The language of categories

Mariusz Wodzicki March 15, 2011

1 Universal constructions

1.1 Initial and inal objects 1.1.1 Initial objects An object 푖 of a category 풞 is said to be initial if for any object 푐 ∈ Ob 풞, there exists a unique morphism 푖 → 푐. Any two initial objects are isomorphic and the isomorphism is unique.

1.1.2 Final objects An object 푓 of a category 풞 is said to be inal if for any object 푐 ∈ Ob 풞, there exists a unique morphism 푐 → 푓. Any two inal objects are isomorphic and the isomorphism is unique.

1.1.3 Example: the category of sets In the category of sets, denoted hereafter Set, the empty set ∅ is the unique initial object. A set is a inal object in Set precisely when it has one element. The same holds in the category of topological spaces Top.

1.1.4 Example: a partially ordered set Given a partially ordered set (푆, ≤), consider the category 풮 whose objects are elements of 푆 and, for any pair 푠, 푡 ∈ 푆, there is exactly one morphism 푠 → 푡 if

푠 ≤ 푡, and Hom풮(푠, 푡) = ∅ otherwise. In what follows we shall not distinguish a partially ordered set and the corresponding category.

1 The empty set, 0 = ∅, corresponds to the empty category:

Ob 0 = ∅. (1)

An element 푠 ∈ 푆 is an initial object of 풮, if 푠 = min 푆; it is inal, if 푠 = max 푆. In particular, the initial object is unique when it exists. The same ap- plies also to the inal object.

1.1.5 Example: the category of all (small) categories A category Γ is said to be small if its objects and arrows form sets. Small categories form a category denoted Cat and functors Γ ⇝ Γ are morphisms from Γ to Γ:

HomCat(Γ, Γ ) = Funct(Γ, Γ ) =∶ Γ .

The empty category, 0, is the unique initial object in Cat. Category 1 is a inal object.

1.1.6 Skeletal categories The category 풮 associated with an arbitrary partially ordered set (푆, ≤) is an example of a skeletal category: a category is said to be skeletal if there is at most one morphism between any pair of objects. The deinition of 풮 only requires that the relation on set 푆 be relexive and transitive. Any small skeletal category arises this way and small skeletal cat- egories form a subcategory of Cat which is isomorphic to the category of sets equipped with a relation that is both relexive and transitive.

1.1.7 Discrete categories A category 풞 is said to be discrete if the only morphisms in 풞 are the identity morphisms

id (푐 ∈ Ob 풞). Small discrete categories form a subcategory in Cat which is isomorphic to the category of sets, Set. Note that the initial and inal objects of Set are also initial and, respectively, inal objects in Cat.

2 1.1.8 Example: the category of 푅-modules An 푅-module is an initial as well as a inal object in the category of 푅-modules precisely when it has only one element. We speak of this module as the zero module even though it is not unique: any one-element set can be made into an 푅-module. This applies both to the category of left 푅-modules, which is denoted 푅-mod, and to the category of right 푅-modules which is denoted mod-푅.

A special case is provided by the category of vector spaces, Vect, over a ield 퐾.

1.2 Direct and inverse limits 1.2.1 Diagrams Let Γ be a small category. For any category 풞, functors Γ ⇝ 풞 are often called Γ-diagrams (in 풞). This usage is particularly frequent when Γ has very few objects and morphisms.

1.2.2 Example: the empty diagram For any category 풞, there is just one 0-diagram in 풞, where 0 denotes the empty category, cf. (1). We shall refer to it as the empty diagram.

1.2.3 Example: an object An object in a category 풞 is the same as a Γ-diagram in 풞 where Γ is the par- tially ordered set 2 (the set of all subsets of 0 = ∅).

1.2.4 Example: an arrow An arrow in a category 풞 is the same as a Γ-diagram in 풞 where Γ is the linearly ordered set 2 (the set of all subsets of 1 = {0}).

3 1.2.5 Example: a commutative square A commutative square in a category 풞

/ 푐 푐 (2)

↺   / 푐 푐 is the same as a Γ-diagram in 풞 where Γ is the partially ordered set 2 (the set of all subsets of 2 = {0, 1}).

1.2.6 Example: a composable pair of arrows A composable pair of arrows in a category 풞

/ / 푐 푐 푐 (3) is the same as a Γ-diagram in 풞 where Γ is the linearly ordered set 3 = {0, 1, 2}.

