nLab
strict morphism

Contents

Idea

A strict morphism is a morphism for which the notion of image and coimage coincide.

Compare with strict epimorphism.

Definition

In a category with limits and colimits

Let CC be a category with finite limits and colimits. Let f:cdf : c \to d be a morphism in CC.

Recall that the image of ff is the limit

Imflim(dd cd), Im f \simeq lim( d \rightrightarrows d \sqcup_c d ) \,,

i.e. the equalizer of dd cdd \rightrightarrows d \sqcup_c d,

while the coimage is the colimit

Coimfcolim(c× dcc). Coim f \simeq colim( c \times_d c \rightrightarrows c) \,.

By the various universal properties, there is a unique morphism

u:CoimfImf u : Coim f \to Im f

such that

c f d Coimf u Imf. \array{ c &\stackrel{f}{\to}& d \\ \downarrow && \uparrow \\ Coim f &\stackrel{u}{\to}& Im f } \,.

The morphism ff is called a strict morphism if uu is an isomorphism.

Examples

Examples of categories in which every morphism is strict include

Revised on July 9, 2010 07:23:40 by Urs Schreiber (87.212.203.135)