New PDF release: Toposes, triples, and theories (draft)

Nonfiction 3

By Michael Barr, Charles Wells

ISBN-10: 0387961151

ISBN-13: 9780387961156

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Extra resources for Toposes, triples, and theories (draft)

Example text

The converse is easy. The preceding argument is typical of many arguments making use of the uniqueness part of the definition of universal element. We will usually leave arguments like this to the reader. A consequence of Proposition 1 is that a pullback of a diagram in a category is not determined uniquely but only up to a “unique isomorphism which makes everything commute”. 7. 32 1 Categories Notation for pullbacks We have defined the pullback P of Diagram 1 so that it objectifies the set {(x, y) | f (x) = g(y)}.

This construction defines a contravariant functor from C to Func(C , Set) called the Yoneda map or Yoneda embedding. It is straightforward and very much worthwhile to check that this construction really does give a natural transformation for each arrow f and that the resulting Yoneda map really is a functor. Because Nat(−, −) is contravariant in the first variable (it is a special case of Hom), the map which takes an object B of C and a functor F : C −→ Set to Nat(Hom(B, −), F ) is a functor from C × Func(C , Set) to Set.

In general, an arrow in some category from the terminal object to some object is called a 24 1 Categories global element of that object, for reasons which will become apparent in the next paragraph. In most categories which arise in practice, except Set, an object is not determined by its global elements. For example, in Grp, each group has exactly one global element. A more interesting example arises in connection with continuous functions. This example is worth studying in detail because it illustrates and motivates much of sheaf theory.

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Toposes, triples, and theories (draft) by Michael Barr, Charles Wells

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