1.2.7 Example: a parallel pair Consider the category with just two objects, 푠 and 푡, and two morphisms 훾, 훾 ∶ 푠 → 푡. We shall denote this category by ⇉ .A ⇉-diagram in a category 풞 is the same as a pair of morphisms in 풞 with the common source and target

/ 푐 / 푐 . (4)

1.2.8 Example: a 퐺-object The category of monoids Mon is naturally isomorphic with the full subcate- gory of Cat consisting of categories with a single object:

퐺 ↭ Γ, End(∗) = Hom(∗, ∗) = 퐺, where ∗ denotes the only object of Γ. From now on we shall identify monoids with categories having a single object. Given a monoid 퐺 and a category 풞, a 퐺-diagram in 퐶 is the same as an

4 object 푐 ∈ Ob 풞 equipped with the action of 퐺 on 푐, i.e., with a homomorphism of monoids

퐺 → End풞(푐). We shall refer to it as a 퐺-object.

In the case of 풞 = Set, Top, or Vect, we talk of 퐺-sets, 퐺-spaces and, respectively, (퐾-linear) representations of 퐺.

1.2.9 The category of Γ-diagrams Since Γ-diagrams in 풞 are just functors,

Γ ⇝ 풞,

Γ-diagrams in 풞 form a category

풞 ∶= Funct(Γ, 풞) with morphisms being natural transformations of functors.

1.2.10 The diagonal embedding 풞 ↪ 풞 There is a canonical embedding of 풞 onto a full subcategory of 풞 which sends

푐 ∈ 풞 to the constant diagram Δ:

Δ(푔) = 푐 for any 푔 ∈ Ob Γ (5) and

Δ(훾) = id for any 훾 ∈ Arr Γ (6) In what follows we shall identify 풞 with its image in 풞 under the diagonal embedding.

1.2.11 Direct limits For a given Γ-diagram 퐷 ∶ Γ ⇝ 풞 in a category 풞, consider the category 풞 whose objects are families of morphisms

휙 = {휙 ∶ 퐷(푔) → 푐}∈Ob

5 with common target 푐 ∈ Ob 풞 which are compatible with the diagram, i.e., such that 휙 ∘ 퐷(훾) = 휙 (훾 ∈ Hom(푔, 푔 )). Morphisms 휙 → 휙 in category 풞 are deined as morphisms 훼 ∶ 푐 → 푐 between the targets such that

훼 ∘ 휙 = 휙 (푔 ∈ Ob Γ).

An initial object in 풞 is called the direct limit of diagram 퐷.1 The direct limit of 퐷 is denoted lim 퐷. −−→ In use there are also terms: the inductive limit of 퐷 and the push-out of 퐷. The latter usage is generally conined to diagrams with few vertices.

Exercise 1 Denote by 픻 the partially ordered set of positive integers ordered by the divisibility relation:

푚 ≤ 푛 if 푚|푛.

Consider the following 픻-diagram in the category of abelian groups Ab 푛 퐷 = ℤ, 퐷(푚|푛) ∶= multiplication by , (푚, 푛 ∈ ℤ ). 푚 Show that lim 퐷 = ℚ −−→ with the morphisms 퐷 → ℚ being the homomorphisms of abelian groups 푖 ℤ ⟶ ℚ, 푖 ↦ (푖 ∈ ℤ). 푛

1.2.12 Inverse limits For a given Γ-diagram 퐷 ∶ Γ ⇝ 풞 in a category 풞, consider the category 퐷풞 whose objects are families of morphisms

휓 = {휓 ∶ 푐 → 퐷(푔)}∈Ob

1Note that we are using the deinite article even though direct limit is usually not unique; it is unique only up to a unique isomorphism.

6 with common source 푐 ∈ Ob 풞 which are compatible with the diagram, i.e., such that i.e., such that

퐷(훾) ∘ 휓 = 휓 (훾 ∈ Hom(푔, 푔 )).

Morphisms 휓 → 휓 in category 퐷풞 are deined as morphisms 훼 ∶ 푐 → 푐 between the sources such that

훼 ∘ 휙 = 휙 (푔 ∈ Ob Γ).

