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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, objects, 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 1784 times total.

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