Re: Straw man terminology

Re: Straw man terminology

[Urs Schreiber]

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/ ]

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




[For admin and other information see: http://www.mta.ca/~cat-dist/ ]

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
>

[For admin and other information see: http://www.mta.ca/~cat-dist/ ]

Re: Straw man terminology

[Joyal, André]

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/ ]

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


[For admin and other information see: http://www.mta.ca/~cat-dist/ ]

Re: Straw man terminology

[Joyal, André]

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/ ]

Re: Straw man terminology

[Urs Schreiber]

> 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/ ]

Straw man terminology

[Peter May]

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/ ]