Bousfield-Kan completion as a codensity $\infty$-monad
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| Format: | Preprint |
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2025
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| _version_ | 1866912600608997376 |
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| author | Farjoun, Emmanuel Dror Ivanov, Sergei O. |
| author_facet | Farjoun, Emmanuel Dror Ivanov, Sergei O. |
| contents | Working in the setting of $\infty$-categories, we develop a general theory of the codensity monad $T_\mathcal{D}$ associated with a full subcategory $\mathcal{D}\subseteq \mathcal{C}$. We show that $T_\mathcal{D}$ has a canonical monad structure (unique up to a contractible space of choices), and characterize it as a terminal monad preserving all objects of $\mathcal{D}$. For a monad $\mathcal{M}$ on an $\infty$-category $\mathcal{C}$, we consider the $\mathcal{M}$-completion functor defined as the totalization of the cosimplicial resolution associated with $\mathcal{M}$. We show that the $\mathcal{M}$-completion functor is the codensity monad associated with the full subcategory of $\mathcal{C}$ spanned by objects that admit a structure of $\mathcal{M}$-algebra. In particular, the $\mathcal{M}$-completion functor is the terminal monad preserving all objects that admit a structure of an $\mathcal{M}$-algebra. This gives a full $\infty$-categorical characterization of the classical Bousfield-Kan $R$-completion functor as the terminal monad on the category of spaces preserving the empty space and all products of Eilenberg-MacLane spaces $K(A,n)$, where $A$ is an $R$-module. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2507_08414 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Bousfield-Kan completion as a codensity $\infty$-monad Farjoun, Emmanuel Dror Ivanov, Sergei O. Algebraic Topology Working in the setting of $\infty$-categories, we develop a general theory of the codensity monad $T_\mathcal{D}$ associated with a full subcategory $\mathcal{D}\subseteq \mathcal{C}$. We show that $T_\mathcal{D}$ has a canonical monad structure (unique up to a contractible space of choices), and characterize it as a terminal monad preserving all objects of $\mathcal{D}$. For a monad $\mathcal{M}$ on an $\infty$-category $\mathcal{C}$, we consider the $\mathcal{M}$-completion functor defined as the totalization of the cosimplicial resolution associated with $\mathcal{M}$. We show that the $\mathcal{M}$-completion functor is the codensity monad associated with the full subcategory of $\mathcal{C}$ spanned by objects that admit a structure of $\mathcal{M}$-algebra. In particular, the $\mathcal{M}$-completion functor is the terminal monad preserving all objects that admit a structure of an $\mathcal{M}$-algebra. This gives a full $\infty$-categorical characterization of the classical Bousfield-Kan $R$-completion functor as the terminal monad on the category of spaces preserving the empty space and all products of Eilenberg-MacLane spaces $K(A,n)$, where $A$ is an $R$-module. |
| title | Bousfield-Kan completion as a codensity $\infty$-monad |
| topic | Algebraic Topology |
| url | https://arxiv.org/abs/2507.08414 |