weak homotopy double groupoid

Definition 0.1.

a weak homotopy double groupoid (WHDG) of a compactly–generated space $X_{cg}$, (weak Hausdorff space) is defined through a construction method similar to that developed by R. Brown (ref. [1]) for the homotopy double groupoid of a Hausdorff space. The key changes here involve replacing the regular homotopy equivalence relation from the cited ref. with the weak homotopy equivalence relation in the definition of the fundamental groupoid, as well as replacing the Hausdorff space by the compactly-generated space $X_{cg}$. Therefore, the weak homotopy data for the weak homotopy double groupoid of $X_{cg}$, ${𝝆}^{\mathrm{\square }}(X_{cg})$, will now be:

$$\begin{array}{c}\hfill ({𝝆}_2^{\mathrm{\square }}(X),{𝝆}_1^{\mathrm{\square }}(X),{\partial }_1^{-},{\partial }_1^{+},{+}_1,{\epsilon }_1),{𝝆}_2^{\mathrm{\square }}(X),{𝝆}_1^{\mathrm{\square }}(X),{\partial }_2^{-},{\partial }_2^{+},{+}_2,{\epsilon }_2)\hfill \\ \hfill ({𝝆}_1^{\mathrm{\square }}(X),X,{\partial }^{-},{\partial }^{+},+,\epsilon ).\hfill \end{array}$$

Title	weak homotopy double groupoid
Canonical name	WeakHomotopyDoubleGroupoid
Date of creation	2013-03-22 18:15:12
Last modified on	2013-03-22 18:15:12
Owner	bci1 (20947)
Last modified by	bci1 (20947)
Numerical id	16
Author	bci1 (20947)
Entry type	Definition
Classification	msc 55N33
Classification	msc 55N20
Classification	msc 55P10
Classification	msc 55U40
Classification	msc 18B40
Classification	msc 18D05
Synonym	homotopy double groupoid
Related topic	WeakHomotopyAdditionLemma
Related topic	OmegaSpectrum
Related topic	FEquivalenceInCategory
Defines	higher dimensional weak homotopy