An Introduction to Topos Theory

Introduction......Page 3
Categories......Page 6
Basic types of arrows......Page 9
Generalised elements......Page 11
Basic types of diagrams......Page 13
Functors......Page 20
Functor categories......Page 25
Yoneda's lemma......Page 28
Limits and colimits......Page 32
Adjoints and (co)units......Page 35
Exponentials and cartesian closed categories......Page 41
Subobject classifier......Page 44
Subobjects in Set......Page 45
Subobjects in C......Page 46
The Lawvere–Tierney elementary topos E......Page 48
Sieves......Page 50
The topos of presheaves SetCop......Page 52
An example: sieves and subobjects in the topos Set2op......Page 55
Presheaves over posets......Page 60
Logic and language......Page 62
Language and interpretation......Page 63
Algebras of logic......Page 67
Classical and intuitionistic logic......Page 70
Categorical logic of Heyting algebra......Page 72
Categorical logic in a topos......Page 74
Language of a topos......Page 76
Semantics of a topos......Page 77
The Grothendieck topos Sh(C) of sheaves over category C......Page 78
From infinitesimals to microlinear spaces......Page 79
Well-adapted topos models of differential geometry and smooth analysis......Page 82
References......Page 84
Index......Page 86