Difference between revisions of "Contravariant functor/Definition"

Latest revision as of 16:53, 2 February 2016

A covariant functor, $S:C\leadsto D$ (for categories $C$ and $D$ ) is a pair of mappings^[1]:

$S:\left\{\begin{array}{rcl}\text{Obj}(C) & \longrightarrow & \text{Obj}(D)\\ X & \longmapsto & SX \end{array}\right.$
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 \circs 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 $\begin{xy}\xymatrix{X \ar[rr]^{gf} \ar[dr]_{f} & & Z \\ & Y \ar[ur]_{g} & }\end{xy}$

@@ Line 2: / Line 2: @@
 ==Definition==
 </noinclude>
-A ''covariant functor'', {{M|T:C\leadsto D}} (for [[category|categories]] {{M|C}} and {{M|D}}) is a pair of [[mapping|mappings]]{{rAIRMACCF}}:
+A ''covariant functor'', {{M|S:C\leadsto D}} (for [[category|categories]] {{M|C}} and {{M|D}}) is a pair of [[mapping|mappings]]{{rAIRMACCF}}:
 * {{M|1=S:\left\{ $\begin{array}{rcl}\text{Obj}(C) & \longrightarrow & \text{Obj}(D)\\ X & \longmapsto & SX \end{array}$ \right. }}
 * {{M|1=S:\left\{ $\begin{array}{rcl}\text{Mor}(C) & \longrightarrow & \text{Mor}(D)\\ f & \longmapsto & Sf \end{array}$ \right. }}