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By Emily Riehl

Category idea has supplied the principles for plenty of of the 20 th century's maximum advances in natural arithmetic. This concise, unique textual content for a one-semester advent to the subject is derived from classes that writer Emily Riehl taught at Harvard and Johns Hopkins Universities. The remedy introduces the fundamental techniques of classification conception: different types, functors, ordinary adjustments, the Yoneda lemma, limits and colimits, adjunctions, monads, Kan extensions, and different topics.
Suitable for complicated undergraduates and graduate scholars in arithmetic, the textual content offers instruments for realizing and attacking tricky difficulties in algebra, quantity concept, algebraic geometry, and algebraic topology. Drawing upon a huge variety of mathematical examples from the specific point of view, the writer illustrates how the suggestions and buildings of class thought come up from and remove darkness from extra simple mathematical ideas. While the reader might be rewarded for familiarity with these historical past mathematical contexts, crucial necessities are restricted to easy set conception and logic.

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Thus, F is faithful, and by symmetry, so is G. 10, Gk and the isomorphisms ηc and ηc define a unique h : c → c for which both Gk and GFh make the diagram c ηc Gk or GFh h  c G GFc ηc  G GFc 38The reader is strongly encouraged to stop reading here and attempt to prove this result on their own. 10. 32 1. CATEGORIES, FUNCTORS, NATURAL TRANSFORMATIONS commute. 10 again, GFh = Gk, whence Fh = k by faithfulness of G. Thus, F is full, faithful, and essentially surjective. For the converse, suppose now that F : C → D is full, faithful, and essentially surjective on objects.

What is a functor between groups, regarded as one-object categories? ii. What is a functor between preorders, regarded as categories? iii. Find an example to show that the objects and morphisms in the image of a functor F : C → D do not necessarily define a subcategory of D. iv. 11 are functorial. v. What is the difference between a functor Cop → D and a functor C → Dop ? What is the difference between a functor C → D and a functor Cop → Dop ? vi. , so that f · Fh = Gk · f . Define a pair of projection functors dom : F ↓G → D and cod : F ↓G → E.

5. EQUIVALENCE OF CATEGORIES 31 • and essentially surjective on objects if for every object d ∈ D there is some c ∈ C such that d is isomorphic to Fc. 8. ” A faithful functor need not be injective on morphisms; neither must a full functor be surjective on morphisms. A faithful functor that is injective on objects is called an embedding and identifies the domain category as a subcategory of the codomain; in this case, faithfulness implies that the functor is (globally) injective on arrows. A full and faithful functor, called fully faithful for short, that is injective-on-objects defines a full embedding of the domain category into the codomain category.

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