Covariant functor

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Definition

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

• [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: [ilmath]=Tgf=Tg\circ Tf[/ilmath]	[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] under [ilmath]T[/ilmath]

Discussion

Given

• 3 objects, [ilmath]X[/ilmath], [ilmath]Y[/ilmath] and [ilmath]Z[/ilmath] in a category [ilmath]\mathscr{C} [/ilmath]
• a (covariant) functor from [ilmath]\mathscr{C} [/ilmath] to another category, [ilmath]\mathscr{D} [/ilmath]
  • [ilmath]T:\mathscr{C}\leadsto\mathscr{D} [/ilmath]
• morphisms [ilmath]f:X\rightarrow Y[/ilmath], [ilmath]g:Y\rightarrow Z[/ilmath] and the morphism [ilmath]gf:X\rightarrow Z[/ilmath] corresponding to the composition [ilmath]g\circ f[/ilmath]

The functor gives us "the same" diagram (in terms of objects and arrows) in the target category [ilmath]\mathscr{D} [/ilmath], as shown by the following diagram:

[math] \xymatrix{ X \ar@{-->}@(u,ul)[rrrr] \ar[rr]^{gf}="gf" \ar[dr]_f="f" \ar& & Z \ar@{-->}@(u,ul)[rrrr] & & TX \ar[rr]^{Tgf}="tgf" \ar[dr]_{Tf}="tf" & & TZ\\ & Y \ar[ur]_{g}="g" \ar@{-->}@(d,dl)[rrrr] & & & & TY \ar[ur]_{Tg}="tg" \ar@{.>}@/^/ "gf";"tgf" \ar@{.>}@/_/ "f";"tf" \ar@{.>}@/^/ "g";"tg" } [/math]
The dashed lines represent [ilmath]T[/ilmath]'s image of objects The dotted lines are the image of morphisms under [ilmath]T[/ilmath]

• In this diagram the objects [ilmath]TX[/ilmath], [ilmath]TY[/ilmath] and [ilmath]TZ[/ilmath] are in a different category.