Abstract
? | It is known that if $\mathcal{V}$ is a monoidal model category in which every
object is cofibrant, then any cofibrantly generated $\mathcal{V}$-enriched model
category has a $\mathcal{V}$-enriched cofibrant replacement comonad; conversely,
if a monoidal model category $\mathcal{V}$ (with cofibrant unit object) has a
$\mathcal{V}$-enriched cofibrant replacement comonad, then every object of
$\mathcal{V}$ must be cofibrant. These results leave open the question of what
extra structure, if not an enrichment in the ordinary sense, is naturally
possessed by the cofibrant replacement comonad of an enriched model category
when not every object of the base monoidal model category is cofibrant.
In this talk I will introduce the notions of locally weak comonad and of
monoidal and enriched algebraic weak factorisation system, and will propose an
answer to the above question by showing that the cofibrant replacement comonad
of an enriched algebraic weak factorisation system is a locally weak comonad.
Special attention will be given to the monoidal model category of 2-categories
with the Gray tensor product, in which not every object is cofibrant. |
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