Difference between revisions of "Functor"
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Latest revision as of 14:59, 2 February 2016
Definition
Functors come in two flavours:
Typically "functor" refers to a covariant functor when used by itself[1].
Covariant functor
A covariant functor, [ilmath]T:C\leadsto D[/ilmath] (for categories [ilmath]C[/ilmath] and [ilmath]D[/ilmath]) is a pair of mappings[1]:
- [ilmath]T:\left\{\begin{array}{rcl}\text{Obj}(C) & \longrightarrow & \text{Obj}(D)\\ X & \longmapsto & TX \end{array}\right.[/ilmath]
- [ilmath]T:\left\{\begin{array}{rcl}\text{Mor}(C) & \longrightarrow & \text{Mor}(D)\\ f & \longmapsto & Tf \end{array}\right.[/ilmath]
Which preserve composition of morphisms and the identity morphism of each object, that is to say:
- [ilmath]\forall f,g\in\text{Mor}(C)[Tfg=T(f\circ g)=Tf\circ Tg=TfTg][/ilmath] (I've added the [ilmath]\circ[/ilmath]s in to make it more obvious to the reader what is going on)
- Where such composition makes sense. That is [ilmath]\text{target}(g)=\text{source}(f)[/ilmath].
- and [ilmath]\forall A\in\text{Obj}(C)[T1_A=1_{TA}][/ilmath]
Thus if [ilmath]f:X\rightarrow Y[/ilmath] and [ilmath]g:Y\rightarrow Z[/ilmath] are morphisms of [ilmath]C[/ilmath], then the following diagram commutes:
| [ilmath]\begin{xy}\xymatrix{TX \ar[rr]^{Tgf} \ar[dr]_{Tf} & & TZ \\ & TY \ar[ur]_{Tg} & }\end{xy}[/ilmath]
Thus the diagram just depicts the requirement that:
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[ilmath]\ [/ilmath] | Note that the diagram is basically just the "image" of [ilmath]\begin{xy}\xymatrix{X \ar[rr]^{gf} \ar[dr]_{f} & & Z \\ & Y \ar[ur]_{g} & }\end{xy}[/ilmath]
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Contravariant functor
A covariant functor, [ilmath]S:C\leadsto D[/ilmath] (for categories [ilmath]C[/ilmath] and [ilmath]D[/ilmath]) is a pair of mappings[1]:
- [ilmath]S:\left\{\begin{array}{rcl}\text{Obj}(C) & \longrightarrow & \text{Obj}(D)\\ X & \longmapsto & SX \end{array}\right.[/ilmath]
- [ilmath]S:\left\{\begin{array}{rcl}\text{Mor}(C) & \longrightarrow & \text{Mor}(D)\\ f & \longmapsto & Sf \end{array}\right.[/ilmath]
- Note that if [ilmath]f:A\rightarrow B[/ilmath] then [ilmath]Sf:B\rightarrow A[/ilmath]
Which preserves only the identity morphism of each object - it reverses composition of morphisms, that is to say:
- [ilmath]\forall f,g\in\text{Mor}(C)[Sgf=S(g\circ f)=Sf\circ Sg=SfSg][/ilmath] (I've added the [ilmath]\circ[/ilmath]s in to make it more obvious to the reader what is going on)
- Where such composition makes sense. That is [ilmath]\text{target}(f)=\text{source}(g)[/ilmath].
- and [ilmath]\forall A\in\text{Obj}(C)[S1_A=1_{SA}][/ilmath]
Thus if [ilmath]f:X\rightarrow Y[/ilmath] and [ilmath]g:Y\rightarrow Z[/ilmath] are morphisms of [ilmath]C[/ilmath], then the following diagram commutes:
| [ilmath]\begin{xy}\xymatrix{SX & & SZ \ar[ll]_{Sgf} \ar[dl]^{Sg}\\ & SY \ar[ul]^{Sf} & }\end{xy}[/ilmath]
Thus the diagram just depicts the requirement that:
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[ilmath]\ [/ilmath] | Note that the diagram is similar to [ilmath]\begin{xy}\xymatrix{X \ar[rr]^{gf} \ar[dr]_{f} & & Z \\ & Y \ar[ur]_{g} & }\end{xy}[/ilmath] |
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Discussion
TODO: Flesh this out
