Re: Straw man terminology

[jim stasheff <jds@math.upenn.edu>]

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
>

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