Tootfinder

Bousfield-Kan completion as a codensity $\infty$-monad
Emmanuel Dror Farjoun, Sergei O. Ivanov
https://arxiv.org/abs/2507.08414 https://

Bousfield-Kan completion as a codensity $\infty$-monad
We develop a theory of codensity monads associated with full subcategories in the setting of $\infty$-categories. For an $\infty$-monad $\mathcal{M}$ on an $\infty$-category $\mathcal{C}$ that admits homotopy totalizations, we consider the $\mathcal{M}$-completion functor defined as the homotopy totalization of the canonical cosimplicial functor associated with $\mathcal{M}$. We prove that the $\mathcal{M}$-completion is the codensity $\infty$-monad of a full subcategory $\mathcal{A}(\mathcal{M…

A list is a #monad
https://alexyorke.github.io//2025/06/29/a-list-is-a-monad/

Ultracategories via Kan extensions of relative monads
Umberto Tarantino, Joshua Wrigley
https://arxiv.org/abs/2506.09788 https://arxi…

Ultracategories via Kan extensions of relative monads
Many structured categories of interest are most naturally described as algebras for a relative monad, but turn out nonetheless to be algebras for an ordinary monad. We show that, under suitable hypotheses, the left oplax Kan extension of a relative 2-monad on categories yields a pseudomonad having the same category of colax algebras. In particular, we apply this to the study of ultracategories to recover the 'ultracompletion' pseudomonad.

This https://arxiv.org/abs/2311.15919 has been replaced.
link: https://scholar.google.com/scholar?q=a

What Monads Can and Cannot Do with a Few Extra Pages
The delay monad provides a way to introduce general recursion in type theory. To write programs that use a wide range of computational effects directly in type theory, we need to combine the delay monad with the monads of these effects. Here we present a first systematic study of such combinations. We study both the coinductive delay monad and its guarded recursive cousin, giving concrete examples of combining these with well-known computational effects. We also provide general theorems stati…

Replaced article(s) found for math.CT. https://arxiv.org/list/math.CT/new/
[1/1]:
A factorization of the Giry monad on standard Borel spaces using $[0,\infty]$-generalized points

This https://arxiv.org/abs/2111.14796 has been replaced.
link: https://scholar.google.com/scholar?q=a

Familial Monads as Higher Category Theories
Categories, n-categories, double categories, and multicategories (among others) all have similar definitions as collections of cells with composition operations. We give an explicit description of the information required to define any higher category structure which arises as algebras for a familially representable monad on a presheaf category, then use this to describe several examples relating to higher category theory and cubical sets. The proof of this characterization avoids tedious natur…