# 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 $X_{cg}$, $\boldsymbol{\rho}^{\square}(X_{cg})$, will now be:

 $\begin{array}[]{c}(\boldsymbol{\rho}^{\square}_{2}(X),\boldsymbol{\rho}_{1}^{% \square}(X),\partial^{-}_{1},\partial^{+}_{1},+_{1},\varepsilon_{1}),% \boldsymbol{\rho}^{\square}_{2}(X),\boldsymbol{\rho}^{\square}_{1}(X),\partial% ^{-}_{2},\partial^{+}_{2},+_{2},\varepsilon_{2})\\ (\boldsymbol{\rho}^{\square}_{1}(X),X,\partial^{-},\partial^{+},+,\varepsilon)% .\end{array}$

## References

• 1 R. Brown, K.A. Hardie, K.H. Kamps and T. Porter, A homotopy double groupoid of a Hausdorff space, Theory and Applications of Categories 10,(2002): 71-93.
 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