Sunday, January 12, 2014

Categorical Limits and Colimits

Many types of universal constructions that appear in a wide variety of mathematical contexts can be realized as categorical limits and colimits. To define a limit we first need the notion of a cone of a diagram. A diagram of type $J$ in a category $\mathcal C$ is a simply a functor $F \colon J \to \mathcal C$. We imagine $J$ as an indexing for the objects and morphisms in $\mathcal C$ under consideration. When $J$ is a finite category it can be visualized as a directed graph.

A cone of $F$ is a pair $(N, \Psi)$ where $N$ is an object of $\mathcal C$ and $\Psi$ is a collection of $\mathcal C$-morphisms $\Psi_X \colon N \to F(X) $ (one for each object $X$ in $J$) such that $\Psi_Y = F(f) \circ \Psi_X$ for every $J$-morphism $f \colon X \to Y$. Given two cones $(N, \Psi)$ and $(M, \Phi)$ we call a $\mathcal C$-morphism $u \colon N \to M$ a cone morphism from $(N, \Psi)$ to $(M, \Phi)$ provided that $u$ respects the cone property, which is to say that for every object $X$ in $J$ the morphism $\Psi_X$ factors through $u$, i.e., $\Psi_X = \Phi_X \circ u$. We will write $u \colon (N, \Psi) \to (M, \Phi)$ to denote that $u$ is a cone morphism.