5
votes
2answers
529 views

Monadicity theorem in homotopy theory.

Let $\mathbf{C}$ be a cofibrantly generated model category (assume for simplicity that all objects are fibrant) and $\mathbf{C}^{\mathrm{T}}$ the category of $\mathrm{T}$-algebras with the induced ...
10
votes
1answer
800 views

The reflexive free-category comonad-resolution is a cofibrant replacement of the discrete simplicial category associated with an ordinary category in the Bergner model structure on the category of small simplicial categories?

Let $X$ be the category of reflexive quivers, and let $Cat$ be the category of small categories. There exists an evident forgetful functor $U:Cat\to X$ sending a category $A$ to its underlying ...
4
votes
2answers
458 views

What are the algebras over $\Omega^k\Sigma^k$ ?

Let $Ho(Spc)$ be the homotopy category of spaces. There is an adjoint pair $$ \Sigma^k \colon Ho(Spc) \leftrightarrows Ho(Spc)\colon \Omega^k, $$ where $\Sigma^k$ is the $k$-th supension functor and ...