### Koudenburg on Algebraic Weighted Colimits

#### Posted by Simon Willerton

My student Roald Koudenburg recently successfully defended his thesis and has yesterday put his thesis on the arXiv.

Roald Koudenburg, Algebraic weighted colimits

I will give a rough caricature of what he does. For a much nicer overview, I suggest you read the well-written introduction to the thesis! (Relating to the Café, there’s even an example including Simon Wadsley’s Theorem into Coffee.)

Roald was originally thinking about Ezra Getzler’s approach to operads in Operads revisited, and needed to generalize a result of Getzler’s on when the left Kan extension of a symmetric monoidal functor along a symmetric monoidal functor is itself a symmetric monoidal functor.

If you are of that kind of persuasion, you will be aware that symmetric monoidal categories are the pseudo-algebras for the free symmetric strict monoidal category $2$-monad on ${Cat}$, the $2$-category of categories, and that symmetric monoidal functors are the algebra maps for this $2$-monad.

You might then consider the situation of a $2$-monad $T$ on a $2$-category, and ask when the left Kan extension of a map of $T$-algebras along a map of $T$-algebras is again a $T$-algebra map.

Here are three examples of algebras for a $2$-monad that Roald considers.

Ordered compact Hausdorff spaces are algebras for the ultrafilter $2$-monad on $2$-Cat. In this case the question becomes, when is the left Kan extension of a continuous order preserving map along another such map also continuous and order preserving?

Double categories can be considered as algebras for a certain $2$-monad on a $2$-category of internal categories in a specific presheaf category. (Yes, I find that a mouthful.)

Similarly, monoidal globular categories of Batanin are algebras for some $2$-monad.

Going back to the symmetric strict monoidal category monad, this monad actually extends from the $2$-category $Cat$ to the the proarrow equipment of categories, functors and profunctors. Mike wrote a nice post here at the Café on Equipments and what they have to do with limits.

It turns out that there are several examples of $2$-monads on equipments, with interesting algebras and Roald looks at conditions necessary for when, given such a $2$-monad $T$, the left Kan extension of a map of $T$-algebras along map of $T$-algebras is a map of $T$-algebras.

I should at this point say something precise about the main result: given a ‘right suitable normal’ monad $T$ on a closed equipment $K$, Roald defines ‘right colax $T$-promorphisms’ which, together with the usual colax T-algebra maps, form an equipment $T\text{-rcProm}$. The main theorem is the following.

Theorem:Let $T$ be a ‘right suitable normal’ monad on a closed equipment $K$. The forgetful functor $UT: T\text{-rcProm} \to K$ ‘lifts’ all weighted colimits. Moreover its lift of a weighted colimit $colim_J d: B\to M$, where $d: A \to M$ is a pseudomorphism and $J : A \to B$ is a right pseudopromorphism, is a pseudomorphism whenever the canonical vertical cell $colim_{T} (m \circ Td) \Rightarrow m\circ T(colim_J d)$ is invertible, where $m: TM \to M$ is the structure map of $M$.

## Re: Koudenburg on Algebraic Weighted Colimits

Congratulations, both!