added pointer to:

- Saunders MacLane, Section VII.1 of:
*Categories for the Working Mathematician*, Graduate texts in mathematics, Springer (1971, second ed. 1997) [doi:10.1007/978-1-4757-4721-8]

added publication data for this item:

- Ross Street,
*Monoidal categories in, and linking, geometry and algebra*, Bull. Belg. Math. Soc. Simon Stevin**19**5 (2012) 769-820 [arXiv:1201.2991, doi:10.36045/bbms/1354031551]

Added a subsection on the rig of scalars $\mathcal{C}[I,I]$.

]]>Added a very explicit definition of a strict monoidal category.

]]>added pointer to:

- Francis Borceux, Section 6.1 of:
*Handbook of Categorical Algebra*Vol. 2:*Categories and Structures*$[$doi:10.1017/CBO9780511525865$]$, Encyclopedia of Mathematics and its Applications**50**, Cambridge University Press (1994)

Nowhere in what I wrote was I suggesting that André had not put in a lot of hard work in developing the theory, and I was agreeing with you, Urs, that there were some in the 1980s and 90s who were still trying to do the inductive process. You are remembering 2007, I am remembering 15 to 20 years earlier, so there is no inconsistency between what you are saying and what I wrote. What is disappointing is that after that 24 year period, André still felt he had to justify that higher category theory existed, especially after the Minnesota conference of 2004, where a large number of people had met to discuss the state of the theory, and there were many talks about the various approaches. It was not 100% certain at that time which of the many versions were going to survive the race, nor if they were all equivalent.

]]>Joyal did not just have an “approach” (nor just a “pursuit” “towards” a goal) as many had. He had seen and then worked out the *theory*, essentially what is now called $(\infty,1)$-category theory.

It wasn’t as widely known as it should have. I remember him opening a talk on quasi-categories in 2007 at the Fields Institute with the words “In this talk I want to convince you that higher category theory exists.” An innocent sounding statement, but somewhat damning to a room full of people supposedly all working on higher categories.

]]>I was intrigued by the above and for the historical record, I looked back at my letters to Grothendieck from 1983. I pointed out there that Kan complexes were a good model for infinity groupoids and that there were several good candidates for infinity categories. (I do not seem to have explicitly mentioned weak Kan complexes / quasi-categories, but about that time Cordier and I started working on both fibrant SSet-categories and on quasicategories. We did not seem to appreciated the importance of the (infty,1)-idea however.) We had a sketch of the theory of weak Kan complexes to include the analogues of limits and colimits, ends and coends, but never wrote that up, as Jean-Marc felt that the SSet-categories would be more acceptable to both homotopy theorists and category theorists. Our write up of the ends and coends stuff in that latter setting took a lot longer that we had expected due to health issues and excessive teaching loads. We put that SSet-category view forward in the paper *Homotopy Coherent Category Theory*, Trans. Amer. Math. Soc. 349 (1997) 1-54, but that paper, which had been essentially finished several years earlier, was initally rejected by another journal on the basis that ‘homotopy theorists did not need such a categorical way of looking at homotopy coherence’, or some such wording. It received a good report from the referee for TAMS however.

There were thus people who were looking at what eventually became quasi-category theory at about the same time as Joyal’s lovely approach was being developed, and with the Bangor approach to strict omega categories etc. the idea of doing all dimensions at once was pushed quite firmly. It should be also mentioned that, of course, Ross Street, Dominic Verity , Michael Batanin, and others in Sydney were putting forward a parallel vision at that time; (Edit) see for instance here for the Australian view in 2004. In the category theory conferences of the time there were talks which were more top-down, doing all dimensions at one by concentrating on the coherence questions, as well as those which were approaching the definition from the bottom-up.

I also remember, I think it was Maxim Kontsevich. giving a talk (probably 1992), which used A_infty categories and this was clearly linked in his mind and for many of the category theorists in the audience, to that of ’doing infinity category theory in all dimensions’ albeit for him it was based on a more algebraic dg-cat like structure.

