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I would like to know in anyone has developed method for constructing colimits in the category of algebra for a monad in the $(\infty,1)$-categorical framework, using transfinite constructions.

I have in mind an analogue of Kelly's construction of such colimits in the $\infty$-categorical framework. Though any explicit construction of colimits in the category of algebra for an accessible monad on a presentable categories would be of interest.

Lurie is proving the existence of colimits in the special case of algebras for operads, but I don't think its proof can be extended to more general monads, but there might some things to be done...

My motivations: I want to prove that a certain class of morphisms between algebras for a monad are stable under colimits. This doesn't seems to follow from formal properties, so I need a concrete construction of these colimits to prove it. If I naively assume that Kelly's construction works in $\infty$-categories, then I can prove what I want by induction on the transfinite construction. But of course Kelly's construction doesn't work without modification in $\infty$-category theory, so I'm wondering if someone has developed some tools to explicitly construct such colimits.

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  • $\begingroup$ How about Higher Algebra 4.2.3.5 and 4.2.3.7 in the case where C is the category of endofunctors for M? $\endgroup$
    – Dylan Wilson
    Commented Sep 17, 2018 at 10:31
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    $\begingroup$ @DylanWilson : These shows that for a monad M preserving colimits of shape K, the category of M-algebras has colimits of shape K and these are preserved by the forgetful functor to C. This case doesn't require transfinite induction. I'm interested in the case where the monad M does not preserves colimits (only filtered enough colimits) so that one needs to do some iterated colimits/application of M to gradually correct the colimits in C into a colimits in M-algebras. $\endgroup$
    – Simon Henry
    Commented Sep 17, 2018 at 11:04
  • $\begingroup$ Interesting ! So I'm completely fine with those assumptions, but what I'm after is an 'explicit' construction of these colimits using some transfinite construction (in the spirit of Kelly's construction). Something that can be used to prove concrete properties of the colimits in concrete situation. Your remarks does seems to qualifies as a proof that colimits exists, but I'm not really convinced yet that this give an explicit description of them... (but maybe it does) $\endgroup$
    – Simon Henry
    Commented Sep 17, 2018 at 14:16
  • $\begingroup$ Hmm... well are we allowed to assume that the category $\mathcal{C}$ is presentable and that the monad $T$ is accessible? Then $\mathsf{Alg}_T$ is at least accessible using the above arguments. But it has all limits, so it's automatically presentable as well (has a fully faithful functor to presheaves on compact generators (take free algebras on compact things to get those) and the functor is accessible. Since the target is presentable, the adjt functor theorem supplies a left adjoint, so we've expressed $\mathsf{Alg}_T$ as an accessible localization of a presheaf category). $\endgroup$
    – Dylan Wilson
    Commented Sep 17, 2018 at 14:19
  • $\begingroup$ re-added it, but the thing about being able to apply the adjoint functor theorem doesn't immediately work I think. Should still be true, but need to apply a more general version of the adjoint functor theorem than the one in HTT. In the 1-categorical case this is the "reflection theorem" in Adamek-Rosicky $\endgroup$
    – Dylan Wilson
    Commented Sep 17, 2018 at 14:20

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