As a small point, I'm quite sure Boardman and Vogt never used the term ``weak Kan complex'' that Andre rightly opposes: they referred to the ``restricted Kan condition'' (see SLN 347, p. 102). I have no idea who is guilty of ``weak'' in this context, but I think it was never in common use. In fact, Boardman and Vogt were right away sensitive to size issues, referring to ``simplicial classes'' rather than simplicial sets satisfying the inner horn condition. They did not think of them as special kinds of Kan complexes, which arguably they are not since Kan complexes are generally understood to be simplicial sets. Certainly that was how Kan complexes were understood when Boardman and Vogt were writing (published 1973, mostly written earlier). They already knew then that their notion gave them, in their words, ``good substitutes for categories''. I also like Andre's term ``quasi-categories'' and prefer it to the infinity alternatives, for the reasons he gives. Like Kan complex, it has a fixed and evocative definite meaning, unlikely to be confused with anything else. Now if only he would publish ... Responding to another part of this (endless) string, I also regret how words with a nice ring to them get so modified that they no longer have a clear precise meaning attached to them. Everyone can guess one word I'm thinking of (my mother was an opera singer). Peter May [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
I also like Andre's term ``quasi-categories'' and prefer it to the infinity alternatives, for the reasons he gives. Like Kan complex, it has a fixed and evocative definite meaning, unlikely to be confused with anything else.
The term "(infinity,1)-category" is not so much meant as an alternative for "quasi-category", but as a intentionally less specific term that subsumes concepts that are different from, but equivalent to, quasi-categories. Such as Kan-complex-enriched categories or complete Segal spaces, or algebraic quasi-categories, or categories with weak equivalences, or... When doing abstract higher category theory it is useful to be able to speak, for instance, of the (infinity,1)-category of all small infinity-groupoids and its abstract properties, without having to specifically fix a concrete model in terms of which this entity may be brought to paper. [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
Dear Urs, You wrote:
The term "(infinity,1)-category" is not so much meant as an alternative for "quasi-category", but as a intentionally less specific term that subsumes concepts that are different from, but equivalent to, quasi-categories. Such as Kan-complex-enriched categories or complete Segal spaces, or algebraic quasi-categories, or categories with weak equivalences, or...
When doing abstract higher category theory it is useful to be able to speak, for instance, of the (infinity,1)-category of all small infinity-groupoids and its abstract properties, without having to specifically fix a concrete model in terms of which this entity may be brought to paper.
I agree that the terminology (infinity,1)-terminology can be useful. Can I point out that Lurie is calling a quasi-category an infinity-category? There is a clash of terminology. Best, André [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
Dear Andre,
I agree that the terminology (infinity,1)-terminology can be useful.
Okay.
Can I point out that Lurie is calling a quasi-category an infinity-category?
Okay, let's look at Lurie's use of terminology then. Notice that just a little later in On the classification of TFTs http://arxiv.org/PS_cache/arxiv/pdf/0905/0905.0465v1.pdf#page=31 In the remark 2.1.26 he speaks of "the various models of the theory of (oo,1)-categories" referring to Julie Bergner's article which shows that quasi-categories, sSet-categories, Segal categories and complete Segal spaces give four equivalent such models. Then still a bit later in (oo,2)-Categories and the Goodwillie calculus http://www.math.harvard.edu/~lurie/papers/GoodwillieI.pdf he uses terminology exactly as I have been suggesting in my previous messages: starting in the third sentence: "Let us use the term (oo,n)-category to indicate a higher category in which all k-morphisms are assumed to be invertible for k> n. [...] The theory of (oo,1)-categories is also quite well understood, though in this case there is a variety of possible approaches. [...] These are known as quasicategories in the literature; we will follow the terminology of [HTT] and refer to them simply as oo-categories." So, for what it's worth, Lurie adopts the convention that I was talking about, it seems to me: to say (oo,n)-category for the general concept and use other terms for concrete models. He just happens to have the extra convention that "oo-category" (without the ",1") is his term for the model that you called quasi-category. Maybe in this context it is noteworthy that in this last article alone, there is presented literally a dozen of different and equivalent models for (oo,2)-categories. Best, Urs [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
Dear Urs, I agree that Lurie is using the infinity-n-category terminology. I am not questioning that. I am observing that he is calling a quasi-category an infinity-category. In his terminology, an infinity-category is a special kind of infinity-one-category. I believe that the name infinity-category should apply to all "infinity" categories, inculding the infinity-one-category. No? Best, Andre -------- Message d'origine-------- De: Urs Schreiber [mailto:urs.schreiber@googlemail.com] Date: mer. 26/05/2010 13:59 À: Joyal, André Cc: categories@mta.ca Objet : Re: RE : categories: Re: Straw man terminology Dear Andre,
I agree that the terminology (infinity,1)-terminology can be useful.
Okay.
Can I point out that Lurie is calling a quasi-category an infinity-category?
