Naive homotopy theories in cartesian closed categories
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866914786822848512 |
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| author | Hernández, Enrique Ruiz Solórzano, Pedro |
| author_facet | Hernández, Enrique Ruiz Solórzano, Pedro |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_03793 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Naive homotopy theories in cartesian closed categories Hernández, Enrique Ruiz Solórzano, Pedro Category Theory Algebraic Topology Primary 18B25, Secondary 55U40 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. |
| title | Naive homotopy theories in cartesian closed categories |
| topic | Category Theory Algebraic Topology Primary 18B25, Secondary 55U40 |
| url | https://arxiv.org/abs/2405.03793 |