Categorical Foundations: Special Topics in Order, Topology, Algebra, and Sheaf Theory, Volume 97Maria Cristina Pedicchio, Walter Tholen The book offers categorical introductions to order, topology, algebra and sheaf theory, suitable for graduate students, teachers and researchers of pure mathematics. |
Contents
Chapter | 2 |
Locales | 41 |
A Functional Approach to General Topology | 103 |
References | 160 |
Regular Protomodular and Abelian Categories | 165 |
Epimorphisms and regular categories | 169 |
Normal monomorphisms and protomodular categories | 181 |
Regular protomodular categories | 196 |
Applications of Lawveres Theorem | 287 |
Chapter VII | 295 |
Locally finitely presentable categories | 298 |
References | 308 |
Sheaf Theory | 311 |
Sheaves on a topological space | 314 |
Topologies closure operators and localizations | 322 |
Extensive categories | 341 |
The Short Five Lemma and the Tierney equation | 203 |
References | 210 |
Aspects of Monads | 213 |
Conditions for monadicity | 222 |
Chapter VI | 234 |
Conditions for the unit of a monad to be a monomorphism | 239 |
The Kleisli triples of Manes and their generated monads | 249 |
Monads idempotent monads and commutative monads | 257 |
Algebraic Categories | 269 |
The Lawvere Theorem for algebraic categories | 279 |
References | 355 |
Effective Descent Morphisms | 359 |
Generalization of the kernelcokernel correspondence | 367 |
Elementary descent theory | 373 |
B Grothendiecks descent theory | 381 |
E Fibrations pseudofunctors cosmoi and 2dimensional | 389 |
objects that | 395 |
References | 404 |
Index | 407 |
Common terms and phrases
2-category abelian groups adjunction algebraic category arrow f bijection Boolean canonical characterization closure operator coequalizer cokernel commutative compact composition consider coproduct Corollary defined Definition denote descent theory discrete fibration effective descent morphism Eilenberg-Moore element elementary topos equivalence relation exact category exact completion example Exercise exists extension F-closed F-compact F-open F-perfect F-separated factorization FamC fibration filtered colimits finite limits finitely complete following diagram forgetful functor full subcategory function given Grothendieck topology Hausdorff hence Heyting homomorphism implies kernel pair Kleisli lattice left adjoint Lemma locally finitely presentable Mathematics monad mono monoid monomorphism morphism morphism f notion object ordered set phism preserves presheaf presheaf category Proof Proposition prove pullback quotient reflexive regular category regular epi regular projective cover right adjoint sheaf sheaves small category stable under pullback sublocale subobject subset surjective Theorem topological space trivial unique universal closure operator