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v • d • e Category Theory

Overview of the concepts of Category Theory

Key objects	Category, Functor (Covariant, Contravariant), Subcategory	[ilmath]\xymatrix{ & \text{Arrow} \\ \text{Monic} \ar@{^{(}->}[ur] & & \text{Epic} \ar@{^{(}->}[ul] \\ & \text{Bimorphism} \ar@{^{(}->}[ur] \ar@<-0.5ex>@{^{(}->}[ul] \\ {\begin{array}{c}\text{Section}\\ \text{(Split monic)} \end{array} } \ar@{^{(}->}[uu] & & {\begin{array}{c}\text{Retraction}\\ \text{(Split epic)} \end{array} } \ar@<-0.75ex>@{^{(}->}[uu] \\ & \text{Isomorphism} \ar@{^{(}->}[ur] \ar@<-0.5ex>@{^{(}->}[ul] \ar@{^{(}->}[uu] }[/ilmath]

Typical morphism types (see diagram on right)	Arrow (AKA: Morphism), Monic, Epic, Bimorphism, Section (AKA: Split monic), Retraction (AKA: Split epic), Isomorphism

Key objects	[ilmath]\text{Initial} [/ilmath], [ilmath]\text{Final} [/ilmath] (compared)

Primitive constructs	Wedge (AKA Cone & Cocone)

Key constructs	Product / Coproduct (Product and coproduct compared), Limit / Colimit, Equaliser / Coequaliser

Important examples	Demonstrating why category arrows are best thought of as arrows and not functions

Trivial category examples	Category induced by a monoid, Category induced by a poset

Common categories	[ilmath]\mathrm{SET} [/ilmath], [ilmath]\mathrm{Pfn} [/ilmath], [ilmath]\mathrm{GROUP} [/ilmath]

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