We give a completion theorem for ordered magmas (i.e. ordered algebras with monotone operations) in a general form. Particular instances of this theorem are already known, and new results follow. The semantics of programming languages is the motivation of such investigations. Key wordscomplete partial orders, semantics of programming languages. Introduction. When defining recursive functions by...

We associate to any (strict) multiple category C three homology theories : the rst one is called the globular homology and it contains the oriented loops of C ; both other ones are called corner homology, the negative one and the positive one, which contain the corners included in C. We make explicit the link between these homology theories and the homotopy of paths in multiple category. We con...

A new relation between morphisms in a category is introduced – roughly speaking (accurately in the categories Set and Top), f ‖ g iff morphisms w : dom(f) → dom(g) never map subobjects of fibres of f non-constantly to fibres of g. (In the algebraic setting replace fibre with kernel.) This relation and a slight weakening of it are used to define “connectedness” versus “disconnectedness” for morp...

We study compactifications of very affine varieties defined by imposing a polyhedral structure on the non-archimedean amoeba. These com-pactifications have divisorial boundary with combinatorial normal croosings. We consider some examples including M 0,n ⊂ M 0,n (and more generally log canonical models of complements of hyperplane arrangements) and tropical recompactifications of Chow quotients...

We define a unital $A\_\infty$-category ${\mathit{Fuk}}({\mathbb R}\times Y)$ whose objects are exact Lagrangian cobordisms in the symplectization of $Y=P\times {\mathbb R}$, with negative cylindrical ends over Legendrians equipped augmentations. The morphism spaces ${\mathrm{hom}}{{\mathit{Fuk}}({\mathbb Y)}(\Sigma\_0,\Sigma\_1)$ given terms Floer complexes ${\mathrm{Cth}}+(\Sigma\_0,\Sigma\_1...

The categorical relativity is a groupoid category of massive bodies in mutual motions. The relative velocity is defined to be the basis-free and coordinate-free binary morphism. In categorical relativity there is no need to distinguish separately the constant relative velocities (special relativity) from the variable accelerated velocities, hence the categorical relativity goes beyond boundary ...

Working in the fragment of Martin-L6fs extensional type theory ]-12] which has products (but not sums) of dependent types, we consider two additional assumptions: firstly, that there are (strong) equality types; and secondly, that there is a type which is universal in the sense that terms of that type name all types, up to isomorphism. For such a type theory, we give a version of Russell's para...

We show that the class of separable morphisms in the sense of G. Janelidze and W. Tholen in the case of Galois structure of second order coverings of simplicial sets due to R. Brown and G. Janelidze coincides with the class of covering maps of simplicial sets. Introduction Separable morphisms were introduced in [3] by A. Carboni and G. Janelidze for lextensive categories. In the way of [3] one ...

Our work over the past years shows that not only the collection of (for instance) all topological spaces gives rise to a category, but also each topological space can be seen individually as a category by interpreting the convergence relation x −→ x between ultrafilters and points of a topological space X as arrows in X. Naturally, this point of view opens the door to the use of concepts and id...