I think the idea that one could do all dimensions at once was therefore well represented in talks during the 1980s and 90s, but some people preferred to be cautious and to try to understand the low dimensional weak categories (bicategories, tricategories, etc) which were combinatorially very tricky, and were therefore avoided by some (I would say that if one uses homotopy coherence and in particular higher operads (which we missed completely in our approach in the 1980s) , the combinatorics becomes more manageable, but can be hard work!)

By the way, the Grothendieck correspondence is due to be published some time next year I think.

]]>Interesting that old quote. Yes, that’s the point.

I have a vague memory of digging out, in a similar conversation years ago, quotes that explicitly make the error mentioned in #87. I am pretty sure where to look for them, but would have to search again. Maybe it’s not worthwhile.

I suppose if A. Joyal had been more into publishing his insights, the drama could have been shortcut by about two decades.

I felt this was all well-understood by now, but it wouldn’t hurt to have an $n$Lab entry on it. I might try to start something later on the weekend.

]]>We’re wondering about such matters in a conversation from 2012 beginning here:

When I was learning about the higher dimensional program from John Baez all those years ago, I took it that n-categories were to be the basic entity. Then n-groupoids were to be thought of a special case of n-categories, particularly useful because homotopy theorists had worked out very powerful theories to deal with the former. The trick was to extend what they’d done, but to an environment with no inverses.

Do you think that what you’re finding here about the difficulty of directed homotopy type theory suggests that in some sense n-groupoids shouldn’t be thought of as a variant of something more basic?

I wonder if we have the points made in #85 and #87 on the nLab anywhere.

]]>The drama of the eventual lifting of the impasse of old-school higher category is also reflected in Voevodsky’s “breakthrough” through his “greatest roadblock” by realizing that (my slight paraphrase): “categories are not higher sets but higher posets; the actual higher sets are groupoids” (here).

This is referring to old-school higher category theory folklore being fond of the fact that “groupoids are just certain categories”. While true, it mislead people into not recognizing that homotopy theory is the foundation of higher category theory, not the other way around. Only when this was turned around and put on its feet did higher category theory start to run.

]]>Regarding serious attention: This began with the use of $(\infty,n)$-categories by Lurie in the classification of TQFTs and the article on Goodwillie calculus.

I remember the revelation when opening this, having been brought up with the old-school ideas forever “towards an $n$-category of cobordisms” (tac:18-10). Suddenly there was a definition that worked.

]]>Certainly by Lectures on n-Categories and Cohomology, but I think it was much earlier.

]]>On the history lesson, when did the idea of $n$-categories get refined into the idea of $(n,m)$-categories? When I was first casually reading about higher categories, it took a long time before I really encountered the latter being given any serious attention, but that could very well just be an artifact of what I was reading.

]]>Thank you for that chunk of wisdom! I was definitely on track to falling into that way of thinking. In response to #80, I wonder if certain combinatorial species (those closed under product, so not trees, but forests, for example) are monoidal category objects in the monoidal 2-category of combinatorial species, with product given by the “star product” of combinatorial species. I’ll have to think about it a bit more in detail.

]]>Re #85, #86:

It was a wide-spread mistake of old-school higher category theorists to think that to obtain a good theory of $n$-categories one needs to first define $(n+1)$-categories, because, so the logic went, the collection of all $n$-categories is bound to form an $(n+1)$-category which is needed to provide the ambient context for dealing with $n$-categories, notably to discuss their coherence laws.

This perceived infinite regression was arguably one of the reasons why the field of higher category theory was, by and large, stuck and fairly empty, before the revolution.

The error in the above thinking was to miss the fact that coherences only ever take value in *invertible* higher morphisms, so that a decent theory of $n$-categories is available already inside the $(\infty,1)$-category of $n$-categories.

This insight breaks the impasse: First define $(\infty,1)$-categories all at once, and then find the tower of $(\infty,n)$-categories on that homotopy-theoretic foundation.

