nLab
congruence

Congruences

Definitions

Definition

In a finitely complete category CC, a congruence on an object XX is an internal equivalence relation on XX.

This means that it consists of a subobject R(p 1,p 2)X×XR\stackrel{(p_1,p_2)}\hookrightarrow X \times X equipped with the following morphisms: * internal reflexivity: r:XRr: X \to R which is a section both of p 1p_1 and of p 2p_2; * internal symmetry: s:RRs: R \to R which interchanges p 1p_1 and p 2p_2, namely p 1s=p 2p_1\circ s = p_2 and p 2s=p 1p_2\circ s = p_1; * internal transitivity: t:R× XRRt: R \times_X R \to R; where with the notation for the projections in the cartesian square

R× XR q 2 R q 1 p 2 R p 1 X\array{ R \times_X R & \stackrel{q_2}\rightarrow & R\\ \downarrow^{q_1} && \downarrow^{p_2}\\ R & \stackrel{p_1}\rightarrow & X }

the following holds: p 1q 1=p 1tp_1\circ q_1 = p_1\circ t and p 2q 2=p 2tp_2\circ q_2 = p_2\circ t.

Remark

Since (p 1,p 2)(p_1,p_2) is a monomorphism, the maps rr, ss, and tt are necessarily unique if they exist.

Example

Every kernel pair is a congruence.

Definition

A congruence which is the kernel pair of some morphism is called effective.

Definition

The coequalizer of a congruence is called a quotient object.

Proposition

An effective congruence is always the kernel pair of its quotient if that quotient exists.

Definition

The quotient of an effective congruence is an effective quotient.

Definition

A regular category is called an exact category if every congruence is effective.

Examples

Example

An equivalence relation is precisely a congruence in Set.

Example

The eponymous example is congruence modulo nn (for a fixed natural number nn), which can be considered a congruence on \mathbb{N} in the category of rigs, or on \mathbb{Z} in the category of rings.

Example

A quotient group by a normal subgroup KGK \hookrightarrow G is the quotient of the relation G×K(p 1,p 2)G×GG \times K \stackrel{(p_1,p_2)}{\hookrightarrow} G \times G, where p 1p_1 is projection on the first factor and p 2p_2 is multiplication in GG (these are source and target maps in the action groupoid GKG \sslash K).

A special case of this is that of a quotient module.

Revised on September 11, 2012 10:10:02 by Urs Schreiber (82.169.65.155)