weak homotopy double groupoid

Definition 0.1.

a weak homotopy double groupoid (WHDG) of acompactly–generated space $X_{cg}$, (weak Hausdorff space) isdefined 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:

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