Okay, let's look at Lurie's use of terminology then. Notice that just a little later in On the classification of TFTs http://arxiv.org/PS_cache/arxiv/pdf/0905/0905.0465v1.pdf#page=31 In the remark 2.1.26 he speaks of "the various models of the theory of (oo,1)-categories" referring to Julie Bergner's article which shows that quasi-categories, sSet-categories, Segal categories and complete Segal spaces give four equivalent such models. Then still a bit later in (oo,2)-Categories and the Goodwillie calculus http://www.math.harvard.edu/~lurie/papers/GoodwillieI.pdf he uses terminology exactly as I have been suggesting in my previous messages: starting in the third sentence: "Let us use the term (oo,n)-category to indicate a higher category in which all k-morphisms are assumed to be invertible for k> n. [...] The theory of (oo,1)-categories is also quite well understood, though in this case there is a variety of possible approaches. [...] These are known as quasicategories in the literature; we will follow the terminology of [HTT] and refer to them simply as oo-categories." So, for what it's worth, Lurie adopts the convention that I was talking about, it seems to me: to say (oo,n)-category for the general concept and use other terms for concrete models. He just happens to have the extra convention that "oo-category" (without the ",1") is his term for the model that you called quasi-category. Maybe in this context it is noteworthy that in this last article alone, there is presented literally a dozen of different and equivalent models for (oo,2)-categories. Best, Urs [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
Dear Andre,
I agree that Lurie is using the infinity-n-category terminology. I am not questioning that. I am observing that he is calling a quasi-category an infinity-category. In his terminology, an infinity-category is a special kind of infinity-one-category.
I believe that the name infinity-category should apply to all "infinity" categories, inculding the infinity-one-category. No?
Yes, I entirely agree with that. In our discussion I did not promote Jacob Lurie's use of "oo-category" for "quasi-category", What I did and do promote is to use * "oo-groupoid" and "(oo,0)-category" as the term for the abstract concept which is equivalently realized by Kan complexes, simplicial groupoids, topological spaces, etc. and has special strict models by strict omega-groupoids, oo-fold groupoids etc. * "(oo,1)-category" as the term for the abstract concept which is equivalently realized by quasi-categories, Kan-complex enriched categories, complete Segal spaces, Segal categories, categories with weak equivalences, (oo,1)-theta spaces, etc. * "(oo,n)-category" as the term for the abstract concept which is equivalently realized by n-fold complete Segal spaces, (oo,n)-Theta spaces etc. Best, Urs [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
Urs, Andre did not say that Lurie's usage of the term (infty,1)-category distinguishes from yours but that his usage of the term infinity-category (note the difference) is specific to mean quasicategory: his introduction says: "We begin with what we feel is the most intuitive approach to the subject, based on topological categories. This approach is easy to understand, but difficult to work with when one wishes to perform even simple categorical constructions. As a remedy, we will introduce the more suitable formalism of ∞-categories (called weak Kan complexes in [10] and quasi-categories in [43]), which provides a more convenient setting for adaptations of sophisticated category-theoretic ideas. Our goal in §1.1.1 is to introduce both approaches and to explain why they are equivalent to one another." [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
Urs Schreiber wrote:
Dear Andre,
I agree that the terminology (infinity,1)-terminology can be useful.
Okay.
Can I point out that Lurie is calling a quasi-category an infinity-category?
Okay, let's look at Lurie's use of terminology then. Notice that just a little later in
On the classification of TFTs http://arxiv.org/PS_cache/arxiv/pdf/0905/0905.0465v1.pdf#page=31
In the remark 2.1.26 he speaks of
"the various models of the theory of (oo,1)-categories"
referring to Julie Bergner's article which shows that quasi-categories, sSet-categories, Segal categories and complete Segal spaces give four equivalent such models.
Then still a bit later in
(oo,2)-Categories and the Goodwillie calculus http://www.math.harvard.edu/~lurie/papers/GoodwillieI.pdf
he uses terminology exactly as I have been suggesting in my previous messages:
starting in the third sentence:
"Let us use the term (oo,n)-category to indicate a higher category in which all k-morphisms are assumed to be invertible for k> n.
THIS IS MUCH BETTER - A DEFINITION - NOT AN EXAMPLE (AKA MODEL) OR APPROACH
[...]
The theory of (oo,1)-categories is also quite well understood, though in this case there is a variety of possible approaches. [...] These are known as quasicategories in the literature; we will follow the terminology of [HTT] and refer to them simply as oo-categories."
So, for what it's worth, Lurie adopts the convention that I was talking about, it seems to me: to say (oo,n)-category for the general concept and use other terms for concrete models. He just happens to have the extra convention that "oo-category" (without the ",1") is his term for the model that you called quasi-category.
Maybe in this context it is noteworthy that in this last article alone, there is presented literally a dozen of different and equivalent models for (oo,2)-categories.
Best, Urs
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
Maybe in this context it is noteworthy that in this last article alone, there is presented literally a dozen of different and equivalent models for (oo,2)-categories.
Best, Urs
Is not any (oo,2)-category a model for (oo,2)-categories. ? or has model been defined? jim [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
participants (5)
-
jim stasheff -
Joyal, André -
Peter May -
Urs Schreiber -
zoran skoda