Una teoría de homotopía para categorías finitas.

Abstract

We present a construction of categories and functors that allows us to define an analogous theory to the one of homotopy of paths in topological spaces, but over finite categories. For this purpose, we first develop the Classical Theory of Homotopy, in which the fundamental group of a topological space is constructed and several fundamental results are proven about it. Next, a brief introduction to the Theory of Categories is presented from a topological approach, which will be useful for the construction of simplicial sets; mainly with the nerve of a category and thus we will define a geometry for this category, showing its relationship with classical homotopy. Finally, a categorization of the homotopy is carried out, in which the fundamental group of a category is defined, and it is proven that a category whose geometry is homeomorphic to the circle verifies that its fundamental group in the sense of the categories, coincides with the topological fundamental group of the circle. Plus theorems similar to those developed in the respective part of the classic homotopy are proved.