A inal object in 풞 is called the inverse limit of diagram 퐷. The inverse limit of 퐷 is denoted lim 퐷. ←−− In use there are also terms: the projective limit of 퐷 and the pull-back of 퐷. The latter usage is generally conined to diagrams with few vertices whereas the former one is more often applied to diagrams with ininitely many vertices.

1.2.13 Limits and colimits Another usage gaining popularity is: limits—for inverse limits, and colim- its—for direct limits.

Exercise 2 Show that lim ∅ is a inal object of 풞, while lim ∅ is an initial object ←−− −−→ of 풞.

Exercise 3 Let ℕ be the set of natural numbers with the reverse linear order ≥ and let 푝 be a prime. Consider the following ℕ-diagram in the category of rings

퐷 = ℤ/푝 ℤ, 퐷(푚 ≥ 푛) ∶= the canonical epimorphism ℤ/푝 ℤ ↠ ℤ/푝 ℤ, where 푚, 푛 ∈ ℤ. Show that the inverse limit of 퐷 is the ring of 푝-adic numbers,

lim 퐷 = ℤ , ←−− with the morphisms ℤ → 퐷 being the quotient maps

ℤ ⟶ ℤ/푝 ℤ.

7 Exercise 4 Denote by 픻 the partially ordered set introduced in Exercise 1. Con- sider the following 픻op-diagram in the category of rings

op 퐷 = ℤ/푛ℤ, 퐷 (푚|푛) ∶= the canonical epimorphism ℤ/푚ℤ ↠ ℤ/푛ℤ, where 푚, 푛 ∈ ℤ. Show that

lim 퐷 = ℤ (7) ←−− prime with the morphisms lim 퐷 → 퐷 being the products over all primes of the canon- ←−− ical quotient maps () ℤ ⟶ ℤ/푝 ℤ, where 휔(푛) denotes the 푝-order of 푛 ∈ ℤ. Hint: we use here the canonical decomposition of a cyclic group into the product of the corresponding 푝-groups of order 푝(): 푍/푛ℤ ≃ ℤ/푝()ℤ. prime

1.2.14 The formal completion of ℤ The inverse limit lim 퐷 in Exercise 4 is called the formal completion of the ring ←−− of integers. It is denoted ℤ. The abelianization2 of the Galois group

Gal(ℚ) = Galℚalg/ℚ is canonically isomorphic to the additive group of ℤ. This is a deep result of Algebraic Number Theory.

Exercise 5 Show that ℤ is canonically isomorphic to the ring of endomorphisms

EndAb(휇(ℂ)) of the group of complex roots of identity

휇(ℂ) = {휁 ∈ ℂ ∣ 휁 = 1 for some 푛 ∈ ℕ}. 2The abelianization 퐺ab of a group 퐺 is the maximal abelian quotient of 퐺; it coincides with the quotient 퐺/[퐺, 퐺] of 퐺 by its commuattor subgroup.

8 1.2.15 Equalizers and coequalizers The inverse limit of a parallel pair (4) is called the equalizer of 푓 and 푔. The direct limit is called the coequalizer of 푓 and 푔.

1.2.16 Cartesian squares A commutative square 푎 / 푏 (8)

↺   푐 / 푑 in a category 풞 is said to be Cartesian if

푎 / 푏 (9)

 푐 is the inverse limit of 푏 (10)

 푐 / 푑

1.2.17 Cocartesian squares A commutative square (8) is said to be Cocartesian if, vice-versa, (10) is the direct limit of (9).

1.2.18 For a given Γ and a general category 풞, certain Γ-diagrams in 풞 possess either the direct or the inverse limit while others do not. A category 풞 is said to be complete if lim 퐷 exists for any small category Γ ←−− and any Γ-diagram in 풞. If lim 퐷 exists for any Γ-diagram, we say that category 풞 is cocomplete. −−→ Exercise 6 Show that the category of sets is complete.

9 Hint: consider the subset of the product of all 퐷,

퐷, ∈Ob consisting of the tuples 푥 ∈Ob which satisfy the following relations

푥() = 퐷(훾) 푥() (훾 ∈ Arr Γ). Exercise 7 Show that the category of sets is cocomplete.

