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Stream: deprecated: history of ideas

Topic: Lax functors in Span & their Grothendieck construction

Matteo Capucci (he/him) (Oct 26 2022 at 15:28):

TL;DR: Do you know who first studied lax functors in Span(Set) and their 'Grothendieck' construction?

At [[displayed category]] Benabou is credited for the correspondence between functors $D \to C$ and normal lax functors $C \to \bf Prof$, citing Streicher's notes of his lectures in Darmstadt from 2000 (though it is clear from the notes their contents have been known to Benabou & his circle for at least 30 years prior).
However, I can't trace a reference to the fact functors $D \to C$ are lax functor $C \to \bf \mathbb Span(Set)$, especially to the relative Grothendieck construction which I'd like to rename to something less confusing (ideally, 'X construction' where X is the name of the person who figured it out first, but if you have better proposals I'll listen).

John Baez (Oct 26 2022 at 17:29):

I too would love to know more about the history of this.

Alexander Campbell (Oct 27 2022 at 03:30):

In Street's note on *Powerful functors*, the citation for this correspondence reads:

J. Bénabou, Lectures at Oberwolfach (Germany) and other places, 1972 – 1995.

Alexander Campbell (Oct 27 2022 at 03:31):

The abstract for Bénabou's 1972 Oberwolfach talk can be found on pages 6-7 of the Tagungsbericht found here (at the bottom of the page): https://oda.mfo.de/handle/mfo/1575, which shows that his talk was indeed about this constrution, although the abstract does not go as far as to actually give the construction or state the correspondence theorem.

Mike Shulman (Oct 27 2022 at 03:34):

@Alexander Campbell Aren't those references about Prof, not Span?

Alexander Campbell (Oct 27 2022 at 03:35):

Mike Shulman said:

Alexander Campbell Aren't those references about Prof, not Span?

Oops, you're quite right, I didn't read Matteo's question carefully enough!

Mike Shulman (Oct 27 2022 at 03:53):

I made the same mistake myself at first, but caught it before posting... (-:

David Michael Roberts (Oct 27 2022 at 07:21):

Thomas Streicher might know....

Matteo Capucci (he/him) (Oct 27 2022 at 07:29):

Thanks anyway @Alexander Campbell :) I'm happy to have a better reference for the construction for displayed categories anyway (pushing it to nLab now)

Matteo Capucci (he/him) (Oct 27 2022 at 07:30):

David Michael Roberts said:

Thomas Streicher might know....

Good point... If no one comes forward with anything here I'll email him

Jade Master (Oct 27 2022 at 08:50):

I remember when @Joe Moeller and I were writing about this, one author said it was folklore (do you remember which?)

Matteo Capucci (he/him) (Oct 27 2022 at 09:10):

oh no, not the F-word :worried:

Matteo Capucci (he/him) (Oct 27 2022 at 09:11):

though 'folklore construction' sounds good

Matt Earnshaw (Oct 27 2022 at 11:23):

In (1997), Pavlović and Abramsky say they can't even cite folklore, but consider it too basic to be unknown. Surely it was already known to Benabou due to the closely related fact about monads in Span(Set) – if only he had published the promised part II of 'Introduction to bicategories'.

Joe Moeller (Oct 27 2022 at 14:16):

Yes, it was Pavlović and Abramsky we were looking at @Jade Master

Matteo Capucci (he/him) (Oct 27 2022 at 14:33):

Matt Earnshaw said:

In (1997), Pavlović and Abramsky say they can't even cite folklore, but consider it too basic to be unknown. Surely it was already known to Benabou due to the closely related fact about monads in Span(Set) – if only he had published the promised part II of 'Introduction to bicategories'.

I wonder what it means to 'find a reference in [..] folklore' lol

Matteo Capucci (he/him) (Oct 27 2022 at 14:33):

I see, so we get to give it a descriptive name instead of a patronymic

Matt Earnshaw (Oct 27 2022 at 16:25):

Reference in folklore struck me as funny too. I guess "private communication", "mimeographed lecture notes", "unpublished manuscript", etc. would be species of this...

Jason Erbele (Oct 27 2022 at 16:49):

I don't know. "Private communication" etc. don't strike me as forms of folklore. You still have a source for the ideas with private communication, even if the reader cannot (easily) find the actual reference. I think "folklore" indicates a large part of the community knows about X, but nobody knows who first came up with X, often because X is so basic that "anybody" could have come up with it.

Mike Shulman (Oct 27 2022 at 16:54):

cf [[folklore]]

Matt Earnshaw (Oct 27 2022 at 18:37):

Jason Erbele said:

I don't know. "Private communication" etc. don't strike me as forms of folklore.

they are perhaps forms of "references in folklore", referenced in the above reference

Joe Moeller (Oct 27 2022 at 18:49):

Jason Erbele said:

I don't know. "Private communication" etc. don't strike me as forms of folklore. You still have a source for the ideas with private communication, even if the reader cannot (easily) find the actual reference. I think "folklore" indicates a large part of the community knows about X, but nobody knows who first came up with X, often because X is so basic that "anybody" could have come up with it.

