Difference between revisions of "Category (category theory)"

(discrete and opposite categories)
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* For any set <math>S</math>, we can form the ''[[discrete category]] on <math>S</math>'' whose objects are elements of <math>S</math> and such that the only morphisms are the identity morphisms on each object.
 
* For any set <math>S</math>, we can form the ''[[discrete category]] on <math>S</math>'' whose objects are elements of <math>S</math> and such that the only morphisms are the identity morphisms on each object.
 
* For any category <math>\mathcal{C}</math> we can form the ''[[opposite category]]'' or the ''dual category'', <math>\mathcal{C}^{op}</math> whose objects are the objects of <math>\mathcal{C}</math>, but where all the morphisms are 'reversed' (i.e. a morphism from <math>X</math> to <math>Y</math> would be replaced by a morphism from <math>Y</math> to <math>X</math>).
 
* For any category <math>\mathcal{C}</math> we can form the ''[[opposite category]]'' or the ''dual category'', <math>\mathcal{C}^{op}</math> whose objects are the objects of <math>\mathcal{C}</math>, but where all the morphisms are 'reversed' (i.e. a morphism from <math>X</math> to <math>Y</math> would be replaced by a morphism from <math>Y</math> to <math>X</math>).
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Examples which are more specific:
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* The category in which the objects are sets and there is a morphism <math>U \to V</math> if and only if <math>U \subset V</math>.
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* The category in which the objects are positive integers and there is a morphism <math>s \to t</math> if and only if <math>s</math> divides <math>t</math>.
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* The category in which the objects are positive integers and there is a morphism <math>s \to t</math> if and only if <math>s \le t</math>.
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* For a fixed ring <math>R</math>, the category in which the objects are elements of <math>R</math> and there is a morphism <math>s \to t</math> if and only if there exists some <math>u</math> such that <math>su = t</math>.
 
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[[Category:Category theory]]
 
[[Category:Category theory]]

Revision as of 18:38, 7 April 2012

A category, $\mathcal{C}$, is a mathematical object consisting of:

  • A class, $\text{Ob}(\mathcal{C})$ of objects.
  • For every pair of objects $A,B\in \text{Ob}(\mathcal{C})$, a class $\text{Hom}(A,B)$ of morphisms from $A$ to $B$. (We sometimes write $f:A \to B$ to mean $f\in \text{Hom}(A,B)$.)
  • For every three objects, $A,B,C \in \mathcal{C}$, a binary operation $\circ: \text{Hom}(B,C) \times \text{Hom}(A,B) \to \text{Hom}(A,C)$ called composition, which satisfies:
    • (associativity) Given $f:A\to B$, $g:B\to C$ and $h:C \to D$ we have \[h\circ(g\circ f) = (h \circ g)\circ f.\]
    • (identity) For and object $X$, there is an identity morphism $1_X:X\to X$ such that for any $f:A\to B$: \[1_B\circ f = f = f\circ 1_A.\]

The class of all morphisms of $\mathcal{C}$ is denoted $\text{Hom}(\mathcal{C})$.

A category $\mathcal{C}$ is called small if both $\text{Ob}(\mathcal{C})$ and $\text{Hom}(\mathcal{C})$ are sets. If $\mathcal{C}$ is not small, then it is called large. $\mathcal{C}$ is called locally small if $\text{Hom}(A,B)$ is a set for all $A,B\in \text{Ob}(\mathcal{C})$. Most important categories in math are not small, but are locally small.

Intuitively we can think of the objects of $\mathcal{C}$ as being sets (perhaps with some additional structure) and morphisms as being functions between these sets (perhaps satisfying some properties) and composition as being regular function composition, however there are examples of categories which do not satisfy this. Typically when studying category theory we deal with morphisms and composition completely abstractly (similarly to how we study multiplication abstractly in group theory), and never talk about 'plugging things in to' morphisms.

Examples

Some common examples of categories are:

Examples which are more specific:

  • The category in which the objects are sets and there is a morphism $U \to V$ if and only if $U \subset V$.
  • The category in which the objects are positive integers and there is a morphism $s \to t$ if and only if $s$ divides $t$.
  • The category in which the objects are positive integers and there is a morphism $s \to t$ if and only if $s \le t$.
  • For a fixed ring $R$, the category in which the objects are elements of $R$ and there is a morphism $s \to t$ if and only if there exists some $u$ such that $su = t$.

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