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Have added a pointer to Katryn Hess’s article to (infinity,1)-monad
added also a pointer to the article by Emily Riehl and Dominic Verity, also at higher monadic descent
Added under Properties – Homotopy coherence a brief remark:
An (∞,1)-adjunction is uniquely determined already by its image in the homotopy 2-category (Riehl-Verity 13, theorem 5.4.14). This is not in general true for $(\infty,1)$-monads: these being monoids in an (∞,1)-category of endomorphisms, they in general have coherence data all the way up in degree. However, by the previous statement, those $(\infty,1)$-monads which arise from (∞,1)-adjunctions are determined by less coherence data (Riehl-Verity 13, page 6). This should justify the model-category theoretic discussion in (Hess 10) in (∞,1)-category theory.
Does this mean there’s a difference from ordinary monads, where each arises from an adjunction?
But we have at (infinity,1)-monad
These relate to (∞,1)-adjunctions as monads relate to adjunctions.
By the way, monad and adjunction hardly speak to one another. And monadic adjunction seems isolated too. What’s the best way to organise these?
So in the $(\infty, 1)$-monad case, is there anything which plays the role of the EM and Kleisli adjunctions for ordinary monads, something more general than an adjunction?
Maybe I did’t say this well. The $\infty$-EM category of $\infty$-algebras always exists and I suppose the forgetful functor from there always is a right $\infty$-adjoint (this must be in section 3 of HigherAlgebra, I suppose). Certainly if it is, then this gives an $\infty$-adjunction that reproduces the original $\infty$-monad (this is remark 6.2.0.7 in Higher Algebra).
I have now adjusted the text in the entry (infinity,1)-monad as follows (but this is still not optimal, everybody please feel invited to further improve):
{#HomotopyCoherence}
An (∞,1)-adjunction is uniquely determined already by its image in the homotopy 2-category (Riehl-Verity 13, theorem 5.4.14). This is not in general true for $(\infty,1)$-monads: these being monoids in an (∞,1)-category of endomorphisms, they in general have relevant coherence data all the way up in degree. However, by the previous statement, for $(\infty,1)$-monads given as arising from specified (∞,1)-adjunctions are determined by less (further) coherence data (Higher Algebra, remark 6.2.0.7, prop. 6.2.2.3, Riehl-Verity 13, page 6). This should justify the simplicial model category-theoretic discussion in (Hess 10) in (∞,1)-category theory.
{#MonadicityTheorem}
+– {: .num_theorem}
Let $(L \dashv R)$ a pair of adjoint (∞,1)-functors such that
$R$ is a conservative (∞,1)-functor;
the domain (∞,1)-category of $R$ admits geometric realization ((∞,1)-colimit) of simplicial objects;
and $R$ preserves these
then for $T \coloneqq R \circ L$ the essentially uniquely defined $(\infty,1)$-monad structure on the composite endofunctor, there is an equivalence of (∞,1)-categories identifying the domain of $R$ with the (∞,1)-category of algebras over an (∞,1)-monad over $T$ and $R$ itself as the canonical forgetful functor.
=–
This appears as (Higher Algebra, theorem 6.2.0.6, theorem 6.2.2.5, Riehl-Verity 13, section 7)
Sorry, I’m being dim. Take an ∞-monad with coherence data all the way up in degree. Send it to its ∞-EM category of ∞-algebras and so to an ∞-adjunction. Then send this to its ∞-monad. Am I back where I started? But then isn’t this an ∞-monad determined by less coherence data?
I need to look more into this, maybe not right now though. But two remarks:
First, picking that adjunction is already choosing a bit of data.
Second, mentioned above: is for every $\infty$-monad the forgetful functor from its $\infty$-EM-category a right adjoint? I don’t see this stated anywhere (though it might be evident, either way). The relevant general statement would be around corollary 3.1.3.4 in HigherAlgebra, but this needs further unwinding in the present case. Maybe not for every $\infty$-monad that functor is a right adjoint? Then it would not form an adjunction giving that $\infty$-monad.
Need to look more into this…
Implicit in Definition 6.1.15 of the Riehl–Verity paper is the fact that every homotopy-coherent monad induces a homotopy-coherent monadic adjunction. If I understand correctly, the abstract nonsense of weighted limits implies that the homotopy-coherent monad coming from that adjunction is strictly equal to the one we started with.
Right, thanks. That we recover the original monad from this is then also HA remark 6.2.0.7.
I’ll edit the entry a bit more accordingly…
Okay, I have expanded the entry here.
David,
I suppose the way to think about where the coherence data goes is as follows:
The statement is that given an $\infty$-monad with all its coherence data, we can recover it from the $\infty$-adjunction between its $\infty$-category $Alg_{\mathcal{C}}(T)$ of algebras and the underlying $\infty$-category $\mathcal{C}$. Now here there is not much relevant “compositional” coherence data, in that this $\infty$-adjunction is fixed on its shadow in the homotopy 2-category. On the other hand, now the $\infty$-category $Alg_{\mathcal{C}}(T)$ is part of the data, and that clearly is a lot of information about $T$.
So the statement is that once you choose an $\infty$-category $\mathcal{D}$ over which your $\infty$-adjunction $\mathcal{C} \leftrightarrow \mathcal{D}$ factors, then the coherence data of the $\infty$-monad on $\mathcal{C}$ is uniquely induced form the 2-categorical coherence data of the adjunction. But this uses the data provided by $\mathcal{D}$.
Re #13, that makes sense thanks.
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