \otimes \alpha_A) \rho\inv\)
4 Examples
- locally small category: \((\cSet, \times, \set{*})\)
- preorder: \((\cTwo, \land, \top)\)
- strict 2-category: \(\cCat\)
- Lawvere metric space: \(\overline{R}_+ = [0, \infty]\)