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Stream: theory: category theory

Topic: When is the left Kan extension of a monad again a monad?

Patrick Nicodemus (Feb 02 2025 at 02:20):

Let $C, D$ be categories. Let $T : C\to C$ be a monad. Let $i : C \to D$ be a functor.
Under what, if any, known conditions is the left Kan extension of $iT$ along $i$ a monad?

The right Kan extension of $iT$ along $i$ is always a monad.

Patrick Nicodemus (Feb 02 2025 at 02:34):

Let $A$ be the category whose objects are natural numbers and where a morphism $n \to m$ is a subset inclusion ${k, k+1, k+2, ..., k+n-1}$ of $n$ consecutive natural numbers in the set ${0,...,m-1}$.

Let $S : A \to \mathbf{Graph}$ be the functor that sends $n$ to the graph with $n$ nodes that has an edge from $k$ to $k+1$ for each $k$. Let $K :A \to\mathbf{Graph}$ be the functor that sends $n$ to the graph with $n$ nodes that has an edge from $k$ to $k'$ whenever $k \leq k'$. It should be the case (I haven't computed this in detail) that the left Kan extension of $K$ along $S$ is the free category monad on $\mathbf{Graph}$.

If $\mathbf{Rel}$ is defined to be the subcategory of $\mathbf{Graph}$ where the edge set $E(x,y)$ is either empty or a singleton for all $x,y$ in the graph, then we can define $T$ to be the reflexive-transitive closure monad on $\mathbf{Rel}$ and it seems plausible that the left Kan extension of this to $\mathbf{Graph}$ is also the free category monad. This is what is motivating my question.

Patrick Nicodemus (Feb 02 2025 at 02:37):

Maybe the answer comes from asking : when is the left Kan extension operation $T \mapsto Lan_i(iT)$ strong monoidal with respect to endofunctor composition as opposed to merely being colax monoidal?

Kenji Maillard (Feb 02 2025 at 09:27):

In the settings of your question, iT will always be a relative monad with respect to i. Theorem 4.6 of monads need not be endofunctors gives sufficient conditions for the resulting left kan extension to be amonad.

Patrick Nicodemus (Feb 05 2025 at 06:42):

Thanks Kenji! Very helpful answer.
(I no longer think my comment about the inclusion of Rel into Graph is correct.)

Ivan Di Liberti (Feb 05 2025 at 10:40):

Patrick Nicodemus said:

Let $C, D$ be categories. Let $T : C\to C$ be a monad. Let $i : C \to D$ be a functor.
Under what, if any, known conditions is the left Kan extension of $iT$ along $i$ a monad?

The right Kan extension of $iT$ along $i$ is always a monad.

https://mathoverflow.net/questions/321909/extending-monads-along-dense-functors