The first part of the talk will be devoted to showing that Moerdijk-Weiss’s category of dendrices \Omega is a Lawvere theory with arities for the free-operad monad on coloured symmetric Set-valued collections. This demonstration is due to J. Kock, following Weber, and generalises the known example of the category \Delta of simplices as a Lawvere theory with arities for the free-category monad on graphs.

If time permits, the second part will try to generalise this setting to simplicial operads and simplicial (i.e. sSet-enriched) model categories by replacing Set with the category sSet of simplicial sets, following Moerdijk-Weiss and Cisinski-Moerdijk.

The underlying theme is a variant of the small object argument that produces orthogonal factorisation systems (i.e. whose left and right maps lift uniquely against each other). This variant has been presented in a previous talk by M. Anel, and its generalisation to simplicial model categories is left Bousfield localisation.