The aim of this paper is to construct some new fuzzy categories related to many-valued sets and many valued topologies and to consider their properties. Keywords— Fuzzy categories, GL-monoids, L-valued equalities, L-topologies and L-fuzzy topologies.

I point out that (BPS saturated) A-type D-branes in superstring compactification on Calabi-Yau threefolds correspond to graded special Lagrangian submanifolds, a particular case of the graded Lagrangian submanifolds considered by M. Kontsevich and P. Seidel. Combining this with the categorical formulation of cubic string field theory in the presence of D-branes, I consider a collection of topol...

We prove that the algebra of singular cochains on a smooth manifold, equipped with the cup product, is equivalent to the A∞ structure on the Lagrangian Floer cochain group associated to the zero section in the cotangent bundle. More generally, given a pair of smooth manifolds Q1 and Q2 of the same dimension with embeddings of a submanifold B with isomorphic normal bundles, we construct a differ...

We construct a simplicial locally convex algebra, whose weak dual is the standard cosimplicial topological space. The construction is carried out in a purely categorical way, so that it can be used to construct (co)simplicial objects in a variety of categories — in particular, the standard cosimplicial topological space can be produced.

The paper proposes the notions of topological platform and quantalic topological theory for the presentation and investigation of categories of interest beyond the realm of algebra. These notions are nevertheless grounded in algebra, through the notions of monad and distributive law. The paper shows how they entail previously proposed concepts with similar goals.

The Madsen-Tillmann spectra defined by categories of threeand four-dimensional Spin manifolds have a very rich algebraic structure, whose surface is scratched here. For Michael Atiyah, in deep gratitude. 1. Cobordism categories 1.1 Many variations and generalizations are possible, but to begin, consider the topological two-category DCobord whose objects are oriented smooth closed d-manifolds (D...

1. Preliminaries about topological model categories 5 2. Preliminaries about equivalences of model categories 9 3. The level model structure on D-spaces 10 4. Preliminaries about π∗-isomorphisms of prespectra 14 5. Stable weak equivalences of D-spectra 18 6. The stable model structure on D-spectra 23 7. Comparisons among P , ΣS , IS , and W T 27 8. Model categories of ring and module spectra 30...

The definition of a category of (T,V)-algebras, where V is a unital commutative quantale and T is a Set-monad, requires the existence of a certain lax extension of T. In this article, we present a general construction of such an extension. This leads to the formation of two categories of (T,V)-algebras: the category Alg(T,V) of canonical (T,V)-algebras, and the category Alg(T′,V) of op-canonica...