category theory

# Contents

## Definition

Traditionally, a wide subcategory of a category $C$ is a subcategory containing all the objects of $C$.

Equivalently, it is a subcategory through which the canonical functor $disc(Obj(C)) \to C$ (from the discrete category on the collection of objects) factors, or whose inclusion functor is bijective on objects.

Notice that the condition to contain all the object is not invariant under equivalence of categories and so the definition of wide subcategory above violates the principle of equivalence. A variant of the definition which fixes this is:

an essentially wide subcategory contains at least one object from each isomorphism class of objects; that is, its inclusion functor is essentially surjective on objects.

A wide subcategory is also called a lluf subcategory (“lluf” being “full” spelled backwards).

Revised on November 13, 2012 12:36:29 by Urs Schreiber (82.169.65.155)