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 \colon X \to R which is a section both of p 1p_1 and of p 2p_2;

  • internal symmetry: s:RRs \colon 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 1 R p 2 X\array{ R \times_X R & \stackrel{q_2}\rightarrow & R \\ \downarrow^{\mathrlap{q_1}} && \downarrow^{\mathrlap{p_1}} \\ R & \stackrel{p_2}\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.

Remark

Equivalently, a cogruence on XX is an internal category with XX the object of objects, such that the (source,target)-map is a monomorphism and such that if there is there is a morphism x 1x 2x_1 \to x_2 then there is also a morphism x 2x 1x_2 \to x_1 (internally).

Definition

A congruence which is the kernel pair of some morphism (example 1) is called effective.

Definition

The coequalizer of a congruence is called a quotient object.

The quotient of an effective congruence is called an effective quotient.

Definition

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

Properties

Proposition

An effective congruence, def. 2, is always the kernel pair of its quotient, def. 3, if that quotient exists.

Examples

Example

Every kernel pair is a congruence.

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 November 9, 2014 19:12:05 by Urs Schreiber (87.183.162.48)