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EXERCISES on LIMITS & COLIMITS Exercise 1. Prove That Pullbacks Of EXERCISES ON LIMITS & COLIMITS PETER J. HAINE Exercise 1. Prove that pullbacks of epimorphisms in Set are epimorphisms and pushouts of monomorphisms in Set are monomorphisms. Note that these statements cannot be deduced from each other using duality. Now conclude that the same statements hold in Top. Exercise 2. Let 푋 be a set and 퐴, 퐵 ⊂ 푋. Prove that the square 퐴 ∩ 퐵 퐴 퐵 퐴 ∪ 퐵 is both a pullback and pushout in Set. Exercise 3. Let 푅 be a commutative ring. Prove that every 푅-module can be written as a filtered colimit of its finitely generated submodules. Exercise 4. Let 푋 be a set. Give a categorical definition of a topology on 푋 as a subposet of the power set of 푋 (regarded as a poset under inclusion) that is stable under certain categorical constructions. Exercise 5. Let 푋 be a space. Give a categorical description of what it means for a set of open subsets of 푋 to form a basis for the topology on 푋. Exercise 6. Let 퐶 be a category. Prove that if the identity functor id퐶 ∶ 퐶 → 퐶 has a limit, then lim퐶 id퐶 is an initial object of 퐶. Definition. Let 퐶 be a category and 푋 ∈ 퐶. If the coproduct 푋 ⊔ 푋 exists, the codiagonal or fold morphism is the morphism 훻푋 ∶ 푋 ⊔ 푋 → 푋 induced by the identities on 푋 via the universal property of the coproduct. If the product 푋 × 푋 exists, the diagonal morphism 훥푋 ∶ 푋 → 푋 × 푋 is defined dually. Exercise 7. In Set, show that the diagonal 훥푋 ∶ 푋 → 푋 × 푋 is given by 훥푋(푥) = (푥, 푥) for all 푥 ∈ 푋, so 훥푋 embeds 푋 as the diagonal in 푋 × 푋, hence the name. The codiagonal 훻푋 ∶ 푋 ⊔ 푋 → 푋 is a bit more mysterious. Give a description of 훻푋 (still in the category Set). Exercise 8. Let 퐶 be a category and 푋, 푌 ∈ 퐶. Suppose that the coproducts 푋⊔ 푋 and 푌⊔ 푌 exist, and let 훻푋 ∶ 푋 ⊔ 푋 → 푋 and 훻푌 ∶ 푌 ⊔ 푌 → 푌 denote the codiagonals. Show that for any morphism 푓∶ 푋 → 푌 we have 푓 ∘ 훻푋 = 훻푌 ∘ (푓 ⊔ 푓). What is the dual statement? Date: January 28, 2018. 1 2 PETER J. HAINE Exercise 9. Let 퐶 be a category with pullbacks and 푋1 푋0 푋2 푍1 푍0 푍2 푌1 푌0 푌2 a commutative diagram in 퐶. Prove that we have a natural isomorphism (푋1 ×푍 푌1) ×푋 × 푌 (푋2 ×푍 푌2) ≅ (푋1 ×푋 푋2) ×푍 × 푍 (푌1 ×푌 푌2) . 1 0 푍0 0 2 0 1 푍0 2 0 Definition. Let 퐶 be a category with pullbacks and 푓∶ 푋 → 푌 a morphism in 퐶. The diagonal of 푓 is the morphism 훥푓 ∶ 푋 → 푋 ×푌 푋 induced via the universal property of the pullback by the square 푋 푋 푓 푋 푌 . 푓 Remark. Note that 훥 = 훥 . id푋 푋 Exercise 10 (magic square). Let 퐶 be a category and let 푓1 ∶ 푋1 → 푌, 푓2 ∶ 푋2 → 푌, and 푔∶ 푌 → 푍 morphisms in 퐶. Assuming that 퐶 has pullbacks, prove that the square 푋1 ×푌 푋2 푋1 ×푍 푋2 푌 푌 ×푍 푌 훥푔 is a pullback square in 퐶 (where the unlabeled morphisms are the morphisms naturally induced by the universal property of the pullback). Definition. Let 퐶 be a category with pullbacks, 푍 ∈ 퐶, and 푓∶ 푋 → 푌 a morphism in the slice category 퐶/푍. The graph morphism of 푓 is the morphism 푓∶ 푋 → 푋 ×푍 푌 induced via the universal property of the pullback by the square 푓 푋 푌 푋 푍 . Exercise 11. Show that if 푓∶ 