Difference between revisions of "Category (category theory)"

(motivation for definition)
Line 10: Line 10:
  
 
A category <math>\mathcal{C}</math> is called '''small''' if both <math>\text{Ob}(\mathcal{C})</math> and <math>\text{Hom}(\mathcal{C})</math> are [[sets]]. If <math>\mathcal{C}</math> is not small, then it is called '''large'''. <math>\mathcal{C}</math> is called '''locally small''' if <math>\text{Hom}(A,B)</math> is a set for all <math>A,B\in \text{Ob}(\mathcal{C})</math>. Most important categories in math are not small, but are locally small.
 
A category <math>\mathcal{C}</math> is called '''small''' if both <math>\text{Ob}(\mathcal{C})</math> and <math>\text{Hom}(\mathcal{C})</math> are [[sets]]. If <math>\mathcal{C}</math> is not small, then it is called '''large'''. <math>\mathcal{C}</math> is called '''locally small''' if <math>\text{Hom}(A,B)</math> is a set for all <math>A,B\in \text{Ob}(\mathcal{C})</math>. Most important categories in math are not small, but are locally small.
 +
 +
Intuitively we can think of the objects of <math>\mathcal{C}</math> 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.
 
{{stub}}
 
{{stub}}
  
 
[[Category:Category theory]]
 
[[Category:Category theory]]

Revision as of 00:34, 2 September 2008

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. This article is a stub. Help us out by expanding it.

Invalid username
Login to AoPS