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arXiv · 2405.03793

Naive homotopy theories in cartesian closed categories

Enrique Ruiz Hernández·Pedro Solórzano
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Abstract

An elementary notion of homotopy can be introduced between arrows in a cartesian closed category $E$. The input is a finite-product-preserving endofunctor $Π_0$ with a natural transformation $p$ from the identity which is surjective on global elements. As expected, the output is a new category $E_p$ with objects the same objects as $E$. Further assumptions on $E$ provide a finer description of $E_p$ that relates it to the classical homotopy theory where $Π_0$ could be interpreted as the ``path-connected components'' functor on convenient categories of topological spaces. In particular, if $E$ is a 2-value topos the supports of which split and is furthermore assumed to be precohesive over a boolean base, then the passage from $E$ to $E_p$ is naturally described in terms of explicit homotopies -- as is the internal notion of contractible space.

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Enrique Ruiz Hernández, Pedro Solórzano. 2024-05-06. Naive homotopy theories in cartesian closed categories. https://arxiv.org/abs/2405.03793

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