Box-homotopy and the reduction of $\mathit{pro}^*\textit{-}\mathit{HTop}$ category

A new generalized definition of homotopy is proposed, including a new class of continuous mappings that we call box-homotopies. It turns out that to be box-homotopic is an equivalence relation on $\mathit{Top}(X,Y)$ and $\mathit{HTop}(X,Y)$, and that it is well adjusted with composition, which allows us to introduce a new category $H _{\square}\mathit{Top}$, the corresponding quotient category of the category $\mathit{HTop}$, and consequently, $\mathit{pro}\textit{-}H _{\square}\mathit{Top}$. We propose a new functor $\tilde{R}$ from $\mathit{pro}^{*}\textit{-}\mathit{HTop}$ to $\mathit{pro}\textit{-}H _{\square}\mathit{Top}$, which represents morphisms in $\mathit{pro}^{*}$-category as morphisms in $\mathit{pro}$-category between more complex objects.

Keywords

reduced product, $P$-space, box-homotopy, $\mathit{pro}^*$-category

2010 Mathematics Subject Classification

55N99, 55P55, 55Q05

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Received 3 December 2018

Accepted 24 May 2019

Published 23 October 2019