nLab congruence

Congruences

Congruences

Idea

In Euclidean geometry, by congruence one means the equivalence relation on the collection of subsets of a Euclidean space which regards two of these as equivalent if one is carried into the other by an isometry (of the ambient Euclidean space).

Similarly, in algebra, by a congruence one means certain equivalence relations on elements of algebraic structures, such as groups or rings (cf. e.g. the multiplicative group of integers modulo n).

Therefore, in category theory the term congruence is used in the broad generality of equivalence relations on (the generalized elements of) any object internal to any finitely complete category.

Definitions

Definition

In a category CC with pullbacks, a congruence on an object XX is an internal equivalence relation on XX (i.e.: an internal groupoid — hence an internal category with all morphisms being isomorphisms — but with no non-identity automorphisms).

This means that it consists of a subobject

(1)i:R(p 1,p 2)X×X i \;\colon\; R\stackrel{(p_1,p_2)}\hookrightarrow X \times X

of the Cartesian products of XX with itself, 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, i.e., p 1r=p 2r=1 Xp_1 r = p_2 r = 1_X;

  • internal symmetry: s:RRs \colon R \to R which interchanges p 1p_1 and p 2p_2, i.e., p 1s=p 2p_1\circ s = p_2 and p 2s=p 1p_2\circ s = p_1;

  • internal transitivity:

    t:R× XRRt \,\colon\, R \times_X R \to R

    (where on the left we have the fiber product of RX×Xpr 2XR \hookrightarrow X \times X \overset{pr_2}{\to} X with RX×Xpr 1R \hookrightarrow X \times X \overset{pr_1}{\to}, i.e. the subobject of pairs of composable pairs in relation)

    which factors the left/right projection map R× XRX×XR \times_X R \to X \times X through RR, i.e., the following diagram commutes

    R t R× XR (p 1q 1,p 2q 2) X×X, \array{ && R \\ & {}^{\mathllap{t}}\nearrow & \big\downarrow \\ R \times_X R & \underset{(p_1 q_1,p_2 q_2)}{\longrightarrow} & X \times X \mathrlap{\,,} }

    where q 1q_1 and q 2q_2 are the projections defined by the pullback diagram

    R× XR q 2 R q 1 p 1 R p 2 X\array{ R \times_X R & \overset{q_2}\longrightarrow & R \\ \big\downarrow{\mathrlap{{}^{q_1}}} && \big\downarrow{\mathrlap{{}^{p_1}}} \\ R & \underset{p_2}\longrightarrow & X }
Remark

Since ii is a monomorphism, the maps rr, ss, and tt are necessarily unique if they exist.

Remark

Equivalently, a congruence 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 a morphism x 1x 2x_1 \to x_2 then there is also a morphism x 2x 1x_2 \to x_1 (internally).

Remark

We can equivalently define a congruence RR as (a representing object of) a representable sub-presheaf of hom(,X×X)\hom(-, X \times X) so that for each object YY, the composite of R(Y)hom(Y,X×X)hom(Y,X)×hom(Y,X)R(Y) \hookrightarrow \hom(Y, X \times X) \cong \hom(Y, X) \times \hom(Y, X) exhibits R(Y)R(Y) as an equivalence relation on the set hom(Y,X)\hom(Y, X). The upshot of this definition is that it makes sense even when CC is not finitely complete.

Definition

A congruence which is the kernel pair of some morphism (example ) 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. , is always the kernel pair of its quotient, def. , if that quotient exists.

Examples

Example

Every diagonal morphism on an object is a congruence and has a quotient object isomorphic to the original object.

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 congruence R={(x,y):xy 1K}R = \{(x,y) : xy^{-1} \in K \}.

Alternatively, a quotient group by a normal subgroup KGK \hookrightarrow G is the quotient of the congruence 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.

See also

References

In common mathematics

In category theory

References using terminology as above:

But, for what it’s worth, a different use of the term “congruence” in category theory appears in Def. 3.5.1 on p. 89 in:

  • Michael Barr, Charles Wells, Category theory for computing science, Prentice-Hall International Series in Computer Science (1995); reprinted in: Reprints in Theory and Applications of Categories 22 (2012) 1-538 [pdf, tac:tr22]

Last revised on November 4, 2023 at 10:28:23. See the history of this page for a list of all contributions to it.