Homotopy-theoretic aspects of 2-monads

By Stephen Lack. 2006

70 citations recorded by the source corpus.

We study 2-monads and their algebras using a Cat-enriched version of Quillen model categories, emphasizing the parallels between the homotopical and 2-categorical points of view. Every 2-category with finite limits and colimits has a canonical model structure in which the weak equivalences are the equivalences; we use these to construct more interesting model structures on 2-categories, including a model structure on the 2-category of algebras for a 2-monad T, and a model structure on a 2-category of 2-monads on a fixed 2-category K .

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/fb059e1d8fcf952118c5495e9c7a4eb8279382b8

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In Algebra and Number Theory

In Geometry and Topology

In Mathematical Physics

Citations in the corpus

A 2-Categories Companion (2007, background): “The theory of model 2-categories, the model structure on the category of monads, and its relation to structure and semantics, all come from [37].”
Algebraic model structures (2009, background): “lows the “generators” to be a category, rather than simply a set. One place where such algebraic weak factorization systems appear is in Lack’s trivial model structure on certain diagram 2-categories [Lac07], and consequently, these algebraic model structures are coﬁbrantly generated in the new sens”
Enhanced 2-categories and limits for lax morphisms (2011, methodology): “In this case, q : QpA → A is a cofibrant replacement in a suitable Quillen model structure on T -Algs whose homotopy 2-category is T -Algp; see [Lac07].”
Relative pseudomonads, Kleisli bicategories, and substitution monoidal structures (2016, methodology): “Furthermore, we introduce and study the notion of a lax idempotent relative pseudomonad, which appears to be the appropriate counterpart in our setting of the notion of a lax idempotent 2-monad (often called Kock-Zöberlein doctrines) [37, 38, 65] and pseudomonad [48, 52, 60].”
Categorical Homotopy Theory (2014, background): “The 2-category CatD of functors, natural transformations, and modifications inherits a model structure with its fibrations and weak equivalences defined representably [Lac07].”