Hint: consider the quotient of the disjoint union of all 퐷,

퐷, ∈Ob by the equivalence relation generated by the following relation: 푥 ∼ 푥 if 푥 = 퐷(훾) (푥 ) , 푥 ∈ 퐷() and 푥 ∈ 퐷(), for some 훾 ∈ Arr Γ. Exercise 8 Show that the category 풮 associated with a partially ordered set (푆, ≤) is complete if and only if (푆, ≤) is inf-complete, i.e., every subset of 푆 has in- imum. Similarly, show that 풮 is cocomplete if and only if (푆, ≤) is sup-complete, i.e., every subset of 푆 has supremum.

1.2.19 Functorial direct limits It is not infrequent that every Γ-diagram in a given category may have the cor- responding limits, and that those limits depend functorially on the diagram. We say that a category 풞 possesses functorial direct limits (for Γ-diagrams), if there exists a functor lim ∶ 풞 ⇝ 풞 (11) −−→ such that, for any object 푐 ∈ Ob 풞 and Γ-diagram 퐷 ∈ Ob 풞, there exists a natural in 푐 and 퐷 bijection

Hom lim 퐷, 푐 ↭ Hom (퐷, Δ ). (12) 풞 −−→ 풞 where Δ denotes the diagonal embedding functor introduced in 1.2.10.

10 1.2.20 Functorial inverse limits We say that a category 풞 possesses functorial inverse limits (for Γ-diagrams), if there exists a functor lim ∶ 풞 ⇝ 풞 (13) ←−− such that, for any object 푐 ∈ Ob 풞 and Γ-diagram 퐷 ∈ Ob 풞, there exists a natural in 푐 and 퐷 bijection

Hom 푐, lim 퐷 ↭ Hom (Δ , 퐷). (14) 풞 ←−− 풞 where Δ denotes the diagonal embedding functor introduced in 1.2.10.

1.3 Adjoint functors 1.3.1 In (12) and (14) we encounter a fundamentally important concept: a pair of adjoint functors. Suppose that, for a pair of functors,

o o/ o/ o/ 풞 /o /o /o / 풟 , (15) be given. If, for any 푐 ∈ Ob 풞 and 푑 ∈ Ob 풟, there exists a bijective correspon- dence Hom풞(퐹푑, 푐) ↭ Hom풟(푑, 퐺푐) (16) which is natural both in 푐 and 푑, then we say that 퐹 is left adjoint to 퐺, and 퐺 is right adjoint to 퐹.

1.3.2 If two functors, 퐹 and 퐹, are left adjoint to 퐺, then they are isomorphic.

1.3.3 Example: functorial direct and inverse limits A category 풞 possesses functorial direct limits if the diagonal embedding Δ ∶ 풞 ⇝ 풞 admits a left adjoint. Similarly, if Δ admits a right adjoint, then 풞 possesses functorial inverse limits.

11 Exercise 9 Let 퐺 be a monoid. Show that the functor,

() ∶ Set ⇝ Set, 푋 ↦ 푋 ∶= Fix(푋) = {푥 ∈ 푋 ∣ 푔푥 = 푥 for any 푔 ∈ 퐺},

which associates with a 퐺-set 푋 the ixed-point set, 푋 = Fix(푋), is right adjoint to the diagonal embedding functor Δ ∶ Set ⇝ Set. In particular, lim 푋 = 푋 = Fix (푋) ←−− for any 퐺-set 푋.

Exercise 10 Let 퐺 be a monoid. Show that the functor,

() ∶ Set ⇝ Set, 푋 ↦ 푋 ∶= 푋/퐺, which associates with a 퐺-set 푋 the set of orbits of 퐺 on 푋, i.e., the set of equiv- alence classes of the equivalence relation

푥 ∼ 푥 if 푥 = 푔푥 and 푥 = 푔푥 for some 푥 ∈ 푋 and 푔, 푔 ∈ 퐺. is left adjoint to the diagonal embedding functor Δ ∶ Set ⇝ Set. In particular, lim 푋 = 푋 = 푋 −−→ /퐺 for any 퐺-set 푋.

1.3.4 The sets of 퐺-invariants and 퐺-coinvariants Notation 푋 and 푋 is standard in representation theory where 푋 is called the set of 퐺-invariants, and 푋 is called the set of 퐺-coinvariants of 퐺-represen- tation 푋. Notation Fix(푋) and 푋/퐺, and the corresponding terminology: the set of ixed-point set and the set of orbits, are primarily used for “non-linear” actions, i.e., for 퐺-actions on sets, topological spaces, algebraic varieties.

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