Re: Straw man terminology

["Joyal, André" <joyal.andre@uqam.ca>]

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




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