Difference between revisions of "Covariant functor"
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{{:Covariant functor/Definition}} | {{:Covariant functor/Definition}} | ||
==Discussion== | ==Discussion== | ||
| + | Given | ||
| + | * 3 objects, {{M|X}}, {{M|Y}} and {{M|Z}} in a [[category]] {{M|\mathscr{C} }} | ||
| + | * a (covariant) functor from {{M|\mathscr{C} }} to another category, {{M|\mathscr{D} }} | ||
| + | ** {{M|T:\mathscr{C}\leadsto\mathscr{D} }} | ||
| + | * morphisms {{M|f:X\rightarrow Y}}, {{M|g:Y\rightarrow Z}} and the morphism {{M|gf:X\rightarrow Z}} corresponding to the [[composition]] {{M|g\circ f}} | ||
| + | The functor gives us "the same" diagram (in terms of objects and arrows) in the target [[category]] {{M|\mathscr{D} }}, as shown by the following [[diagram]]: | ||
| + | {| class="wikitable" border="1" | ||
| + | |- | ||
| + | |<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 {{M|T}}'s image of objects<br/>The dotted lines are the image of morphisms under {{M|T}} | ||
| + | |} | ||
| + | * In this diagram the objects {{M|TX}}, {{M|TY}} and {{M|TZ}} are in a different category. | ||
| + | |||
==References== | ==References== | ||
<references/> | <references/> | ||
{{Definition|Category Theory}} | {{Definition|Category Theory}} | ||
Latest revision as of 16:27, 2 February 2016
TODO: Flesh this page out
Definition
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:
|
[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]
|
|---|
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.
