Functor

Definition

Functors come in two flavours:

1. Covariant functor
2. Contravariant functor

Typically "functor" refers to a covariant functor when used by itself.

Covariant functor

A covariant functor, $T:C\leadsto D$ (for categories $C$ and $D$) is a pair of mappings:

• $T:\left\{\begin{array}{rcl}\text{Obj}(C) & \longrightarrow & \text{Obj}(D)\\ X & \longmapsto & TX \end{array}\right.$
• $T:\left\{\begin{array}{rcl}\text{Mor}(C) & \longrightarrow & \text{Mor}(D)\\ f & \longmapsto & Tf \end{array}\right.$

Which preserve composition of morphisms and the identity morphism of each object, that is to say:

• $\forall f,g\in\text{Mor}(C)[Tfg=T(f\circ g)=Tf\circ Tg=TfTg]$ (I've added the $\circ$s in to make it more obvious to the reader what is going on)
  • Where such composition makes sense. That is $\text{target}(g)=\text{source}(f)$.
• and $\forall A\in\text{Obj}(C)[T1_A=1_{TA}]$

Thus if $f:X\rightarrow Y$ and $g:Y\rightarrow Z$ are morphisms of $C$, then the following diagram commutes:

$\begin{xy}\xymatrix{TX \ar[rr]^{Tgf} \ar[dr]_{Tf} & & TZ \\ & TY \ar[ur]_{Tg} & }\end{xy}$ Thus the diagram just depicts the requirement that: $=Tgf=Tg\circ Tf$	$\ $	Note that the diagram is basically just the "image" of $\begin{xy}\xymatrix{X \ar[rr]^{gf} \ar[dr]_{f} & & Z \\ & Y \ar[ur]_{g} & }\end{xy}$ under $T$

Contravariant functor

A covariant functor, $S:C\leadsto D$ (for categories $C$ and $D$) is a pair of mappings:

• $S:\left\{\begin{array}{rcl}\text{Obj}(C) & \longrightarrow & \text{Obj}(D)\\ X & \longmapsto & SX \end{array}\right.$
• $S:\left\{\begin{array}{rcl}\text{Mor}(C) & \longrightarrow & \text{Mor}(D)\\ f & \longmapsto & Sf \end{array}\right.$
  • Note that if $f:A\rightarrow B$ then $Sf:B\rightarrow A$

Which preserves only the identity morphism of each object - it reverses composition of morphisms, that is to say:

• $\forall f,g\in\text{Mor}(C)[Sgf=S(g\circ f)=Sf\circ Sg=SfSg]$ (I've added the $\circ$s in to make it more obvious to the reader what is going on)
  • Where such composition makes sense. That is $\text{target}(f)=\text{source}(g)$.
• and $\forall A\in\text{Obj}(C)[S1_A=1_{SA}]$

Thus if $f:X\rightarrow Y$ and $g:Y\rightarrow Z$ are morphisms of $C$, then the following diagram commutes:

$\begin{xy}\xymatrix{SX & & SZ \ar[ll]_{Sgf} \ar[dl]^{Sg}\\ & SY \ar[ul]^{Sf} & }\end{xy}$ Thus the diagram just depicts the requirement that: $=Sgf=Sf\circ Sg$	$\ $	Note that the diagram is similar to $\begin{xy}\xymatrix{X \ar[rr]^{gf} \ar[dr]_{f} & & Z \\ & Y \ar[ur]_{g} & }\end{xy}$

Discussion

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TODO: Flesh this out

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