Modelling and computing homotopy types: I

Abstract

The aim of this article is to explain a philosophy for applying the 1-dimensional and higher dimensional Seifert-van Kampen Theorems, and how the use of groupoids and strict higher groupoids resolves some foundational anomalies in algebraic topology at the border between homology and homotopy. We explain some applications to filtered spaces, and special cases of them, while a sequel will show the relevance to n-cubes of pointed spaces.

Some Remarks on MCI Crossed Modules

“…Crossed modules are introduced by Whitehead [16] as a model of homotopy 2-types and used to classify higher dimensional cohomology groups. See [5] for more details on crossed modules. Afterwards, crossed modules are also studied for various algebraic structures such as in the categories of (commutative) algebras, dialgebras, Lie and Leibniz algebras, etc.…”

Braided Hopf Crossed Modules Through Simplicial Structures

“…In other words, they can be seen as a way to encode a strict group [12]. For more details on crossed modules especially from the topological, algebraic and geometric point of view, we refer [10,20,31]. Moreover, the notion and some applications of crossed modules are extended to various algebraic structures such as monoids [6,7], Lie algebras [17], groupoids [11], groups with operations [33], modified categories of interest [9], shelves (via generalized Yetter-Drinfeld modules) [24], etc.…”

Computing 3-Dimensional Groups : Crossed Squares and Cat$^2$-Groups

“…The impetus for the study of higher-dimensional groups comes from algebraic topology [9]. Crossed modules are algebraic models of connected (weak homotopy) 2-types, while crossed squares model connected 3-types.…”