Homotopy structures realizing algebraic kk-theory
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866917134735507456 |
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| author | Ellis, Eugenia Cirone, Emanuel Rodríguez |
| author_facet | Ellis, Eugenia Cirone, Emanuel Rodríguez |
| contents | Algebraic $kk$-theory, introduced by Cortiñas and Thom, is a bivariant $K$-theory defined on the category $\mathrm{Alg}$ of algebras over a commutative unital ring $\ell$. It consists of a triangulated category $kk$ endowed with a functor from $\mathrm{Alg}$ to $kk$ that is the universal excisive, homotopy invariant and matrix-stable homology theory. Moreover, one can recover Weibel's homotopy $K$-theory $\mathrm{KH}$ from $kk$ since we have $kk(\ell,A)=\mathrm{KH}(A)$ for any algebra $A$. We prove that $\mathrm{Alg}$ with the split surjections as fibrations and the $kk$-equivalences as weak equivalences is a stable category of fibrant objects, whose homotopy category is $kk$. As a consecuence of this, we prove that the Dwyer-Kan localization $kk_\infty$ of the $\infty$-category of algebras at the set of $kk$-equivalences is a stable infinity category whose homotopy category is $kk$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_19936 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Homotopy structures realizing algebraic kk-theory Ellis, Eugenia Cirone, Emanuel Rodríguez K-Theory and Homology Category Theory 19D55, 18N45, 18N60, 19K35 Algebraic $kk$-theory, introduced by Cortiñas and Thom, is a bivariant $K$-theory defined on the category $\mathrm{Alg}$ of algebras over a commutative unital ring $\ell$. It consists of a triangulated category $kk$ endowed with a functor from $\mathrm{Alg}$ to $kk$ that is the universal excisive, homotopy invariant and matrix-stable homology theory. Moreover, one can recover Weibel's homotopy $K$-theory $\mathrm{KH}$ from $kk$ since we have $kk(\ell,A)=\mathrm{KH}(A)$ for any algebra $A$. We prove that $\mathrm{Alg}$ with the split surjections as fibrations and the $kk$-equivalences as weak equivalences is a stable category of fibrant objects, whose homotopy category is $kk$. As a consecuence of this, we prove that the Dwyer-Kan localization $kk_\infty$ of the $\infty$-category of algebras at the set of $kk$-equivalences is a stable infinity category whose homotopy category is $kk$. |
| title | Homotopy structures realizing algebraic kk-theory |
| topic | K-Theory and Homology Category Theory 19D55, 18N45, 18N60, 19K35 |
| url | https://arxiv.org/abs/2412.19936 |