The cubical sets model of Homotopy Type Theory was introduced by Bezem, Coquand and Huber using a particular category of presheaves. We show that this category is equivalent to a category of sets whose elements have a finite support property with respect to an action of a monoid of name substitutions; and that this is isomorphic to a category of nominal sets equipped with source and target maps...

Let ∆∞ = End([∞]) be the monoid of convergent monotonic sequences in N∪{∞}. The category Pre(∆∞) of sequential sets is the category whose objects are sets equipped with an action of this monoid, and whose morphisms are equivariant maps. We call a category C sequentially generated if it is a localization of Pre(∆∞). In this paper we show that several important categories, including the category ...

Michalski gave a short and elegant proof of theorem A. Kumar which states that for each set A⊆R, there exists B⊆A is full in A such no distance between points B rational number. He also proved similar sets R2. In this paper, we generalize these results some special types category bases.

Staton has shown that there is an equivalence between the category of presheaves on (the opposite of) finite sets and partial bijections and the category of nominal restriction sets: see [2, Exercise 9.7]. The aim here is to see that this extends to an equivalence between the category of cubical sets introduced in [1] and a category of nominal sets equipped with a ‘01-substitution’ operation. I...

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 simplicial identities and give a si...

We investigate the existence of normalizers of subobjects in pointed categories defined in the expected way, as motivated by the standard definition used in the category of groups. We show that, for a semi-abelian category C: (a) if the category C2 of morphisms in C is action representable, then so is the category Mon(C) of monomorphisms in C; (b) if Mon(C) is action representable, then normali...