By Michael Barr, Charles Wells

CONTENTS

========

Preface

1. Categories

1 Definition of category

2 Functors

three traditional transformations

four parts and Subobjects

five The Yoneda Lemma

6 Pullbacks

7 Limits

eight Colimits

nine Adjoint functors

10 Filtered colimits

eleven Notes to bankruptcy I

2. Toposes 63

1 easy rules approximately Toposes

2 Sheaves on a Space

three homes of Toposes

four The Beck Conditions

five Notes to bankruptcy 2

3. Triples 83

1 Definition and Examples

2 The Kleisli and Eilenberg-Moore Categories

three Tripleability

four houses of Tripleable Functors

five Suficient stipulations for Tripleability

6 Morphisms of Triples

7 Adjoint Triples

eight historic Notes on Triples

4. Theories 123

1 Sketches

2 The Ehresmann-Kennison Theorem

three Finite-Product Theories

four Left detailed Theories

five Notes on Theories

5. houses of Toposes 148

1 Tripleability of P

2 Slices of Toposes

three Logical Functors

four Toposes are Cartesian Closed

five Exactness houses of Toposes

6 The Heyting Algebra constitution on

6. Permanence homes of Toposes 170

1 Topologies

2 Sheaves for a Topology

three Sheaves shape a topos

four Left precise cotriples

five Left distinct triples

6 different types in a Topos

7 Grothendieck Topologies

eight Giraud's Theorem

7. illustration Theorems 207

1 Freyd's illustration Theorems

2 The Axiom of Choice

three Morphisms of Sites

four Deligne's Theorem

five common quantity Objects

6 Countable Toposes and Separable Toposes

7 Barr's Theorem

eight Notes to bankruptcy 7

8. Cocone Theories 240

1 general Theories

2 Finite Sum Theories

three Geometric Theories

four homes of version Categories

9. extra on Triples 252

1 Duskin's Tripleability Theorem

2 Distributive Laws

three Colimits of Triple Algebras

four loose Triples

Bibliography 275

Index of exercises

Index

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**Sample text**

An atom in a Boolean algebrais an element greater than 0 but with no elements between it and 0. A Boolean algebra is atomic if every element x of the algebra is the join of all the atoms smaller than x. A Boolean algebra is complete if every subset has an infimum and a supremum. A CABA is a complete atomic Boolean algebra. A CABA homomorphism is a Boolean algebra homomorphism between CABA’s which preserves all infs and sups (not just finite ones, which any Boolean algebra homomorphism would do).

CCD) . Show that if D is left exact and F : D comma category (C , F ) is left exact. G C preserves finite limits, then the ♦ (LIMISO) . Prove Proposition 1. ♦ (TOP) . Let A be a topological space and let O (A) denote the set of open sets of A partially ordered by inclusion considered as a category. Show that O (A) has finite limits. Does O (A) have all limits? ♦ (REGMON) . A monomorphism is regular if it is the equalizer of two arrows. (The dual notion is called regular epi, not ”coregular”).

Evaluation of a natural transformation at a functor, or as examples of an application operation where the name of a functor is used to stand for the identity natural transformation. ) d. Show that application as defined above is associative in the sense that if (µκ)β is defined, then so is µ(κβ) and they are equal. e. Show that the following rules hold, where ◦ denotes the composition of natural transformations defined earlier in this chapter. These are called Godement’s rules. In each case, the meaning of the rule is that if one side is defined, then so is the other and they are equal.