By Tom Leinster

ISBN-10: 1107044243

ISBN-13: 9781107044241

On the middle of this brief creation to classification concept is the assumption of a common estate, vital all through arithmetic. After an introductory bankruptcy giving the fundamental definitions, separate chapters clarify 3 ways of expressing common homes: through adjoint functors, representable functors, and boundaries. a last bankruptcy ties all 3 jointly. The e-book is acceptable to be used in classes or for autonomous research. Assuming fairly little mathematical heritage, it's excellent for starting graduate scholars or complex undergraduates studying class concept for the 1st time. for every new specific proposal, a beneficiant provide of examples is equipped, taken from various elements of arithmetic. At issues the place the bounce in abstraction is very nice (such because the Yoneda lemma), the reader will locate cautious and large causes. Copious routines are integrated.

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Extra info for Basic Category Theory

Sample text

When f = 1A , this map is αA . 3), and ‘exactly one’ implies that the square commutes. 3 Natural transformations 29 (b) We write F A 11 11  α 8 VB G to mean that α is a natural transformation from F to G. 3 Let A be a discrete category, and let F, G : A → B be functors. Then F and G are just families (F(A))A∈A and (G(A))A∈A of objects of αA B. A natural transformation α : F → G is just a family F(A) −→ G(A) A∈A of maps in B, as claimed above in the case ob A = N. 3) for every map f in A ; but the only maps in A are the identities, and when f is an identity, this axiom holds automatically.

A) Let F : A × B → C be a functor. Prove that for each A ∈ A , there is a functor F A : B → C defined on objects B ∈ B by F A (B) = F(A, B) and on maps g in B by F A (g) = F(1A , g). Prove that for each B ∈ B, there is a functor F B : A → C defined similarly. 3 Natural transformations 27 (b) Let F : A × B → C be a functor. With notation as in (a), show that the families of functors (F A )A∈A and (F B )B∈B satisfy the following two conditions: • if A ∈ A and B ∈ B then F A (B) = F B (A); • if f : A → A in A and g : B → B in B then F A (g) ◦ F B ( f ) = F B ( f ) ◦ F A (g).

There is also an identity natural transformation F A 11 11  1F @ TB F on any functor F, defined by (1F )A = 1F(A) . So for any two categories A and B, there is a category whose objects are the functors from A to B and whose maps are the natural transformations between them. This is called the functor category from A to B, and written as [A , B] or B A . 7 Let 2 be the discrete category with two objects. A functor from 2 to a category B is a pair of objects of B, and a natural transformation is a pair of maps.