# When does the forgetful functor from algebras over a monad commute with homotopy geometric realizations?

Let $\mathcal{C}$ be a combinatorial model category and $\mathrm{T}$ a monad on $\mathcal{C}.$

Assume that the model structure on $\mathcal{C}$ lifts to a model structure on the category of $\mathrm{T}$-algebras $\mathrm{Alg}_{\mathrm{T} }(\mathcal{C}),$ where the weak equivalences and fibrations are those of $\mathcal{C}.$

Assume that $\mathrm{T}: \mathcal{C} \to \mathcal{C}$ preserves homotopy colimits indexed by $\Delta^{\mathrm{op}}.$

Does the forgetful functor $\mathrm{Alg}_{\mathrm{T} }(\mathcal{C}) \to \mathcal{C}$ preserve homotopy colimits indexed by $\Delta^{\mathrm{op}}?$

More generally one can ask the question replacing $\Delta^{\mathrm{op}}$ by an arbitrary category.

Remark: If $\mathrm{T}: \mathcal{C} \to \mathcal{C}$ preserves colimits indexed by some category $\mathrm{K},$ the forgetful functor $\mathrm{Alg}_{\mathrm{T} }(\mathcal{C}) \to \mathcal{C}$ preserves colimits indexed by $\mathrm{K}.$

This also holds for $\infty$-categories with the appropriate notion of monad and algebras over a monad.

• A few years ago, I read a draft of a paper by Gutierrez, Rondigs, Spitzweck, and Ostvaer with a section related to your question. I believe the claim was that for any simplicial monoidal model category $M$, any admissible $C$-colored operad in $M$, and any simplicial object $A_\bullet$ from $Alg_O(M)$, then there is a natural iso $|U(A_\bullet)|_{M^C} \cong U(|A_\bullet|_{Alg_O(M)})$. Maybe you can write to Javier Gutierrez to ask about this. The draft never made it to arxiv, but I remember we checked the correctness of the relevant section. – David White Apr 9 '18 at 22:31
• One thing to note is that an isomorphism of geometric realization does not imply anything about homotopy colimits yet because one needs reedy cofibrant diagrams (in algebras and underlying). – Christian Wimmer Apr 16 '18 at 9:21

In Theorem 1.6 in the paper Bar constructions and Quillen homology of modules over operads, John Harper gave a positive answer to your question when T is the monad of a one-colored symmetric operad O and the underlying category is either symmetric spectra or unbounded chain complexes over a field of characteristic zero. It is not hard to extend his argument to cover the case of colored symmetric operads. I would not be surprised if his argument actually works for simplicial monoidal model categories, but I have not checked it myself.

• Thanks a lot for your comment. Good to know about this work. – Hadrian Heine Apr 14 '18 at 19:18

Denote by U: Alg_T(C)→C and Free: C→Alg_T(C) the adjoint functors between Alg_T(C) and C. Suppose U(j) is a cofibration in C for all j, where j is a cobase change of Free(i) in Alg_T(C), where i is a generating cofibration in C.

In this case the preservation of sifted homotopy colimits follows from the preservation of sifted colimits by the forgetful functor and the fact that sifted homotopy colimits can be computed by replacing the diagram by a weakly equivalent projectively cofibrant diagram.

This follows from the key fact that the forgetful functor from sifted diagrams of T-algebras in C to sifted diagrams in C preserves projectively cofibrant diagrams. Indeed, the forgetful functor preserves sifted colimits and sends cobase changes of free morphisms on generating projective cofibrations to projective cofibrations by assumption on T.

This condition is satisfied in many situations of interest, e.g., when T is induced by a colored operad in a symmetroidal model category, as explained in Theorem 6.6 of arXiv:1410.5675. The model categories of simplicial sets, simplicial symmetric spectra, and chain complexes in characteristic 0 are symmetroidal. If we replace symmetric operads with nonsymmetric operads, then a tractable monoidal model category will suffice, which includes almost all important examples.

• That sounds very interesting to me. – Hadrian Heine May 13 '18 at 15:58
• Let's assume that T preserves cofibrant objects and let J be a small sifted category and $H: J \to Alg_T(\mathcal{C})$ a functor that is projectively-cofibrant in $Alg_T(\mathcal{C})^J.$ Denote $V: Alg_T(\mathcal{C}) \to \mathcal{C}$ the forgetful functor. Why is $V(hocolim H) \simeq V(colim H) \simeq colim(VH)$ the homotopy colimit of VH? Is VH projectively cofibrant in $\mathcal{C}^J$? If yes, why? Does the forgetful functor preserve projectively cofibrant diagrams if T preserves cofibrant objects? If yes, how does one show this? – Hadrian Heine May 13 '18 at 16:17
• @HadrianHeine: I misstated the condition, it's actually a bit more complicated. I adjusted my writeup. Do you have a specific monad T in mind? – Dmitri Pavlov May 13 '18 at 20:36
• I am especially interested in the case of a monad arising from an admissible operad $\mathcal{O}$ in a symmetric monoidal combinatorial model category, where all objects are cofibrant. In the easiest example let $\mathcal{C}$ be the category of chain complexes over a field of char 0 and $\mathcal{O}$ the Lie-operad. – Hadrian Heine May 13 '18 at 23:26
• @HadrianHeine: I see, this precise question is addressed in Proposition 7.8 in the cited arXiv paper. (See also Theorem 7.10, where it is applied to obtain a comparison result.) In particular, the answer to your question for characterstic 0 chain complexes is positive. – Dmitri Pavlov May 14 '18 at 5:48