Subfunctor - Definition

Definition

Let C be a category, and let F be a functor from C to the category of sets Set. A functor G from C to Set is a subfunctor of F if

  1. For all objects c of C, G(c) ⊆ F(c), and
  2. For all arrows f:c′→c of C, G(f) is the restriction of F(f) to G(c′).

This relation is often written as GF.

For example, let 1 be the category with a single object and a single arrow. A functor F:1Set maps the unique object of 1 to some set S and the unique identity arrow of 1 to the identity function 1S on S. A subfunctor G of F maps the unique object of 1 to a subset T of S and maps the unique identity arrow to the identity function 1T on T. Notice that 1T is the restriction of 1S to T. Consequently, subfunctors of F correspond to subsets of S.

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