푋 → 푌 is a map of sets, then the graph 훤푓 ∶ 푋 → 푋 × 푌 is given by 푥 ↦ (푥, 푓(푥)). EXERCISES ON LIMITS & COLIMITS 3 Exercise 12. Let 퐶 be a category with pullbacks, 푍 ∈ 퐶, and 푓∶ 푋 → 푌 a morphism in the slice category 퐶/푍. Write 푔∶ 푌 → 푍 for the structure morphism. Prove that the square 훤푓 푋 푋 ×푍 푌 푓 푓×푍id푌 푌 푌 ×푍 푌 훥푔 is a pullback in 퐶. Definition. Let 퐶 be a category. We say that a collection of morphisms 푃 ⊂ Mor(퐶) is stable under pullback if for any pullback square 푝̄ 푋 × 푌 푌 푍 ⌟ ̄푞 푞 푋 푝 푍 in 퐶, if 푝 ∈ 푃 then 푝̄ ∈ 푃. Exercise 13. Let 퐶 be a category with pullbacks and 푃 ⊂ Mor(퐶) a collection of morphisms in 퐶 stable under composition and pullback. Prove that given any commutative triangle 푓 푋 푌 푝 푔 푍 in 퐶, if 푝 ∈ 푃 and 훥푔 ∈ 푃, then 푓 ∈ 푃. Definition. A functor 퐹∶ 퐶 → 퐷 is left cofinal if for all categories 퐸 and diagrams 퐺∶ 퐷 → 퐸, the colimit colim퐷 퐺 exists if and only if colim퐶 퐺퐹 exists, in which case the natural morphism colim퐶 퐺퐹 → colim퐷 퐺 is an isomorphism. Dually, 퐹∶ 퐶 → 퐷 is right cofinal if 퐹op ∶ 퐶op → 퐷op is left cofinal. Remark. The “co” in “cofinal” uses the non-mathematical English prefix meaning “jointly” — there’s no duality involved here. Exercise 14. Show that equivalences of categories are both left and right cofinal. Exercise 15. Show that if a category 퐶 has a terminal object ∗, then the inclusion {∗} ↪ 퐶 of the full subcategory of 퐶 spanned by ∗ is left cofinal. Exercise 16. Let 퐹∶ 퐶 → 퐷 be a functor. Show that 퐹 is left cofinal if and only if for all diagrams 퐺∶ 퐷 → Set the natural morphism colim퐶 퐺퐹 → colim퐷 퐺 is an isomorphism. Notation. For a positive integer 푛, write 횫≤푛 for the full subcategory of 횫 spanned by those sets of cardinality at most 푛. 4 PETER J. HAINE Exercise 17. Show that the inclusion 횫≤2 ↪ 횫 is right cofinal. Notation. Write 횫inj ⊂ 횫 for the wide subcategory, i.e., subcategory containing all of the objects, of 횫 where the morphisms are injective maps of linearly ordered finite sets. For a inj inj positive integer 푛, write 횫≤푛 for the full subcategory of 횫 spanned by those sets of cardi- nality at most 푛. inj inj Exercise 18. Show that the inclusion 횫≤2 ↪ 횫 is right cofinal. Now deduce that the inj inclusion 횫≤2 ↪ 횫 is right cofinal. Exercise 19. Let 퐶 be a category with pullbacks and 푓∶ 푋 → 푌 a morphism in 퐶. Construct a functor 퐶(푓)∶̌ 횫op → 퐶 whose value on [푛] = {0 < ⋯ < 푛} is the (푛 + 1)-fold iterated ̌ pullback 푋 ×푌 ⋯ ×푌 푋 (so that the value on [0] is simply 푋). The simplicial object 퐶(푓) is called the Čech nerve of 푓. Exercise 20. Let 푋 be a topological space, 푈 ⊂ 푋 an open set, U = {푈훼}훼∈퐴 an open cover of 푈, and F a presheaf on 푋. Choose a well-ordering of 퐴 (this does not really matter, but is necessary to make the next step well-defined.) Extend the usual “sheaf condition diagram” ∏ F(푈 ) ∏ F(푈 ∩ 푈 ) 훼0 훼0 훼1 훼0∈퐴 훼0,훼1∈퐴 to a diagram 퐶(푈;̌ F)∶ 횫inj → Set of the form ∏ F(푈 ) ∏ F(푈 ∩ 푈 ) ∏ F(푈 ∩ 푈 ∩ 푈 ) ⋯ . 훼0 훼0 훼1 훼0 훼1 훼2 훼0∈퐴 훼0,훼1∈퐴 훼0,훼1,훼2∈퐴 Now reformulate the sheaf condition for a presheaf F in terms of the diagrams 퐶(̌ U; F)..
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