APPENDIX 8 Toolbox

45

10. Limits and Colimits, Products and Equalizers

Limit constructions are a very important tool in category theory. We will introduce the basic facts on limits and colimits in this section.

Definition 8.10.1. A diagramscheme D is a small category (i. e. the class of objects is a set). Let C be an arbitrary category. A diagram in C over the diagramscheme D is a covariant functor F : D ⟶ C.

Example 8.10.2. (for diagramschemes)

1. The empty category D.

2. The category with precisely one object D and precisely one morphism 1_{D}.

3. The category with two objects D_{1}, D_{2} and one morphism f : D_{1} ⟶ D_{2} (apart from the two identities).

4. The category with two objects D_{1}, D_{2} and two morphisms f, g : D_{1} ⟶ D_{2} between them.

5. The category with a family of objects (D_{i}|i ∈ I) and the associated identities.

6. The category with four objects D_{1}, …, D_{4} and morphisms f, g, h, k such that the diagram

[figure: D_{3}; D_{4}; h; D_{1}; D_{2}; f; g; k]

commutes, i. e. kf = hg.

Definition 8.10.3. Let D be a diagramscheme and C a category. Each object C ∈ C defines a constant diagram K_{C} : D ⟶ C with K_{C}(D) := C for all D ∈ D and K(f) := 1_{C} for all morphisms in D. Each morphism f : C ⟶ C′ in C defines a constant natural transformation K_{f} : K_{C} ⟶ K_{C′} with K_{f}(D) = f. This defines a constant functor K : C ⟶ Funct(D, C) from the category C into the category of diagrams Funct(D, C).

Let F : D ⟶ C be a diagram. An object C together with a natural transformation π : K_{C} ⟶ F is called a limit or a projective limit of the diagram F with the projection π if for each object C′ ∈ C and for each natural transformation φ : K_{C′} ⟶ F there is a unique morphism f : C′ ⟶ C such that

[figure: K_{C}; F; π; φ; K_{C′}; K_{f}]

commutes, i. e. the diagrams

[figure: C; F(D_{i}); π_{i}; π_{j}; F(D_{j}); F(g)]

commute for all morphisms g : D_{i} ⟶ D_{j} in D (π is a natural transformation) and the diagrams

[figure: C; F(D_{i}).; π_{i}; φ_{i}; C′; f]

commute for all objects D_{i} in D.

A category C has limits for diagrams over a diagramscheme D if for each diagram F : D ⟶ C over D there is a limit in C. A category C is called complete if each diagram in C has a limit.

Example 8.10.4. 1. Let D be a diagramscheme consisting of two objects D_{1}, D_{2} and the identities. A diagram F : D ⟶ C is defined by giving two objects C_{1} and C_{2} in C. An object C_{1} × C_{2} together with two morphisms π_{1} : C_{1} × C_{2} ⟶ C_{1} and π_{2} : C_{1} × C_{2} ⟶ C_{2} is called a product of the two objects if C_{1} × C_{2}, π : K_{C_{1}×C_{2}} ⟶ F is a limit, i. e. if for each object C′ in C and for any two morphisms φ_{1} : C′ ⟶ C_{1} and φ_{2} : C′ ⟶ C_{2} there is a unique morphism f : C′ ⟶ C_{1} × C_{2} such that

[figure: φ_{1}; φ_{2}; C′; f; C_{1}; C_{1} × C_{2}; π_{1}; C_{2}; π_{2}]

commutes. The two morphisms π_{1} : C_{1} × C_{2} ⟶ C_{1} and π_{2} : C_{1} × C_{2} ⟶ C_{2} are called the projections from the product to the two factors.

2. Let D a diagramscheme consisting of a finite (non empty) set of objects D_{1}, …, D_{n} and the associated identities. A limit of a diagram F : D ⟶ C is called a finite product of the objects C_{1} := F(D_{1}), …, C_{n} := F(D_{n}) and is denoted by C_{1} × … × C_{n} = ∏_{i=1}^{n} C_{i}.

3. A limit over a discrete diagram (i. e. D has only the identities as morphisms) is called product of the C_{i} := F(D_{i}), i ∈ I and is denoted by ∏_{I} C_{i}.

4. Let D be the empty diagramscheme and F : D ⟶ C the (only possible) empty diagram. The limit C, π : K_{C} ⟶ F of F is called the final object. It has the property that for each object C′ in C (the uniquely determined natural transformation φ : K_{C′} ⟶ F does not have to be mentioned) there is a unique morphism f : C′ ⟶ C. In Set the one-point set is a final object. In Ab, Gr, Vec the zero group 0 is a final object.

