Difference between revisions of "Category (category theory)"

(New page: A category, <math>\mathcal{C}</math>, is a mathematical object consisting of: * A class, <math>\text{Ob}(\mathcal{C})</math> of objects. * For every pair of objects <math>A,B\in \text...)
 
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** (associativity) Given <math>f:A\to B</math>, <math>g:B\to C</math> and <math>h:C \to D</math> we have <cmath>h\circ(g\circ f) = (h \circ g)\circ f.</cmath>
 
** (associativity) Given <math>f:A\to B</math>, <math>g:B\to C</math> and <math>h:C \to D</math> we have <cmath>h\circ(g\circ f) = (h \circ g)\circ f.</cmath>
 
** (identity) For and object <math>X</math>, there is an identity morphism <math>1_X:X\to X</math> such that for any <math>f:A\to B</math>: <cmath>1_B\circ f = f = f\circ 1_A.</cmath>
 
** (identity) For and object <math>X</math>, there is an identity morphism <math>1_X:X\to X</math> such that for any <math>f:A\to B</math>: <cmath>1_B\circ f = f = f\circ 1_A.</cmath>
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[[Category:Category theory]]

Revision as of 00:09, 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 $f:A\to B$: \[1_B\circ f = f = f\circ 1_A.\]

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