nLab
quotient object

Context

Category theory

Limits and colimits

Contents

Definition

The quotient object QQ of a congruence (an internal equivalence relation) EE on an object XX in a category CC is the coequalizer QQ of the induced pair of maps EXE \to X.

If EE is additionally the kernel pair of the map XQX \to Q, then QQ is called an effective quotient (and EE is called an effective congruence, with the map XQX \to Q being an effective epimorphism).

Sometimes the term is used more loosely to mean an arbitrary coequalizer. It may also refer to a co-subobject of XX (that is, a subobject of XX in the opposite category C opC^\op), without reference to any congruence on XX. Note that in a regular category, any regular epimorphism (i.e. a “regular quotient” in the co-subobject sense) is in fact the quotient (= coequalizer) of its kernel pair (actually, we can prove this under weaker hypotheses; see below).

Galois connection between quotients and relations

As we have said, there are various notions of quotient object. Let us consider the most general one, so that Quot(X)Quot(X) of an object XX denotes the poset of co-subobjects of XX, in other words the posetal reflection of the preorder of epis XQX \to Q which is a full subcategory of the co-slice category XCX \downarrow \mathbf{C}. A regular quotient then refers to a regular epi XQX \to Q.

On the other hand, let Rel(X)Rel(X) be the poset of relations on XX, i.e., the poset of subobjects of X×XX \times X, or in other words the posetal reflection of the preorder of monos i=e 1,e 2:EX×Xi = \langle e_1, e_2 \rangle: E \rightarrowtail X \times X which is a full subcategory of the slice category CX×X\mathbf{C} \downarrow X \times X.

Between Quot(X)Quot(X) and Rel(X)Rel(X) there is a relation \perp where qe 1,e 2q \perp \langle e_1, e_2 \rangle means exactly qe 1=qe 2q \circ e_1 = q \circ e_2. If the coequalizer coeq(e 1,e 2)coeq(e_1, e_2) of the parallel pair e 1,e 2:EXe_1, e_2: E \rightrightarrows X exists, then by definition we have coeq(e 1,e 2)qcoeq(e_1, e_2) \leq q iff qe 1=qe 2q e_1 = q e_2. On the other hand, if the kernel pair ker(q)\ker(q) of qq exists, then by definition we have qe 1=qe 2q e_1 = q e_2 iff e 1,e 2ker(q)\langle e_1, e_2 \rangle \leq \ker(q).

This indicates in a category which admits coequalizers and kernel pairs, we have

Proposition

coeq:Rel(X)Quot(X)coeq: Rel(X) \to Quot(X) is left adjoint to ker:Quot(X)Rel(X)\ker: Quot(X) \to Rel(X).

Or, in other words, that ker\ker and coeqcoeq set up a Galois connection between Quot(X) opQuot(X)^{op} and Rel(X)Rel(X).

Restricting consideration to kernel pairs only of epis, or coequalizers only of jointly monic pairs, is no real restriction in the presence of epi-mono factorizations:

Lemma

In a category where every morphism f:ABf: A \to B has an epi-mono factorization f=iqf = i \circ q, we have ker(f)=ker(q)\ker(f) = \ker(q). Similarly, for a pair f,g:XYf, g: X \rightrightarrows Y, we have coeq(f,g)=coeq(e 1,e 2)coeq(f, g) = coeq(e_1, e_2) where f,g:XY×Y\langle f, g \rangle: X \to Y \times Y factors as an epi p:XEp: X \to E followed by a mono e 1,e 2:EY×Y\langle e_1, e_2 \rangle: E \to Y \times Y.

Proof

We prove just the first statement; the second is proven similarly. It suffices to observe that the same class of jointly monic pairs (e 1,e 2)(e_1, e_2) are coequalized by ff as by qq; the kernel pair is by definition the maximum of this class. If qe 1=qe 2q e_1 = q e_2, then by applying ii to both sides we deduce fe 1=fe 2f e_1 = f e_2. If fe 1=fe 2f e_1 = f e_2, i.e., if iqe 1=iqe 2i q e_1 = i q e_2, then qe 1=qe 2q e_1 = q e_2 by monicity of ii.