5. Let D be the diagramscheme from 8.10.2 4. with two objects and two morphisms (different from the two identities). A diagram over D consists of two objects C_{1} and C_{2} and two morphisms g, h : C_{1} ⟶ C_{2}. The limit of such a diagram is called Equalizer of the two morphisms and is given by an object Ker(g, h) and a morphism π_{1} : Ker(g, h) ⟶ C_{1}. The second morphism to C_{2} arises from the composition π_{2} = gπ_{1} = hπ_{1}. The equalizer has the following universal property. For each object C′ and each morphism φ_{1} : C′ ⟶ C_{1} with gφ_{1} = hφ_{1}(= φ_{2}) there is a unique morphism f : C′ ⟶ Ker(g, h) with π_{1}f = φ_{1} (and thus π_{2}f = φ_{2}, i. e. the diagram

[figure: C′; f; φ_{1}; Ker(g, h); C_{1}; π_{1}; g; C_{2}; h]

commutes.

Problem 8.10.1. 1. Let F : D ⟶ Set be a discrete diagram. Show that the cartesian product over F coincides with the categorical product.

2. Let D be a pair of morphisms as in 8.10.4 5. and let F : D ⟶ Set be a diagram. Show that the set {x ∈ F(D_{1})|F(f)(x) = F(g)(x)} with the inclusion map into F(D_{1}) is an equalizer of F : D ⟶ Set.

3. Let F : D ⟶ Set be a diagram. Show that the set

{(x_{D}|D ∈ Ob D, x_{D} ∈ F(D))|∀(f : D ⟶ D′) ∈ D : F(f)(x_{D}) = x_{D′}}

with the projections into the single of the families is the limit of F.

Definition 8.10.5. Let F : D ⟶ C be a diagram. An object C and a natural transformation ι : F ⟶ K_{C} is called colimit or inductive limit of the diagram F with the injection ι if for each object C′ ∈ C and for each natural transformation φ : F ⟶ K_{C′} there is a unique morphism f : C ⟶ C′ such that

[figure: F; K_{C}; ι; φ; K_{C′}; K_{f}]

commutes, i. e. the diagram

[figure: F(D_{j}); C; ι_{j}; ι_{i}; F(D_{i}); F(g)]

commutes for all morphisms g : D_{i} ⟶ D_{j} in D (ι is a natural transformation) and the diagram

[figure: F(D_{i}); C; ι_{i}; φ_{i}; C′.; f]

commutes for all objects D_{i} in D.

The special colimits that can be formed over the diagrams as in Example 8.10.4 are called coproduct, initial object, resp. coequalizer.

Example 8.10.6. In Vec the object 0 is an initial object. In 𝕂-Alg the object 𝕂 is an initial object. In Geom the one-element functor A ↦ {∗} is a final object. In 𝕂-Alg the object {a ∈ A|f(a) = g(a)} is the equalizer of the two algebra homomorphisms f : A ⟶ B and g : A ⟶ B. In 𝕂Alg the cartesian (set of pairs) and the categorical products coincide.

Remark 8.10.7. A colimit of a diagram C is a limit of the corresponding (dual) diagram in the dual category C^{op}. Thus theorems about limits in arbitrary categories automatically also produce (dual) theorems about colimits. However, observe that theorems about limits in a particular category (for example the category of vector spaces) translate only into theorems about colimits in the dual category, which most often is not too useful.

Proposition 8.10.8. Limits and colimits of diagrams are unique up to isomorphism.

Proof. Let F : D ⟶ C be a diagram and let C, π and C̃, π̃ be limits of F. Then there are unique morphisms f : C̃ ⟶ C and g : C ⟶ C̃ with πK_{f} = π̃ and π̃K_{g} = π. This implies πK_{1_{C}} = π id_{K_{C}} = π = π̃K_{g} = πK_{f}K_{g} = πK_{fg} and analogously π̃K_{1C̃} = π̃K_{gf}. Because of the uniqueness this implies 1_{C} = fg and 1C̃ = gf.

Remark 8.10.9. Now that we have the uniqueness of the limit resp. colimit (up to isomorphism) we can introduce a unified notation. The limit of a diagram F : D ⟶ C will be denoted by \underleftarrow{lim}(F), the colimit by \underrightarrow{lim}(F).

Theorem 8.10.10. If C has arbitrary products and equalizers then C has arbitrary limits, i. e. C is complete.

Proof. Let D be a diagramscheme and F : D ⟶ C a diagram. First we form the products ∏_{D∈Ob D} F(D) and ∏_{f∈MorD} F(Codom(f)) where Codom(f) is the codomain (range) of the morphism f : D′ ⟶ D′′ in D so in this case Codom(f) = D′′. We define for each morphism f : D′ ⟶ D′′ two morphisms as follows

p_{f} := π_{F(D′′)} : ∏_{D∈Ob D} F(D) ⟶ F(D′′) = F(Codom(f))

and

q_{f} := F(f)π_{F(D′)} : ∏_{D∈Ob D} F(D) ⟶ F(D′) ⟶ F(D′′) = F(Codom(f)).

