Difference between revisions of "Category"
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==Definition== | ==Definition== | ||
A '''Category {{M|C}}''' consists of 3 things<ref name="EOAT">Elements of Algebraic Topology - James R. Munkres</ref>: | A '''Category {{M|C}}''' consists of 3 things<ref name="EOAT">Elements of Algebraic Topology - James R. Munkres</ref>: | ||
| − | # A [[Class|class]] of ''objects'' {{M|X}} | + | # A [[Class|class]] of ''objects'' {{M|\mathcal{X} }}<ref group="Note">Munkres calls the class of objects {{M|X}} and uses {{M|X}} for specific objects. Not sure why, so checked definition with [[https://en.wikipedia.org/w/index.php?title=Category_%28mathematics%29&oldid=682856484 Wikipedia]]</ref> |
# For every ordered pair, {{M|(X,Y)}} of ''objects'' a set {{M|\hom(X,Y)}} of ''morphisms'' {{M|f}} | # For every ordered pair, {{M|(X,Y)}} of ''objects'' a set {{M|\hom(X,Y)}} of ''morphisms'' {{M|f}} | ||
# A function called ''composition of morphisms'': | # A function called ''composition of morphisms'': | ||
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#* {{M|1=1_X\circ f=f}} and {{M|1=g\circ 1_X=g}} | #* {{M|1=1_X\circ f=f}} and {{M|1=g\circ 1_X=g}} | ||
#: for every {{M|f\in\hom(W,X)}} and {{M|g\in\hom(X,Y)}} where {{M|W}} and {{M|Y}} are any class of ''objects'' | #: for every {{M|f\in\hom(W,X)}} and {{M|g\in\hom(X,Y)}} where {{M|W}} and {{M|Y}} are any class of ''objects'' | ||
| + | |||
==Uniqueness of the identity== | ==Uniqueness of the identity== | ||
{{Todo|Be bothered to prove}} | {{Todo|Be bothered to prove}} | ||
Latest revision as of 00:36, 27 September 2015
Contents
Definition
A Category [ilmath]C[/ilmath] consists of 3 things[1]:
- A class of objects [ilmath]\mathcal{X} [/ilmath][Note 1]
- For every ordered pair, [ilmath](X,Y)[/ilmath] of objects a set [ilmath]\hom(X,Y)[/ilmath] of morphisms [ilmath]f[/ilmath]
- A function called composition of morphisms:
- [ilmath]F_{(X,Y,Z)}:\hom(X,Y)\times\hom(Y,Z)\rightarrow\hom(X,Z)[/ilmath]
- defined for every triple, [ilmath](X,Y,Z)[/ilmath] of objects where
- Where [ilmath]F_{(X,Y,Z)}(f,g)[/ilmath] is denoted [ilmath]g\circ f[/ilmath]
and the following [ilmath]2[/ilmath] properties are satisfied:
- (Associativity) if [ilmath]f\in\hom(W,X)[/ilmath] and [ilmath]g\in\hom(X,Y)[/ilmath] and [ilmath]h\in\hom(Y,Z)[/ilmath] then
- [ilmath]h\circ(g\circ f)=(h\circ g)\circ f[/ilmath]
- (Existence of identities) if [ilmath]X[/ilmath] is an object then there exists a [ilmath]1_X\in\hom(X,X)[/ilmath] such that[Note 2]:
- [ilmath]1_X\circ f=f[/ilmath] and [ilmath]g\circ 1_X=g[/ilmath]
- for every [ilmath]f\in\hom(W,X)[/ilmath] and [ilmath]g\in\hom(X,Y)[/ilmath] where [ilmath]W[/ilmath] and [ilmath]Y[/ilmath] are any class of objects
Uniqueness of the identity
TODO: Be bothered to prove
Left & right inverses
Let [ilmath]f\in\hom(X,Y)[/ilmath] and [ilmath]g,\ g'\in\hom(Y,X)[/ilmath], if[1]:
- [ilmath]g\circ f=1_X[/ilmath] we call [ilmath]g[/ilmath] a left inverse for [ilmath]f[/ilmath] and if
- [ilmath]f\circ g'=1_X[/ilmath] we call [ilmath]g'[/ilmath] a right inverse for [ilmath]f[/ilmath]
See also
Notes
- ↑ Munkres calls the class of objects [ilmath]X[/ilmath] and uses [ilmath]X[/ilmath] for specific objects. Not sure why, so checked definition with [Wikipedia]
- ↑ We denote this as [ilmath]1_X[/ilmath] because it is easy to prove that it is unique, but at this point we do not know it is unique
