In this paper we use Quillen's model structure given by Dwyer-Kan for the category of simplicial groupoids (with discrete object of objects) to describe in this category, in the simplicial language, the fundamental homotopy theoretical constructions of path and cylinder objects. We then characterize the associated left and right homotopy relations in terms of simplicialidentities and give a sim...

We define a type B analogue of the category finite sets with surjections, and we study representation theory this category. show that opposite is quasi-Gröbner, which implies submodules finitely generated modules are again generated. prove generating functions have certain prescribed poles, obtain restrictions on representations Coxeter groups can appear in such modules. Our main example module...

In this paper we look at two categories, the category F of fuzzy subsets and a quotient category F=M of fuzzy sets. The category F=M is an extension of the category of sets, and the standard constructions in fuzzy set theory arise naturally within this category.

In modern mathematics, many concepts and ideas are described in terms of category theory. From this viewpoint, it is desirable to analyze what can be determined if, instead of the basic category of sets, we consider a similar category of fuzzy sets. In this paper, we describe a natural fuzzy analog of the category of sets and functions, and we show that, in this category, fuzzy relations (a nat...

The category of finite cardinals (or, equivalently, of finite sets) is the symmetric analogue of the category of finite ordinals, and the ground category of a relevant category of presheaves, the augmented symmetric simplicial sets. We prove here that this ground category has characterisations similar to the classical ones for the category of finite ordinals, by the existence of a universal sym...