INTRODUCTION TO CATEGORY THEORY, II 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 pull- back if and only if the composite rectangle is a pullback. 4. A partially ordered set is called directed, if for every i; j 2 I there is l 2 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 2 I, there is k 2 K with i ≤ k). Let fMi;'jgI be a direct system over i I, and let fMi;'jgK 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 1 2 I i with i ≤ 1 for all i 2 I. If fMi;'jg is a direct system over I, prove that ∼ lim Mi = M1: −! 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)i2I ; ( i ) j≥i, abbreviated fMi; i g, j where (Mi)i2I is an indexed family of objects in C and ( i : Mj ! Mi)j≥i is an i indexed family of morphisms for which i = 1Mi for all i, and such that the following diagram commutes whenever k ≥ j ≥ i. Mk k k i j _ > M > M : j j i i 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) = i . Definition -1.2. Let I be a partially ordered set, and let C be a category. Let j (Mi)i2I ; ( i ) 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)i2I − such that j (1) i αj = αi whenever i ≤ j, j (2) for every object X of C and all morphisms fi : X ! Mi satisfying i 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 i > < 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)i2I ; ('j) i≤j, abbreviated fMi;'jg, i where (Mi)i2I is an indexed family of objects in C and ('j : Mi ! Mj)i≤j is an i indexed family of morphisms for which 'i = 1Mi for all i, and such that the following diagram commutes whenever i ≤ j ≤ k. Mi i i 'k 'j _ > M > M : j j k 'k 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) = 'j. Definition -1.4. Let I be a partially ordered set, and let C be a category. Let i (Mi)i2I ; ('j) 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)i2I −! such that i (1) αj'j = αi whenever i ≤ j, i (2) for every object X of C and all morphisms fi : Mi ! X satisfying fj'j = 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 'j _ Mj.