These two families of morphisms induce two morphisms into the corresponding product

p, q : ∏_{D∈Ob D} F(D) ⟶ ∏_{f∈MorD} F(Codom(f))

with π_{f}q = q_{f} and π_{f}p = p_{f}. Now we show that the equalizer of these two morphisms

[figure: Ker(p, q); ∏_{D∈Ob D} F(D); ψ; ∏_{f∈MorD} F(Codom(f)); p; q]

is the limit of the diagram F : D ⟶ C. We have pψ = qψ. The morphism ρ(D) := π_{F(D)}ψ : Ker(p, q) ⟶ ∏_{D∈Ob D} F(D) ⟶ F(D) defines a family of morphisms for D ∈ Ob D. If f : D′ ⟶ D′′ is in D then the diagram

[figure: Ker(p, q); F(D′); ρ(D′); ρ(D′′); F(D′′); F(f)]

is commutative because of F(f)ρ(D′) = F(f)π_{F(D′)}ψ = q_{f}ψ = π_{f}qψ = π_{f}pψ = p_{f}ψ = π_{F(D′′)}ψ = ρ(D′′). Thus we have obtained a natural transformation ρ : K_{Ker(p,q)} ⟶ F.

Now let an object C′ and a natural transformation φ : K_{C′} ⟶ F be given. Then this defines a unique morphism g : C′ ⟶ ∏_{D∈Ob D} F(D) with π_{F(D)}g = φ(D) for all D ∈ D. Since φ is a natural transformation we have φ(D′′) = F(f)φ(D′) for each morphism f : D′ ⟶ D′′. Thus we obtain π_{f}pg = p_{f}g = π_{F(D′′)}g = φ(D′′) = F(f)φ(D′) = F(f)π_{F(D′)}g = q_{f}g = π_{f}qg for all morphisms f : D′ ⟶ D′′ hence pg = qg. Thus g can be uniquely factorized through the equalizer ψ : Ker(p, q) ⟶ ∏_{D∈Ob D} F(D) in the form g = ψh with h : C′ ⟶ Ker(p, q). Then we have ρ(D)h = π_{F(D)}ψh = π_{F(D)}g = φ(D) for all D ∈ D hence ρK_{h} = φ.

Finally let another morphism h′ : C′ ⟶ Ker(p, q) with ρK_{h′} = φ be given. Then we have π_{F(D)}ψh′ = ρ(D)h′ = φ(D) = ρ(D)h = π_{F(D)}ψh hence ψh′ = ψh = g. Because of the uniqueness of the factorization of g through ψ we get h = h′. Thus (Ker(p, q), ρ) is the limit of F.

Remark 8.10.11. The proof of the preceding Theorem gives an explicit construction of the limit of F as an equalizer

[figure: Ker(p, q); ∏_{D∈Ob D} F(D); ψ; ∏_{f∈MorD} F(Codom(f)); p; q]

Hence the limit can be represented as a subobject of a suitable product. Dually the colimit can be represented as a quotient object of a suitable coproduct. This construction will be used in chapter 3.

Another fact is very important for us, the fact that certain functors preserve limits resp. colimits. We say that a functor G : C ⟶ C′ preserves limits over the diagramscheme D if \underleftarrow{lim}(GF) ≅ G(\underleftarrow{lim}(F)) for each diagram F : D ⟶ C.

Proposition 8.10.12. Covariant representable functors preserve limits. Contravariant representable functors map colimits into limits.

Proof. We only prove the first assertion. The second assertion is dual to the first one. For a diagram F : D ⟶ Set the set

{(x_{D}|D ∈ Ob D, x_{D} ∈ F(D))|∀(f : D ⟶ D′) ∈ D : F(f)(x_{D}) = x_{D′}}

is a limit of F by Problem 8.1. Now let a diagram F : D ⟶ C be given and let \underleftarrow{lim}(F) be the limit. Furthermore let Mor_{C}(C′, -) : C ⟶ Set be a representable functor. By the definition of the limit of F there is a unique morphism f : C′ ⟶ \underleftarrow{lim}(F) with πK_{f} = φ for each natural transformation φ : K_{C′} ⟶ F. This defines an isomorphism Nat(K_{C′}, F) ≅ Mor_{C}(C′, \underleftarrow{lim}(F)). Hence we have

\underleftarrow{lim}(Mor_{C}(C, F)) ≅

{(φ(D) : C′ ⟶ F(D)|D ∈ D)|∀(f : D ⟶ D′) ∈ D : F(f)φ(D) = φ(D′)} = Nat(K_{C′}, F) ≅ Mor_{C}(C′, \underleftarrow{lim}(F)).

Corollary 8.10.13. Let F : C ⟶ C′ be left adjoint to G : C′ ⟶ C. Then F preserves colimits and G preserves limits.

Proof. For a diagram H : D ⟶ C we have

Mor_{C}(-, \underleftarrow{lim}(GH)) ≅ \underleftarrow{lim} Mor_{C}(-, GH) ≅ \underleftarrow{lim} Mor_{C′}(F-, H) ≅

Mor_{C′}(F-, \underleftarrow{lim}(H)) ≅ Mor_{C}(-, G(\underleftarrow{lim}(H))),

hence \underleftarrow{lim}(GH) ≅ G(\underleftarrow{lim}(H)) as representing objects. The proof for the left adjoint functor is analogous.