A universal property specifies an object by the maps into or out of it. This chapter develops the constructions used later. Fix a category \(\mathcal C\). Write \(\operatorname{Hom}(X,Y)\) for its morphisms from \(X\) to \(Y\).
2.1. Diagrams and universal properties
Definition 2.1. Diagrams, cones, and limits
A small category has a set of objects and a set of morphisms. A small diagram in \(\mathcal C\) is a functor \(D:J\to\mathcal C\) from a small category \(J\): it assigns objects and arrows, preserving identities and composition. A cone consists of an object \(T\) and arrows \(t_j:T\to D(j)\) such that \(D(\alpha)t_j=t_k\) for every \(\alpha:j\to k\). A limit is a cone \((L,p_j)\) for which every cone \((T,t_j)\) factors through a unique arrow \(u:T\to L\): \(p_ju=t_j\) for all \(j\).
A cocone consists of \(t_j:D(j)\to T\) with \(t_kD(\alpha)=t_j\). A colimit is a cocone \((L,i_j)\) through which every cocone factors uniquely: \(ui_j=t_j\). “All small limits” or “all small colimits” means these exist for every small diagram.
Alphabeta Math: Diagrams · Cones and cocones · Limits and colimits
Proposition 2.2. Uniqueness with structure maps
Two limits of the same diagram have a unique isomorphism commuting with their cone maps. The corresponding assertion holds for colimits.
Alphabeta Math: Uniqueness
Proof.
For limits \((L,p_j)\) and \((L',p'_j)\), universality gives unique \(u:L\to L'\), \(v:L'\to L\) with \(p'_ju=p_j\), \(p_jv=p'_j\). Both \(vu\) and \(\mathrm{id}_L\) have composites \(p_j\) with every cone map, so uniqueness gives \(vu=\mathrm{id}_L\); likewise \(uv=\mathrm{id}_{L'}\). For colimits, universality gives \(u i_j=i'_j\), \(v i'_j=i_j\). Equality after every injection gives the same inverse equations.
□Uniqueness is always relative to the specified maps. It permits us to identify two constructions once their universal properties agree.
2.2. Products and coproducts
Definition 2.3. Products and coproducts
For a set-indexed family \((X_i)_{i\in I}\), a product is an object \(\prod_iX_i\) with maps \(p_i:\prod_iX_i\to X_i\) such that any family \(t_i:T\to X_i\) has a unique simultaneous factorization \(p_i u=t_i\). A coproduct is an object \(\coprod_iX_i\) with maps \(i_i:X_i\to\coprod_iX_i\) such that any family \(t_i:X_i\to T\) has a unique factorization \(u i_i=t_i\). These are the limit and colimit of a diagram with objects \(I\) and only identity arrows.
The empty product is terminal: there is exactly one map from each \(T\) into it. The empty coproduct is initial: there is exactly one map from it into each \(T\). A zero object is both initial and terminal.
Alphabeta Math: Products and coproducts
Example 2.4. Products and coproducts of abelian groups
The product of abelian groups \((A_i)\) is the group of all tuples \((a_i)\), with coordinatewise addition. Its projection maps have the product property because \(u(t)=(t_i(t))_i\) is the unique possible homomorphism. The coproduct is the direct sum \(\bigoplus_i A_i\), consisting of the tuples with finite support. Maps \(t_i:A_i\to T\) extend uniquely by \((a_i)\mapsto\sum_i t_i(a_i)\); the sum is finite. For infinitely many nonzero groups the product and direct sum generally differ. In \(\prod_{n\geq1}\mathbb Z\), the tuple \((1,1,\ldots)\) is not in \(\bigoplus_{n\geq1}\mathbb Z\).
Alphabeta Math: Direct sums · Coproduct universal property
2.3. Equalizers and coequalizers
Definition 2.5. Imposing equality of two arrows
For parallel arrows \(f,g:X\to Y\), an equalizer is \(e:E\to X\) with \(fe=ge\), universal among arrows satisfying that equation: whenever \(ft=gt\), there is a unique \(u:T\to E\) with \(eu=t\). A coequalizer is \(q:Y\to Q\) with \(qf=qg\), universal among arrows \(t:Y\to T\) satisfying \(tf=tg\): there is a unique \(u:Q\to T\) with \(uq=t\).
Alphabeta Math: Equalizers and coequalizers
Definition 2.6. Monomorphisms and epimorphisms
A morphism \(m:X\to Y\) is monic, or a monomorphism, if \(ma=mb\) implies \(a=b\) for all pairs of arrows with codomain \(X\). A morphism \(q:X\to Y\) is epic, or an epimorphism, if \(aq=bq\) implies \(a=b\) for all pairs with domain \(Y\). These are cancellation properties; they make sense without elements.
Alphabeta Math: Monomorphisms and epimorphisms
Proposition 2.7. Equalizers are monic; coequalizers are epic
Every equalizer arrow is monic, and every coequalizer arrow is epic.
