Re: Straw man terminology

Re: Straw man terminology

[Urs Schreiber]

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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Re: Re: Straw man terminology

[zoran skoda]

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."


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Re: Straw man terminology

[jim stasheff]

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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Re: Straw man terminology

[jim stasheff]

> 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




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