Thanks for doing this!

]]>Additional context Re: 31.

I have received the entire volume of *Catégories, Algèbres, Esquisses et Néo-Esquisses* from Pierre Ageron. The volume is ~141 pages and contains a number of interesting abstracts in French and English. I intend to scan the entire volume when time allows.

Uploaded PDF scan of the following reference:

- John Stell,
*Modelling Term Rewriting Systems by Sesqui-Categories*, Proc. Catégories, Algèbres, Esquisses et Néo-Esquisses (1994). [(pdf)]

Sounds good. Looks like it’s high time to save this book from being lost to history.

]]>I received the following reply from Pierre:

]]>Dear Bryce,

You take me back many years. But yes, I think there are still a few copies in my office at the university of Caen, and I’ll be happy to send you one of them! However I am in Morocco until November, 5, so you have to wait a little. But I won’t forget, immediately after coming back.

Also, scanning the whole volume is certainly a good idea.

I have sent the email, including a request to scan and upload the volume.

]]>Great. Maybe ask him for permission to scan and upload the whole book. Looks like this would be worthwhile, also for the contributing authors.

]]>I will send him an email and see what happens.

]]>On the very bottom of the webpage here, the page’s author (apprently Pierre Ageron) says that hard copies of the book collection *Catégories, algèbres, esquisses et néo-esquisses* (1994) which he edited may be obtained on request.

That’s the best I could find. The usual pirate sites do not seem to have electronic copies of this book.

]]>I was tidying up the references but couldn’t find any link to the published version of:

- John Stell,
*Modelling Term Rewriting Systems by Sesqui-Categories*, Proc. Categories, Algebres, Esquisses et Neo-Esquisses (1994).

Does anyone know where to find it?

]]>Added a cross-reference to funny tensor product.

]]>I’ve rearranged the Definition section at sesquicategory, to make it clear that there are essentially two definitions and not four.

]]>That’s right, you did; everything’s OK.

]]>but “sometimes regarding” is weaker than “usually think”.

Yes, and I agreed in #8 that #6 was overstated.

]]>Possibly I’ve gotten looser with terminology over time

Possibly. But I realised that the principle was fresh in my mind when you wrote the surprising comment #6 because you had (quite correctly) chastised me with it barely a week earlier here.

I also don’t feel a contradiction between wanting terminology for 2-categories to reduce to existing terminology for locally discrete ones, but sometimes regarding 2-categories as 1-categories with extra stuff.

I agree, but “sometimes regarding” is weaker than “usually think”.

]]>Hmm, interesting. Possibly I’ve gotten looser with terminology over time, although I still do agree with that principle in general. But I also don’t feel a contradiction between wanting terminology for 2-categories to reduce to existing terminology for locally discrete ones, but sometimes regarding 2-categories as 1-categories with extra stuff. In fact, of course, it often happens that “an X with extra stuff” is a generalization of an X, if we can take the extra stuff to be “trivial” in some canonical way. Though I can see that there is some interesting interplay.

]]>OK, you state the principle at n-prefix (michaelshulman). I’ll quote the entire second paragraph:

If X has a meaning for 1-categories, then if X is used without a prefix for 2-categories it should include the existing notion for 1-categories as a special case (when 1-categories are considered as homwise-discrete 2-categories). If we consider a notion related to X but which is not a “conservative categorification” in this sense, we will call it 2-X; cf. subcategory (nlab). For instance, we say regular 2-category since a 1-category is regular as a 2-category iff it is regular as a 1-category, but 2-exact 2-category since an exact 1-category is almost never 2-exact as a 2-category.

For the cited discussion at subcategory, it’s probably better to read a forum comment by you. While functions have images, functors have both $1$-images and $2$-images; as you explain, we know which is which by using the convention that the $1$-image of a functor between discrete categories is the same as the image of the corresponding function between sets. You then apply this to the term ‘subcategory’. (So you don’t limit this principle only to your exposition of $2$-topos theory.)

Also explained at that comment (which I had forgotten) is that it’s important to write $\mathbf{B}G$ when interpreting $G$ as a category, in part because the numbering doesn’t correspond in this case. (And indeed, a group is not a category with extra property but a category with extra structure, since it’s really a *pointed* category with extra property, and $\mathbf{B}G$ is simply its underlying category, an observation also due to you. So in both cases, we get the rule that terms are preserved when interpreting an A as a B with extra property but not when interpreting an A as a B with extra structure.)

It is reading all that, as well as other applications of the same principle, that made me unprepared for #6.

]]>Your first paragraph is again quite surprising to me! Maybe I’m just very mixed up about what you used to say; in the extreme case, maybe I’m mixing you up with somebody else (but I’m sure that it was you, really).

For instance, a span between two groupoids does not reduce to a span between two sets when the groupoids are discrete.

True, but this is because ‘span’ is an extremely general term that makes sense in any $\infty$-category, here applied to $Grpd$ and $Set$. You’re right that it’s an exception to the rule that I stated, however. (So is the profunctor example, although I never heard you apply this —and never meant to extend it myself— to general enriched categories.)

What are you referring to?

It’s spread out over naming discussions that are organised by the thing being named, not the principle at play, but I’ll see if I can find something definitive.

]]>We abandon the historical terms ‘2-functor’ and ‘bifunctor’ for ‘functor’, since we recognise a functor between 1-categories as a special case of a functor between 2-categories

That’s not what I would say. I would say that we abandon those terms because a functor between 2-categories, as we mean it, is the *most appropriate* notion of morphism between 2-categories, *analogously* to functors between 1-categories. It happens that when we regard 1-categories as particular 2-categories, functors of 2-categories reduce to functors of 1-categories, and that’s certainly a good thing that avoids confusion, but it’s not a prerequisite to usage of the unqualified “functor.”

There are other situations in which we use unqualified words in a generalized context that don’t reduce to the previous notions when specialized. For instance, a span between two groupoids does not reduce to a span between two sets when the groupoids are discrete. A profunctor between 2-categories does not reduce to a profunctor between 1-categories when the 2-categories are discrete. And even with ordinary functors, for an arbitrary enriching category V, a functor between V-categories need not reduce to an ordinary functor when the V-categories are “discrete.”

especially given that you were the one who clarified for me how the names ought to work.

What are you referring to? I want to know if I’m being inconsistent… (-:

]]>@ Zoran #12:

Agreed, let’s not rename it.

@ Mike #10:

It’s when things are *not* renamed that we see that people think of things as being (not merely analogous but) the same, in the sense of being special cases (possibly one of the other, possibly both of some more general situation). We abandon the historical terms ‘$2$-functor’ and ‘bifunctor’ for ‘functor’, since we recognise a functor between $1$-categories as a special case of a functor between $2$-categories, so the latter is the same thing as the former, deserving of the same name. In contrast, things which are merely analogous get new names.

I agree that both perspectives are useful. But I have a pretty clear idea in my mind which one is primary, and I’m surprised that you have a different idea, especially given that you were the one who clarified for me how the names ought to work.

]]>12 :) That was instructive, I mean all the comments…

]]>I don’t think anyone was proposing to rename it, just having fun complaining about the existing name. (-:

]]>I think sesquicategory is somewhat standard term in Australian category theory school. It has enough of general feeling that it is remembrable. I mean it reminds to roughly the right thing. So why should we play Bourbaki and correct the Australians (as if most of the terminology in mathematics were entirely logically ordered) ?

]]>I don’t really see how the issue of renaming or not is relevant to the question of how 2-categories are related to 1-categories?

]]>