Proposition

Suppose C\mathbf{C} is a category with coequalizers and kernel pairs and where every morphism has an epi-mono factorization. Then every regular epi qq is the coequalizer of its kernel pair: q=coeqker(q)q = coeq \circ \ker(q). And every kernel pair is the kernel pair of its coequalizer: i=kercoeq(i)i = \ker \circ coeq(i).

Proof

We just prove the first statement; the second is proved similarly. We have of course a counit coeqker(q)qcoeq \circ \ker(q) \leq q. On the other hand, if q=coeq(f,g)q = coeq(f, g) (where we may assume f,g\langle f, g \rangle is monic by the lemma), then we have a unit f,gkercoeq(f,g)=ker(q)\langle f, g \rangle \leq \ker \circ coeq(f, g) = \ker(q); applying coeqcoeq to each side, we have qcoeqker(q)q \leq coeq \circ \ker(q), as desired.

In toposes

Constructing quotient objects in an elementary topos E\mathbf{E}, starting from one or another standard definition (e.g., finitely complete category with power objects) that doesn’t already mention colimits, is not trivial.

The standard approach seen in textbooks (see for example Sheaves in Geometry and Logic), apparently first introduced by C.J. Mikkelsen but first published by Paré, is to prove that the contravariant power object functor P:E opEP \colon \mathbf{E}^{op} \to \mathbf{E} is monadic. It follows that PP reflects finite limits in E op\mathbf{E}^{op} from limits in E\mathbf{E}, but finite limits in E op\mathbf{E}^{op} are of course finite colimits in E\mathbf{E}.

This elegant approach does involve a fair amount of categorical machinery (a monadicity theorem, Beck-Chevalley conditions, and consideration of up to six applications of the power object functor), making it a challenge to get across intuitively in say an undergraduate course.

Other approaches that are closer to naive or common sense set-theoretic reasoning are possible. In the case of quotient objects, to form the coequalizer of a parallel pair f,g:XYf, g: X \rightrightarrows Y, we outline a possible path to take (see also at quotient type – from univalence):

  • Construct enough of the internal logic to make available logical operators ,,\wedge, \Rightarrow, \forall.

  • Construct an internal intersection operator :PPXPX\bigcap: P P X \to P X via the formula Φ={x:X| S:PXΦSSx}\bigcap \Phi = \{x: X\; |\; \forall_{S: P X} \Phi \ni S \Rightarrow S \ni x\}.

  • Construct the image of a map f:XYf: X \to Y by taking the internal intersection of all subobjects of YY through which ff factors.

  • To construct the coequalizer of f,g:XYf, g: X \rightrightarrows Y:

    • Take the image of f,g:XY×Y\langle f, g \rangle: X \to Y \times Y to get a relation p 1,p 2:RY×Y\langle p_1, p_2 \rangle: R \hookrightarrow Y \times Y. According to Lemma 1, a coequalizer of (p 1,p 2)(p_1, p_2) is a coequalizer of (f,g)(f, g).
    • Then take the equivalence relation e 1,e 2:EY×Y\langle e_1, e_2 \rangle: E \hookrightarrow Y \times Y generated by RR, by taking the intersection of all equivalence relations containing RR; a coequalizer of (e 1,e 2)(e_1, e_2) is a coequalizer of (p 1,p 2)(p_1, p_2) (akin to the fact that any coequalizer must be the coequalizer of its kernel pair, as in Proposition 2).
    • Take the image factorization of the map χ E:YPY\chi_E: Y \to P Y that classifies the relation EY×YE \hookrightarrow Y \times Y. That is, factor χ E\chi_E as an epi q:YQq: Y \to Q followed by a mono QPYQ \to P Y. Then qq is the desired coequalizer of (e 1,e 2)(e_1, e_2).

Notice that what these steps collectively do is form the object QQ of equivalence classes of the equivalence relation generated by the relation f(x)g(x)f(x) \sim g(x), which is exactly what we would do in ordinary set theory. Full details will appear elsewhere.

In higher category theory

These notions have generalizations when CC is an (∞,1)-category:

For instance an action groupoid is a quotient of a group action in 2-category theory.

Examples

(quotient norm)

References

Revised on April 10, 2016 12:50:40 by Todd Trimble (67.81.95.215)