## The Monoidal Grothendieck Construction

My grad student Joe Moeller is talking at the 4th Symposium on Compositional Structures this Thursday! He’ll talk about his work with Christina Vasilakopolou, a postdoc here at U.C. Riverside. Together they created a monoidal version of a fundamental construction in category theory: the Grothendieck construction! Here is their paper:

• Joe Moeller and Christina Vasilakopoulou, Monoidal Grothendieck construction.

The monoidal Grothendieck construction plays an important role in our team’s work on network theory, in at least two ways. First, we use it to get a symmetric monoidal category, and then an operad, from any network model. Second, we use it to turn any decorated cospan category into a ‘structured cospan category’. I haven’t said anything about structured cospans yet, but they are an alternative approach to open systems, developed by my grad student Kenny Courser, that I’m very excited about. Stay tuned!

The Grothendieck construction turns a functor

$F \colon \mathsf{X}^{\mathrm{op}} \to \mathsf{Cat}$

into a category $\int F$ equipped with a functor

$p \colon \int F \to \mathsf{X}$

The construction is quite simple but there’s a lot of ideas and terminology connected to it: for example a functor $F \colon \mathsf{X}^{\mathrm{op}} \to \mathsf{Cat}$ is called an indexed category since it assigns a category to each object of $\mathsf{X},$ while the functor $p \colon \int F \to \mathsf{X}$ is of a special sort called a fibration.

I think the easiest way to learn more about the Grothendieck construction and this new monoidal version may be Joe’s talk:

• Joe Moeller, Monoidal Grothendieck construction, SYCO4, Chapman University, 22 May 2019.

Abstract. We lift the standard equivalence between fibrations and indexed categories to an equivalence between monoidal fibrations and monoidal indexed categories, namely weak monoidal pseudofunctors to the 2-category of categories. In doing so, we investigate the relation between this global monoidal structure where the total category is monoidal and the fibration strictly preserves the structure, and a fibrewise one where the fibres are monoidal and the reindexing functors strongly preserve the structure, first hinted by Shulman. In particular, when the domain is cocartesian monoidal, lax monoidal structures on a functor to Cat bijectively correspond to lifts of the functor to MonCat. Finally, we give some indicative examples where this correspondence appears, spanning from the fundamental and family fibrations to network models and systems.

To dig deeper, try this talk Christina gave at the big annual category theory conference last year:

• Christina Vasilakopoulou, Monoidal Grothendieck construction, CT2018, University of Azores, 10 July 2018.

Then read Joe and Christina’s paper!

Here is the Grothendieck construction in a nutshell:

### 9 Responses to The Monoidal Grothendieck Construction

1. Samuel Vidal says:

It is also the construction used by Joyal in more particular context for the category of objects of a given species.

• John Baez says:

Yes! And that’s how we ran into the monoidal Grothendieck construction. We were studying species of a special sort! A species is a functor

$F \colon \mathsf{FinSet}_0 \to \mathsf{Set}$

where $\mathsf{FinSet}_0$ is the groupoid of finite sets and bijections. Applying the Grothendieck construction gives a category $\int F.$

But in our paper on network models, we were interested in lax symmetric monoidal functors

$F \colon \mathsf{FinSet}_0 \to \mathsf{Mon}$

where $\mathsf{Mon}$ is the category of monoids. Many species are actually like this.

We noticed that applying the Grothendieck construction gives a symmetric monoidal category $\int F.$ Joe soon realized that any lax symmetric monoidal functor

$F \colon \mathsf{FinSet}_0 \to \mathsf{Cat}$

gives a symmetric monoidal category $\int F.$ And at a more basic level, any lax monoidal functor

$F \colon \mathsf{FinSet}_0 \to \mathsf{Cat}$

gives a monoidal category $\int F.$

So, all this stuff is closely connected to species, but with a little extra dose of ‘monoidalness’.

• Samuel Vidal says:

Thanks for the reply, I will read the series and the article.

• Raymond Lutz says:

Le monde est petit: I’m writing this 8 km from André Joyal’s birthplace. Sorry for being slightly off topic. 8-)

• John Baez says:

Where is his birthplace?

• arch1 says:

Drummondville, Quebec, says Wikipedia. (And thank you Raymond for writing one of the 4 sentences I can understand using my high school French).

• Raymond Lutz says:

Yes, Drummondville (where I teach college physics) and more precisely Saint-Majorique-de-Grantham village. I didn’t know about André Joyal before I listened to a public radio program for which he was interviewed on the occasion of Grothendieck’s demise.

2. Wikipedia gives the definition as in the post, but in nLab the map into Cat is demanded to be a pseudofunctor (the natural map between 2-categories). I’d wish I had better grasp on when each definition is in order.

• John Baez says:

I avoided saying “pseudofunctor” merely because I didn’t want to scare people. If we apply the Grothendieck construction to a pseudofunctor $F \colon \mathsf{X} \to \mathsf{Cat}$ we get a category $\int F$ that’s equipped with a functor

$p \colon \int F \to \mathsf{X}$

This functor $p$ has a special property: it’s a fibration. This means, very roughly, that any morphism in $\mathsf{X}$ can be lifted to a morphism in $\int F$ that’s a ‘cartesian lift’. Joe’s slides explain what this means.

A special case of a pseudofunctor $F \colon \mathsf{X} \to \mathsf{Cat}$ is a functor: this preserves composition exactly, not just up to isomorphism. If we apply the Grothendieck construction to an actual functor we get a category $\int F$ that’s equipped with a split fibration:

$p \colon \int F \to \mathsf{X}$

In a split fibration, we’ve chosen cartesian lifts such that the lift of a composite is the composite of the lifts. This extra property comes from $F$ preserving composition exactly, not just up to isomorphism.

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