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If a category $C$ has limits of a certain type, then the category of monads on $C$ has the same type of limits, and these limits are computed "levelwise" (i.e. are preserved by the forgetful functor from the category of monads on $C$ to the category of endofunctor on $C$).

I wonder if algebras for such limits have been studied somewhere ?

If I focus on product of monads for simplicity there seems to be some interesting to say:

Given $M$ and $N$ two monads on $C$ the monad $M \times N$ on $C$ can be seen as induced by the adjunction:

$$(M , N) : C \leftrightarrows M\text{-Alg} \times N\text{-Alg} : U \times U'$$

Where $U \times U'$ is the functor sending a pairs of algebra $(X,Y)$ to $U(X) \times U'(Y)$ where $U$ and $U'$ denotes the forgetful functors, and $(M,N)$ is the functor sending $X$ to the pair $(M(X),N(X))$.

This adjunction is unfortunately not always monadic, but it is often not very far from it. If I look at the simplest example: $C= Set$, $M=N=Id$, then I can prove by hand that a non-empty $Id \times Id$-Algebra is the same as a pair of non-empty sets, so that the category of $Id \times Id$-Algebra identifies with a the full subcategory of (Set $ \times $ Set) of pairs of sets such that one is empty if and only if the other is also empty.

I haven't check it in full details, but it seems there are many examples where the $(M \times N)-Algebras$ are really pairs of algebras. For example if $M$ and $N$ are monads on Set whose algebras are always non-empty.

Has this been studied somewhere in more detail ? Is there some assumptions under which we can describe algebra for some limits of monads in terms of algebras for the individual monads ?

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  • $\begingroup$ By the formal theory of monads, the 2-functor $Alg: Mnd \to Cat$ is right 2-adjoint to the 2-functor which sends a category to the identity monad on it (here $Mnd$ is the 2-category of monads in $Cat$). So perhaps one should ask which limits in $Mnd(C)$ are also limits in the larger 2-category $Mnd$. I'm a bit confused, though, because there's some weirdness about which direction the coherence 2-cell points in the definition of a 1-morphism of monads, which might not line up properly for $Mnd(C)$ to actually be a sub-2-category of $Mnd$. $\endgroup$
    – Tim Campion
    Commented Jul 17, 2020 at 21:44
  • $\begingroup$ @TimCampion That's interesting. There are several way to construct a category Mnd of monads, which one are you referring to in your comment ? is it equivalent to the full subcategory of $\mathbf{Cat}^\to$ of monadic right adjoint functors ? I have the impression that the underlying idea of your comment actually say something about colimits of monads instead of limits (due to a hidden opposite category in the convention), for which it is indeed easy to describe the category of algebras... $\endgroup$
    – Simon Henry
    Commented Jul 18, 2020 at 15:05

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For finitary commutative monads on $\mathbf{Set}$, this has been studied in Faro–Kelly's On the canonical algebraic structure of a category. I have reworded Proposition 11 ibid. below in terms of finitary monads rather than algebraic theories.

Proposition (Faro–Kelly).

For an finitary monad $\mathscr T$, write $(\mathscr T\text{-Alg})'$ for the full subcategory of $\mathscr T\text{-Alg}$ given by the non-empty algebras.

Let $\mathscr T$ and $\mathscr S$ be non-degenerate commutative finitary monads on $\mathbf{Set}$; so that $\mathscr T \times \mathscr S$ is another such. If both $\mathscr T$ and $\mathscr S$ have nullary operations; we have $$(\mathscr T \times \mathscr S)\text{-Alg}\cong \mathscr T\text{-Alg}\times \mathscr S\text{-Alg}$$ Otherwise $(\mathscr T \times \mathscr S)\text{-Alg}$ is obtained from $((\mathscr T \times \mathscr S)\text{-Alg})'$ by freely adding an initial object; and $((\mathscr T \times \mathscr S)\text{-Alg})'$ is itself isomorphic to $(\mathscr T\text{-Alg})' \times (S\text{-Alg})'$.

The proof appears to generalise at least to arbitrary commutative monads with rank on $\mathbf{Set}$.

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