INTRODUCTION to CATEGORY THEORY, II Homework

Homework 1, due Tuesday July 31.

For this exercise set, read Sections 3.1 - 3.3 of Emily Riehl’s book ”Category Theory in Context”.

1. (Riehl 3.1. vii) Prove that if k P > C h g ∨ ∨ B > A f is a pullback square and f is a monomorphism, then k is a monomorphism. 2. (Riehl 3.1.vi) Prove that if f,g E →h A ⇒ B is an equalizer diagram (Riehl, Def. 3.1.13), then h is a monomorphism. (Here I use the symbol ⇒ for two parallel arrows since ”rightrightarrows” failed to work with my LaTex package...) 3. (Riehl 3.1.viii) Consider a commutative rectangle

f1 f2 X1 > X2 > X3

h1 h2 h3 ∨ ∨ ∨ g1 g2 Y1 > Y2 > Y3 whose right-hand square is a pullback. Show that the left-hand square is a pullback if and only if the composite rectangle is a pullback. 4. A partially ordered set is called directed, if for every i, j ∈ I there is l ∈ I with i ≤ l and j ≤ l. Let K be a coﬁnal subset of a directed index set I (that is, i for each i ∈ I, there is k ∈ K with i ≤ k). Let {Mi, ϕj}I be a direct system over i I, and let {Mi, ϕj}K be the subdirect system whose indices lie in K. Prove that the direct limit over I is isomorphic to the direct limit over K. 5. A partially ordered set I has a top element if there exists an element ∞ ∈ I i with i ≤ ∞ for all i ∈ I. If {Mi, ϕj} is a direct system over I, prove that ∼ lim Mi = M∞. −→ 6. (Riehl 3.3.i) For any diagram K : J → C and any functor F : C → D:

Date: July 25, 2018. 1 2 INTRODUCTION TO CATEGORY THEORY, II

(1) Deﬁne a canonical map colim FK → F colim K, assuming both colimits exist. (2) Show that the functor F preserves the colimit of K just when this map is an isomorphism.

7. (Riehl 3.3 ii) Prove that a full and faithful functor reﬂects both limits and colimits.

Some deﬁnitions

Deﬁnition -1.1. Let I be a partially ordered set, and let C be a category. An j j inverse system in C is an ordered pair (Mi)i∈I , (ψi ) j≥i, abbreviated {Mi, ψi }, j where (Mi)i∈I is an indexed family of objects in C and (ψi : Mj → Mi)j≥i is an i indexed family of morphisms for which ψi = 1Mi for all i, and such that the following diagram commutes whenever k ≥ j ≥ i.

Mk k k ψi ψj ∨ > M > M . j j i ψi

Consider the partially ordered set I as a category: The objects are the elements i of I. Whenever i ≤ j, there is exactly one morphism kj : i → j. The inverse systems in C are just contravariant functors M : I → C, where M(i) = Mi and i j M(kj) = ψi .

Deﬁnition -1.2. Let I be a partially ordered set, and let C be a category. Let j (Mi)i∈I , (ψi ) j≥i be an inverse system in C over I. The inverse limit is an object

lim Mi ←− and a family of projections

(αi : lim Mi → Mi)i∈I ←− such that j (1) ψi αj = αi whenever i ≤ j, j (2) for every object X of C and all morphisms fi : X → Mi satisfying ψi fj = fi for all i ≤ j, there exists a unique morphism

θ : X → lim Mi ←− INTRODUCTION TO CATEGORY THEORY, II 3

making the diagram commute.

θ lim Mi < X ←− αi fi > < M α i fj j ∧ j ψi > < Mj

Deﬁnition -1.3. Let I be a partially ordered set, and let C be a category. A i i direct system in C is an ordered pair (Mi)i∈I , (ϕj) i≤j, abbreviated {Mi, ϕj}, i where (Mi)i∈I is an indexed family of objects in C and (ϕj : Mi → Mj)i≤j is an i indexed family of morphisms for which ϕi = 1Mi for all i, and such that the following diagram commutes whenever i ≤ j ≤ k.

Mi i i ϕk ϕj ∨ > M > M . j j k ϕk

Consider the partially ordered set I as a category: The objects are the elements i of I. Whenever i ≤ j, there is exactly one morphism kj : i → j. The direct systems in C are just covariant functors M : I → C, where M(i) = Mi and i i M(kj) = ϕj.

Deﬁnition -1.4. Let I be a partially ordered set, and let C be a category. Let i (Mi)i∈I , (ϕj) be a direct system in C over I. The direct limit is an object

lim Mi −→ and a family of insertions

(αi : Mi → lim Mi)i∈I −→ such that

i (1) αjϕj = αi whenever i ≤ j, i (2) for every object X of C and all morphisms fi : Mi → X satisfying fjϕj = fi for all i ≤ j, there exists a unique morphism

θ : lim Mi → X −→ 4 INTRODUCTION TO CATEGORY THEORY, II

making the diagram commute. θ lim Mi > X −→ < < >> αi fi

Mi αj fj i ϕj ∨ Mj