San José State University
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The Hom-set Definition of
Mathematical Categories

The original definition of a category is in terms of a set S of objects and a set A of arrows linking the elements of S. The hom-set of objects x and y in category C is the set of all arrows in C with a domain of x and a codomain of y. It is denoted as

homC(x, y) = {f | f is an arrow f:a → b in C}

Alternatively a category could be defined as:

  1. A set of objects S
  2. A function which assigns to each ordered pair <x, y> of objects a set hom(x, y).
  3. For each ordered triple of objects <x, y, z> a function, called composition,

    hom(y, z)×hom(x, y) → hom(a, c)

    This composition function may be expressed as <f, g> → f°g for f ∈ hom(y, z) and g ∈ hom(x, y).

  4. For each object y there is an element of hom(y y), denoted as 1y which is an identity element of hom(y, y).
  5. The composition function is associative.
  6. If <x, y> ≠ <x', y'> then hom(x, y) and hom(x', y') are disjoint; i.e.,

    hom(x, y) ∩ hom(x', y') = ∅

(To be continued.)

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