I think there's also a pragmatic definition of folklore, which is that if I or almost anybody else cannot easily access the source, it is effectively folklore. Even if there is a theorem in a published paper, but it's never been scanned, and physical copies are very hard to come by, it might as well be folklore to me. I should go read [[folklore]] as Mike is suggesting though.

David Michael Roberts (Oct 28 2022 at 00:47):

One notion of folklore that is fairly robust to the point of being nearly citable is if a bunch of people can point to a conference where a certain idea arose out of multiple interlinked discussions, and then a year or so later people started publishing papers using the idea, but no one can lay claim to have come up with it, and there's no one source to cite, but the idea has a definite origin in time that people can point to.

As opposed to something that no one can remember where they heard it first or it seems obvious and is not that difficult to sit down and prove, and it's used without much comment, as a form of basic fact. By this I don't mean results that seem like they are should be true, but which actually are non-trivial to prove to some degree or other (cf the Caramello affair). That to me falls in a different category.

Nathanael Arkor (Oct 28 2022 at 11:38):

One notion of folklore that is fairly robust to the point of being nearly citable is if a bunch of people can point to a conference where a certain idea arose out of multiple interlinked discussions, and then a year or so later people started publishing papers using the idea, but no one can lay claim to have come up with it, and there's no one source to cite, but the idea has a definite origin in time that people can point to.

If there are actual references for an idea, even if the authors say that the idea was not due to them (but do not necessarily recall to whom the idea is due), then it's not folklore. The primary problem with folklore isn't that the progenitor isn't known, but that there aren't any reference for precise statements and proofs. Sometimes you see authors write something like: "The following lemma is folklore.", which is probably fine (though in some of these situations it does seem to be a a shorthand for "I didn't want to put in the effort to work out to whom the idea should be attributed."), but as soon as the result has been written out, it is no longer folklore.

Matteo Capucci (he/him) (Oct 28 2022 at 18:06):

To me the term 'folklore' stresses more the common knowledge of a fact or praxis than an inaccessibility of a source for it. In a sense also traceable facts can become folkloric in being widely known and used without always referencing the original source. Unfortunately some things are never published, in any form, so the initial source is lost even though word of mouth/praxis popularizes the idea nonetheless.

Matteo Capucci (he/him) (Oct 28 2022 at 18:09):

For some reason category theorists seem to be particularly prone to this phenomenon, having some of the founding fathers avoided publication of lots of non-trivial stuff.

Mike Shulman (Oct 28 2022 at 20:47):

Matteo Capucci (he/him) said:

For some reason category theorists seem to be particularly prone to this phenomenon, having some of the founding fathers avoided publication of lots of non-trivial stuff.

I suspect it's also partly due to the fact that so many facts in category theory have proofs that are straightforward and easy for experts to do in their head, but time-consuming and tedious to write down on paper.

Nathanael Arkor (Oct 28 2022 at 21:10):

In a sense also traceable facts can become folkloric in being widely known and used without always referencing the original source.

This is true, though I find it a rather objectionable practice.

fosco (Oct 29 2022 at 10:53):

Mike Shulman said:

Matteo Capucci (he/him) said:

For some reason category theorists seem to be particularly prone to this phenomenon, having some of the founding fathers avoided publication of lots of non-trivial stuff.

I suspect it's also partly due to the fact that so many facts in category theory have proofs that are straightforward and easy for experts to do in their head, but time-consuming and tedious to write down on paper.

I think this problem originates from the fact that academic life does not reward the practice of writing down explicit arguments because yeah yeah, whatever, it's obvious once you're in the circle of ideas. I'm not saying we should write everything up; I'm saying that when someone writes down something completely elementary at a painstaking level of detail and accuracy (with exactly zero occurrences of "one easily sees that the diagram commutes"), we should at least thank them instead of belittling their effort.

fosco (Oct 29 2022 at 10:56):

Anyway, more in topic: I suspect @Matteo Capucci (he/him) asked this question because of me, so maybe I can explain at least in the main lies what motivated me.

I have a construction that I can turn into either

• a normal oplax functor ${\cal C}\to Span(Cat)$
• a normal lax functor ${\cal C}\to CoSpan(Cat)$

(clearly these constructions arise as dual to each other in a suitable sense, so they are "essentially the same")

My problem is that I don't know if this defines some kind of displayed category over $\cal C$, generalising/dualising/etc the Grothendieck-Bénabou construction: if it doesn't, it's completely useless for my purposes.

What gets in the way is that "oplax into Span" part: it's not lax into Span; and even if I transform it into a lax functor, I fall in cospans. So, the blanket is always too short either way I turn the arrow!

fosco (Oct 29 2022 at 10:58):

(in turn, the question is motivated by me trying to exit a cul-de-sac in a work in progress with @Greta Coraglia , who might be interested in adding a word)

Matteo Capucci (he/him) (Oct 29 2022 at 11:55):

(funnily enough, I started this I was thinking about lax functors in Span bc of a conversation with @Jade Master and you posted about it at the same time!)