By Dirk Hofmann, Gavin J. Seal, Walter Tholen

Monoidal Topology describes an lively study zone that, after quite a few earlier proposals on tips on how to axiomatize 'spaces' when it comes to convergence, started to emerge before everything of the millennium. It combines Barr's relational presentation of topological areas by way of ultrafilter convergence with Lawvere's interpretation of metric areas as small different types enriched over the prolonged genuine half-line. consequently, built with a quantale V (replacing the reals) and a monad T (replacing the ultrafilter monad) laxly prolonged from set maps to V-valued kin, the booklet develops a express conception of (T,V)-algebras that's encouraged at the same time by way of its metric and topological roots. The ebook highlights particularly the prestigious position of equationally outlined buildings in the given lax-algebraic context and offers a variety of new effects starting from topology and procedure concept to area conception. all of the important pre-requisites so as and classification concept are awarded within the e-book.

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**Extra info for Monoidal Topology: A Categorical Approach to Order, Metric, and Topology**

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First, every monoid M becomes a one-object category C when one puts C( , ) = M (with denoting the only object of C) and interprets monoid operation as composition. ” Second, every ordered set X becomes a small category C with ob C = X and C(x, y) = { } if x ≤ y in X , and C(x, y) = ∅ otherwise; the composition and identity operations appear as the transitivity and reflexivity properties in X . Note that in this case the order on C defined above coincides with that of X .

5 will be used only sporadically. Chapter IV provides powerful alternative descriptions of the category (T, V)-Cat, the most striking of which arises from the fact that the quantale V and the monad T on Set laxly extended to V-Rel allow for the construction of a new monad Π = Π(T, V) (read “Pi”) on Set laxly extended to Rel = 2-Rel such that (T, V)-Cat = (Π, 2)-Cat , associativity of the Kleisli convolution granted. 5) monad Π. The fact that all relevant information provided by T and V can be encoded by a new monad Π gives the parameter T some prominence over V.

For an adjunction f g : Y − → X, f g, then g is fully the map g is fully faithful if and only if f · g 1Y . If h faithful if and only if h is fully faithful. C Adjunctions for free. denote by r Z the map For a module r : X − → Y and an ordered set Z , Mod(Z , X ) − → Mod(Z , Y ) , t− →r ·t , where Mod(Z , X ) denotes the ordered set of modules Z − → X . With a module s:Y − → X , show ∀Z (r Z s Z ) ⇐⇒ 1∗X ≤ s · r & r · s ≤ 1∗Y . 1 Ordered sets 39 Conclude that for every monotone map f : X − → Y and every ordered set Z , one ∗ has an adjunction ( f ∗ ) Z ( f ) Z .