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For a model category, we prove that taking the category of coalgebras over a comonad commutes with left Bousfield localization in a suitable sense. Then we prove a general existence result for left-induced model structure on the category of coalgebras over a comonad in a left Bousfield localization. Next we provide several equivalent characterizations of when a left Bousfield localization preserves coalgebras over a comonad. These results are illustrated with many applications in chain complexes, (localized) spectra, and the stable module category.
We prove the equivalence of several hypotheses that have appeared recently in the literature for studying left Bousfield localization and algebras over a monad. We find conditions so that there is a model structure for local algebras, so that localiz
Given a combinatorial (semi-)model category $M$ and a set of morphisms $C$, we establish the existence of a semi-model category $L_C M$ satisfying the universal property of the left Bousfield localization in the category of semi-model categories. Our
It is known by results of Dyckerhoff-Kapranov and of Galvez--Carrillo-Kock-Tonks that the output of the Waldhausen S.-construction has a unital 2-Segal structure. Here, we prove that a certain S.-functor defines an equivalence between the category of
In a previous paper, we showed that a discrete version of the $S_bullet$-construction gives an equivalence of categories between unital 2-Segal sets and augmented stable double categories. Here, we generalize this result to the homotopical setting, b
We give an explicit point-set construction of the Dennis trace map from the $K$-theory of endomorphisms $Kmathrm{End}(mathcal{C})$ to topological Hochschild homology $mathrm{THH}(mathcal{C})$ for any spectral Waldhausen category $mathcal{C}$. We desc