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[parent] alternative definition of small category (Definition)
Definition 0.1   A (small) category $ \mathcal{C}$ consists of a set of objects $ C_0$ and a set of arrows $ C_1$ together with the following structure:

  • a source map $ s: C_1 \to C_0$ assigning an object $ s(f)$ to each arrow $ f \in C_1$ ,
  • a target map: $ t: C_1 \to C_0$ assigning an object t(f) to each arrow $ f \in C_1$ ,
  • an identity map $ 1 : C_0 \to C_1$ assigning to each object $ A$ an arrow $ 1_A$ with

    $\displaystyle s(1_A) = t(1_A) = A,$
  • a composition map $ \circ : C_1 \times C_1 \to C_1$ assigning to each pair of arrows $ f,g$ , such that

    $\displaystyle s(g) = t(f)$
    , a third arrow $ g \circ f$ with $ s(g \circ f) = s(f)$ and $ t(g \circ f) = t(g)$ .
  • The composition thus defined “$ \circ$ ” is associative, that is,

    $\displaystyle h \circ (g \circ f) = (h \circ g) \circ f$
    whenever these compositions make sense.
  • the identity map satisfies $ f \circ 1_A = f $ for any $ f$ such that $ s(f) = A$ and $ 1_A \circ g = g$ , and any $ g$ such that $ t(g) = A$ .

Bibliography

P. A. Zito. 2008. [arXiv: math.CT]. 2-$ C^*$ -Categories with non-simple units. ,(Preprint).



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See Also: category theory

Other names:  small category
Also defines:  small category, source map, target map, identity, composition law, composition map
Keywords:  small category

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Cross-references: composition, category
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This is version 4 of alternative definition of small category, born on 2009-04-29, modified 2009-05-02.
Object id is 695, canonical name is AlternativeDefinitionOfSmallCategory.
Accessed 621 times total.

Classification:
Physics Classification: 00. (GENERAL)
 02. (Mathematical methods in physics)
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