Functor

Revision as of 01:14, 2 September 2008 by Jam (talk | contribs) (New page: A '''functor''' is a type of map between two categories. More precisely, a functor <math>F:\mathcal{C} \to \mathcal{D}</math> is a mapping which * sends eve...)
(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)

A functor is a type of map between two categories.

More precisely, a functor $F:\mathcal{C} \to \mathcal{D}$ is a mapping which

  • sends every object $X$ of $\mathcal{C}$ to and object $F(X)$ of $\mathcal{D}$.
  • sends every morphism $f:X\to Y$ of $\mathcal{C}$ to a morphism $F(f):F(X)\to F(Y)$ of $\mathcal{D}$.

Which satisfies the conditions:

  • $F(1_X) = 1_{F(X)}$ for all $X\in \text{Ob}(\mathcal{C})$.
  • $F(g\circ f) = F(g)\circ F(f)$ for all morphisms $f:X\to Y$ and $g:Y \to Z$ of $\mathcal{C}$.

This article is a stub. Help us out by expanding it.