Alphabeta Math: Cancellation properties
Proof.
If \(ea=eb=t\), then \(ft=gt\), and \(a,b\) are both factorizations of \(t\) through its equalizer. Uniqueness gives \(a=b\). If \(aq=bq=t\) for a coequalizer \(q\), then \(tf=tg\), and \(a,b\) are both factorizations of \(t\) through \(q\), hence equal.
□Example 2.8. An equation and a quotient
For abelian-group homomorphisms \(f,g:A\to B\), the subgroup \(E=\{a:f(a)=g(a)\}\), with its inclusion, is the equalizer: a map on which \(f,g\) agree lands in that subgroup and therefore factors uniquely. The image \(U=\operatorname{im}(f-g)\) is a subgroup because \(f-g\) is additive. The coequalizer is the coset group \(B/U\), with its quotient map. Here \([b]=b+U\) equals \([b']\) exactly when \(b-b'\in U\). The operations \([b]+[c]=[b+c]\), \(-[b]=[-b]\), and zero \([0]\) are well defined: changing representatives changes a sum or negative by an element of \(U\). Their abelian-group laws follow from the laws in \(B\). A homomorphism \(t:B\to T\) equalizes \(f,g\) exactly when it vanishes on \(\operatorname{im}(f-g)\); its factorization is \([b]\mapsto t(b)\), well-defined and unique.
Alphabeta Math: Abelian-group kernels and quotients · Quotient universal property
2.4. Constructing every small limit and colimit
Theorem 2.9. Limits from products and equalizers
If \(\mathcal C\) has all set-indexed products and equalizers, it has all small limits. For \(D:J\to\mathcal C\), form
\[A=\prod_{j\in\operatorname{Ob}J}D(j),\qquad B=\prod_{\alpha:j\to k\text{ in }J}D(k).\]Define \(s,t:A\to B\) by the components \(p_\alpha s=D(\alpha)p_j\) and \(p_\alpha t=p_k\). If \(e:L\to A\) equalizes \(s,t\), the maps \(p_je\) give a limit.
Alphabeta Math: Construction of small limits
Proof.
For each \(\alpha:j\to k\), equality \(se=te\) gives \(D(\alpha)p_je=p_ke\), the cone condition. Any cone \((T,t_j)\) gives a unique \(a:T\to A\) with \(p_ja=t_j\). Its cone equations give \(sa=ta\), since all components in \(B\) agree. There is a unique \(u:T\to L\) with \(eu=a\). This implies \(p_jeu=t_j\). Conversely any such \(u\) must satisfy \(eu=a\) by the product property, so the equalizer property makes it unique. Both index collections are sets because \(J\) is small, including the empty case.
□Theorem 2.10. Colimits from coproducts and coequalizers
If \(\mathcal C\) has all set-indexed coproducts and coequalizers, it has all small colimits. For \(D:J\to\mathcal C\), form
\[A=\coprod_{\alpha:j\to k\text{ in }J}D(j),\qquad B=\coprod_{j\in\operatorname{Ob}J}D(j).\]Let \(s,t:A\to B\) satisfy \(s i_\alpha=i_kD(\alpha)\), \(t i_\alpha=i_j\). If \(q:B\to L\) coequalizes \(s,t\), the maps \(q i_j\) give a colimit.
Alphabeta Math: Construction of small colimits
Proof.
The coequalizer equation gives \(q i_kD(\alpha)=q i_j\), the cocone condition. A cocone \((t_j:D(j)\to T)\) gives a unique \(b:B\to T\) with \(bi_j=t_j\). Its equations give \(bs=bt\) on every summand of \(A\), hence as arrows. A unique \(u:L\to T\) satisfies \(uq=b\), and therefore \(uq i_j=t_j\). Conversely any arrow with those composites satisfies \(uq=b\) by the coproduct property and is thus unique.
□Definition 2.11. Preservation
A functor \(F:\mathcal C\to\mathcal D\) preserves a limit or colimit if applying \(F\) to its object and structure maps again gives the corresponding universal cone or cocone. For coproducts this says that the comparison arrow
\[\coprod_iF(X_i)\longrightarrow F(\coprod_iX_i),\qquad i_i\longmapsto F(i_i),\]defined by the coproduct property, is an isomorphism. Indeed, if it is an isomorphism it transfers the universal property; if the image cocone is universal, Proposition 2.2 makes the comparison an isomorphism. Preserving all coproducts includes the empty coproduct.
Alphabeta Math: Preservation
These constructions need no addition. The next chapter equips morphisms with addition; then equalizers and coequalizers become kernels and cokernels of differences.
Worked examples
Each example checks the hypotheses and the calculation for the linked statements. The links in the statements return here.
Worked example 1. The limit of an arrow
Applies to: Definition 2.1, Proposition 2.2, Theorem 2.9.
