# Toposes, Triples and Theories by Michael Barr, Charles Wells

By Michael Barr, Charles Wells

CONTENTS
========

Preface
1. Categories
1 Definition of category
2 Functors
four parts and Subobjects
five The Yoneda Lemma
6 Pullbacks
7 Limits
eight Colimits
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
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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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.