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
[<hidetoc>]Definition
A Category C consists of 3 things[1]:
- A class of objects X[Note 1]
- For every ordered pair, (X,Y) of objects a set hom(X,Y) of morphisms f
- A function called composition of morphisms:
- F(X,Y,Z):hom(X,Y)×hom(Y,Z)→hom(X,Z)
- defined for every triple, (X,Y,Z) of objects where
- Where F(X,Y,Z)(f,g) is denoted g∘f
and the following 2 properties are satisfied:
- (Associativity) if f∈hom(W,X) and g∈hom(X,Y) and h∈hom(Y,Z) then
- h∘(g∘f)=(h∘g)∘f
- (Existence of identities) if X is an object then there exists a 1X∈hom(X,X) such that[Note 2]:
- 1X∘f=f and g∘1X=g
- for every f∈hom(W,X) and g∈hom(X,Y) where W and Y are any class of objects
Uniqueness of the identity
TODO: Be bothered to prove
Left & right inverses
Let f∈hom(X,Y) and g, g′∈hom(Y,X), if[1]:
- g∘f=1X we call g a left inverse for f and if
- f∘g′=1X we call g′ a right inverse for f
See also
Notes
- <cite_references_link_accessibility_label> ↑ Munkres calls the class of objects X and uses X for specific objects. Not sure why, so checked definition with [Wikipedia]
- <cite_references_link_accessibility_label> ↑ We denote this as 1X because it is easy to prove that it is unique, but at this point we do not know it is unique
References
- ↑ <cite_references_link_many_accessibility_label> 1.0 1.1 Elements of Algebraic Topology - James R. Munkres
