Monoidal Topology: A Categorical Approach to Order, Metric and TopologyDirk Hofmann, Gavin J. Seal, Walter Tholen Cambridge University Press, 2014 M07 31 - 503 pages Monoidal Topology describes an active research area that, after various past proposals on how to axiomatize 'spaces' in terms of convergence, began to emerge at the beginning of the millennium. It combines Barr's relational presentation of topological spaces in terms of ultrafilter convergence with Lawvere's interpretation of metric spaces as small categories enriched over the extended real half-line. Hence, equipped with a quantale V (replacing the reals) and a monad T (replacing the ultrafilter monad) laxly extended from set maps to V-valued relations, the book develops a categorical theory of (T,V)-algebras that is inspired simultaneously by its metric and topological roots. The book highlights in particular the distinguished role of equationally defined structures within the given lax-algebraic context and presents numerous new results ranging from topology and approach theory to domain theory. All the necessary pre-requisites in order and category theory are presented in the book. |
Contents
Monoidal structures | 18 |
Lax algebras | 145 |
Kleisli monoids | 284 |
Lax algebras as spaces | 375 |
| 467 | |
Selected categories | 480 |
Selected symbols | 487 |
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Common terms and phrases
2Q G adjunction approach space Barr extension bijective Cartesian closed characterization closure operation cocontinuous commutative compact Hausdorff complete lattice composition condition consider continuous maps conv convergence coproduct Corollary defined definition denoted Eilenberg–Moore embedding epimorphisms equivalent Example Exercise filter finite finitely first forgetful functor fully faithful G BX G TX given Hausdorff spaces Hence homomorphism identity monad induced injective isomorphism kernel pair Kleisli extension Kleisli monoids lattice lax algebras left adjoint Lemma map f metric spaces monad monad morphism monomorphisms monotone map morphism morphism f natural transformation objects observable realization spaces obtains ordered category ordered set power-enriched monad powerset preserves Proof Proposition pullback pullback diagram quantale reflective reflexive relation right adjoint satisfies satisfying Scott topology Section structure subsets surjective Theorem topological space U-initial ultrafilter unitary V-Cat V-category V-Rel V-relation V)-functor X I Y