The microcosm principle is an archetypical example of the need for this perspective: The coherences (unitor, associator, triangle, pentagon) on a monoidal category are all invertible, hence can be made sense of already inside the $(2,1)$-category of categories, functors, and natural *iso*-morphisms between them.

sufficiency : property :: necessity : structure

By that I mean, there exist a non-monoidal category C and a sense in which one can define a monoid object M in C, by specifying a functor \otimes, associator, unitors, etc for M. In this sense, having C monoidal guarantees that this _can_ be done. So we are referring to "properties" of this object M.

On the other hand, we _need_ C to be a monoidal category in order to be able to define monoid objects whose monoidal structure is canonical. Here M is equipped with structure inherited from C.

If what I wrote makes sense, I wonder if it would be relevant to talk about this nuance between necessity and requirement in the stuff, structure, property page, say with a hyperlink on the word "ability" or "necessity" on this article.

I wonder if the first notion (sufficiency/property) is not so good from the perspective of category theory. In its defense, there are certainly categories where only certain objects have something special about them. For example, elliptic curves, among curves, have an addition law. But there is no biproduct for algebraic curves is there? Nevertheless, the notion of addition on elliptic curves isn't completely arbitrary; we still "can" define group objects here in a meaningful sense.

Thank you both for your replies. This has been really helpful for me (as is the entire website). Also I hope to understand the last remark (#85) by Urs in the not-so-distant future. In the meantime, I am content with the following non-circular recipe: (1) Cat is a category; (2) Cat has products (thinking of pairs of sets as (x,y)={{x},{x,y}} to prove existence, but rarely ever again thinking of pairs like this); (3) monoidal categories are defined in terms of Cat and its finite Cartesian product operation; (4) categories with finite products are monoidal categories with respect to these; (5) define bicategory again using (Cat,\times); (6) in addition to being monoidal under the Cartesian product, Cat is a strict bicategory when using natural transformations for its 2-morphisms; (7) the 2-category Cat is a monoidal 2-category with respect to Cartesian products. Out of curiosity, do people not like this way of thinking about things because of step (2)? What is the reason for preferring a different recipe--using the "(2,1)-category core", as you say? Is there an nlab page that has the answer to this? ]]>

Regarding sufficiency or necessity: In Lurie’s actual realization of the microcosm principle (here) it is both: algebras over an $\infty$-operad $\mathcal{O}$ are defined internal to $\mathcal{O}$-monoidal $\infty$-categories.

Incidentally, the $(\infty,1)$-categorical formulation resolves what in the original formulation of the principle looks like an infinite regression: To define monoidal categories we don’t actually need the monoidal 2-category $Cat$ but just its $(2,1)$-category core (since the coherence 2-morphisms that it is to supply are all invertible).

]]>I said “secretly” because the point of this discussion is (as far as I see) that when one looks at the standard definition of monoidal categories, it is typically not made explicit that an ambient 2-categorical monoidal structure is being used, this happens tacitly or secretly in the background. We are adding a remark highlighting this pedantic subtlety.

]]>I guess the “ability” is implicitly about “sufficiency”, so that with the later “needs” both are covered, but yes there should be better wording. Why “secretly”?

But then we don’t even say this at microcosm principle, which just mentions the sufficiency part:

In higher algebra/higher category theory one can define (generalized) algebraic structures internal to categories which themselves are equipped with certain algebraic structure, in fact with the same kind of algebraic structure. In (Baez-Dolan 97) this has been called the microcosm principle.

So what in fact is the case? Is it both necessary and sufficient that higher structure be in place?

]]>Maybe “necessity” instead of or in addition to “ability”: I think the point being made is that in defining monoidal categories one (secretly, maybe) *needs* to appeal to monoidal 2-category structure.

I’ve fiddled with the wording a little

]]>The ability to define pseudomonoids in any monoidal 2-category is an example of the so-called microcosm principle, where the definition of (higher) algebraic structures uses and requires analogous algebraic structure present on the ambient higher category.

Yes! But if we have examples of (pseudo-)monoids in other monoidal 2-categories, then this would be a good point to mention them/link to them.

]]>