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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: groupoid

Also defines:  source map, target map
Keywords:  small category, source, target maps

Cross-references: composition, composition map, identity, objects, category
There are 9 references to this object.

This is version 2 of small category, born on 2009-03-09, modified 2009-03-09.
Object id is 583, canonical name is SmallCategory.
Accessed 778 times total.

Classification:
Physics Classification: 00. (GENERAL)
 02. (Mathematical methods in physics)
 03.65.Fd (Algebraic methods )
Pending Errata and Addenda
None.
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