nLab
initial object

Context

Category theory

Limits and colimits

Contents

Definition

An initial object in a category CC is an object \emptyset such that for any object xx of CC, there is a unique morphism !:x!:\emptyset\to x. An initial object, if it exists, is unique up to unique isomorphism, so we speak of the initial object.

An initial object may also be called coterminal, universal initial, co-universal, or simply universal.

Initial objects are the dual concept to terminal objects: an initial object in CC is the same as a terminal object in C opC^{op}. An object that is both initial and terminal is called a zero object.

Hence the initial object may also be viewed as the colimit over the empty diagram.

Examples

Strict initial objects

An initial object \emptyset is called a strict initial object if any morphism xx\to \emptyset must be an isomorphism. The initial objects of a poset, of SetSet, CatCat, TopTop, and of any topos (in fact, any extensive category) are strict. At the other extreme, a zero object is only a strict initial object if the category is trivial (equivalent to the terminal category).

Revised on April 21, 2017 10:56:10 by Urs Schreiber (131.220.184.222)