In \(\mathrm{Ab}\), let the indexing category have two objects, their identities, and one arrow, sent to \(f:\mathbb Z\to\mathbb Z\), \(f(n)=2n\). It is small. A cone is a pair \(a,b:T\to\mathbb Z\) with \(b=2a\); it factors uniquely through \(L=\mathbb Z\) with projections \(1,f\), by the map \(a\). The product-equalizer construction gives \(\{(x,y)\in\mathbb Z^2:2x=y\}\). Identity arrows add only tautological equations. The compatible isomorphism from \(L\) to this model is \(n\mapsto(n,2n)\), with inverse first projection; the cone equations force this unique choice. This explicitly exhibits the abstract limit construction and its uniqueness.
Worked example 2. A coproduct and an arrow colimit
Applies to: Definition 2.3, Theorem 2.10.
For the discrete two-object diagram \(\mathbb Z,\mathbb Z/2\) in \(\mathrm{Ab}\), the direct sum is also the product. A pair \(a:T\to\mathbb Z,b:T\to\mathbb Z/2\) gives \(t\mapsto(a(t),b(t))\); a pair of maps out gives \((n,\bar r)\mapsto u(n)+v(\bar r)\). These formulas prove uniqueness in both directions. The empty family gives the trivial group: there is just one map to or from it, so it is terminal, initial, and zero.
For the arrow \(f(n)=2n\) of the previous example, a cocone has maps \(a,b:\mathbb Z\to T\) with \(a=bf\). Its colimit is the target \(\mathbb Z\), with maps \(f,1\). The coproduct-coequalizer construction is \(\mathbb Z^2/\langle(-n,2n):n\in\mathbb Z\rangle\). The map \([(x,y)]\mapsto2x+y\) is well defined, with inverse \(z\mapsto[(0,z)]\), because \((x,y)-(0,2x+y)=(x,-2x)\) is a relation. Thus the formula in Theorem 2.10 really produces this colimit.
Worked example 3. Equalizing two integer multiplications
Applies to: Definition 2.5, Definition 2.6, Proposition 2.7.
Take \(f,g:\mathbb Z\to\mathbb Z\), \(f(n)=2n,g(n)=4n\). Their equalizer is \(0\hookrightarrow\mathbb Z\), since \(2n=4n\) forces \(n=0\). A map \(t\) with \(ft=gt\) is zero and factors uniquely through 0. Their coequalizer is \(q:\mathbb Z\to\mathbb Z/2\): \(qf=qg=0\), and a homomorphism \(t\) with \(tf=tg\) kills \(2\mathbb Z\), so \(\bar n\mapsto t(n)\) is its unique factorization. The inclusion is monic since maps into 0 are unique. The quotient is epic since \(aq=bq\) can be evaluated at a preimage of each residue. This verifies the universal properties and both cancellation claims rather than identifying them solely with injectivity and surjectivity.
Worked example 4. Reduction preserves sums, but can destroy a kernel
Applies to: Definition 2.11.
For \(F(M)=M/2M\) and any set-indexed family of abelian groups, the map \(\bigoplus_iF(M_i)\to F(\bigoplus_iM_i)\) sends finitely supported residue tuples to the residue of that tuple. It is onto by finite support; it is one-to-one because a tuple lies in \(2\bigoplus_iM_i\) exactly when every coordinate lies in \(2M_i\). This includes the empty family. However \(\ker(2:\mathbb Z\to\mathbb Z)=0\), while the induced map on \(\mathbb F_2\) is zero and has kernel \(\mathbb F_2\). Thus preservation refers to the actual universal construction and can hold for coproducts while failing for kernels.
Counterexamples and limits of the hypotheses
Each construction below refutes the explicitly stated stronger claim. It does not refute the related theorem with its original hypotheses.
Counterexample 1. Monic and epic need not mean invertible
Related statements: Definition 2.6.
Claim refuted. In every category, a morphism that is both monic and epic is an isomorphism.
Proof.
Work in the category of commutative unital rings and unital ring maps, and let \(i:\mathbb Z\hookrightarrow\mathbb Q\). It is monic: equality of two maps after the injective function \(i\) gives equality on every element. To prove it epic, let \(a,b:\mathbb Q\to T\) agree on \(\mathbb Z\). For every nonzero integer \(n\), both \(a(1/n)\) and \(b(1/n)\) are inverses of the same element \(a(n)=b(n)\); inverses in a ring are unique, since \(u=u(nv)=(un)v=v\). Therefore \(a(m/n)=a(m)a(1/n)=b(m/n)\) for all rationals, proving cancellation. Nevertheless \(i\) has no inverse: a unital map \(\mathbb Q\to\mathbb Z\) would make 2 a unit in \(\mathbb Z\), which is impossible. This does not contradict Proposition 4.5, which assumes an abelian category; the present category is not additive.
Comments