Makkai's suggestion

Makkai's suggestion

[David Leduc]

In "Towards a categorical foundation of mathematics", Makkai wrote that:

"This suggests that, possibly, the right approach to the definition of weak
n-category is to aim at formulating all coherence conditions at once,
regardless the fact that this might give a very "theoretical" definition. It
would then be a separate, and still very important, project to find a
(hopefully) finite and concise set of coherence conditions that would be
enough to imply all coherence conditions."

It was 15 years ago! Did this approach give something?


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Re: Makkai's suggestion

[John Baez]

David Leduc wrote:

In "Towards a categorical foundation of mathematics", Makkai wrote that:
>
> "This suggests that, possibly, the right approach to the definition of weak
> n-category is to aim at formulating all coherence conditions at once,
> regardless the fact that this might give a very "theoretical" definition.
> It
> would then be a separate, and still very important, project to find a
> (hopefully) finite and concise set of coherence conditions that would be
> enough to imply all coherence conditions."
>
> It was 15 years ago! Did this approach give something?
>

Yes: most or all of the successful approaches to infinity-categories follow
a philosophy of this general sort.  You can find a lot of references here:

http://ncatlab.org/johnbaez/show/Towards+Higher+Categories

Best,
jb


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Re: Makkai's suggestion

[David Leduc]

John Baez <baez@math.ucr.edu> wrote:
> Yes: most or all of the successful approaches to infinity-categories follow
> a philosophy of this general sort.  You can find a lot of references here:
>
> http://ncatlab.org/johnbaez/show/Towards+Higher+Categories

Thank you for the reference. But I don't know where to start. Each
author seems to be working with his own favorite special case of
infinity-categories: (inf,1)-categories, opetopic and multitopic
categories, simple omega-categories, theta-categories,
protocategories... and so on.

Is there a definitive definition of omega-categories somewhere in the
literature or is it still unknown? Can it be stated in elementary
terms (i mean in terms of object, arrows, ... without references to
simplicial sets or topology) ?

In the definition of a bicategory, one could replace the coherence
axioms by the statement that all diagrams built from the canonical
ismorphisms commute. Can it be generalized to n=3, ... , omega.


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Re: Makkai's suggestion

[John Baez]

David wrote:

> http://ncatlab.org/johnbaez/show/Towards+Higher+Categories
>
> Thank you for the reference. But I don't know where to start.


Start by reading the above book together with Cheng and Lauda's "Higher
categories: an illustrated guidebook":

http://www.cheng.staff.shef.ac.uk/guidebook/

and Leinster's "A Survey of Definitions of n-Category":

http://arxiv.org/abs/math/0107188

Then try Lurie's "Higher Topos Theory":

http://arxiv.org/abs/math/0608040

They're all free online!

Expect to spend a decade on this stuff.  Or, wait two decades for people to
polish it up, and then spend half a decade learning the basics and half a
decade learning what people have done in the next two decades.  That may be
more efficient.

Is there a definitive definition of omega-categories somewhere in the
> literature or is it still unknown?


There are *several* definitions that are almost surely "right" and likely to
be studied for many years hence.  There is no particular reason to expect
that one definition will be best for all applications - but there's a lot of
reason to expect that all the "right" definitions will be shown to be
equivalent (in a rather subtle sense).


> Can it be stated in elementary terms (I mean in terms of object, arrows,
> ... without references to simplicial sets or topology) ?
>

You should learn to love simplicial sets - they're way too important to
avoid!

If for some reason you're allergic to simplicial sets, you might like
Batanin's definition of omega-categories.   But then you need to like
operads.  You could state it without operads, but then it becomes quite
long.

The book by Cheng and Lauda takes various definitions and makes them less
scary by illustrating how they work with lots of pictures.


> In the definition of a bicategory, one could replace the coherence
> axioms by the statement that all diagrams built from the canonical
> ismorphisms commute. Can it be generalized to n=3, ... , omega.
>

You could say that's the basic idea behind Batanin's definition.

Best,
jb


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Re: Makkai's suggestion

[Greg Meredith]

Dear John,

i have always taken this situation

There are *several* definitions that are almost surely "right" and likely
> to

be studied for many years hence.  There is no particular reason to expect

that one definition will be best for all applications - but there's a lot
> of

reason to expect that all the "right" definitions will be shown to be

equivalent (in a rather subtle sense).


to be a pretty clear critique of category theory's basic formulation. This
may be too idealistic, but i've always felt that an ideally robust
formulation would admit a meta-theory that makes n-categories "just fall
out". The fact that they are so hard to formulate suggests that the basic
design of the original presentation misses something crucial.

i confess that taken together with the fact that it is exceptionally hard to
get categorical composition to line up with parallel composition (in the
sense of concurrent computation) in a manner that respects Curry-Howard, has
really made me evaluate category theory as still very much a work in
progress.

Personally, i have wondered if there is a presentation that takes monad as
the fundamental building block. i think this might not be too much of a
stretch goal, actually, as monad as polymorphic comprehension is now
well-established.

Best wishes,

--greg

On Mon, Aug 30, 2010 at 9:34 PM, John Baez <baez@math.ucr.edu> wrote:

> David wrote:
>
>> http://ncatlab.org/johnbaez/show/Towards+Higher+Categories
>>
>> Thank you for the reference. But I don't know where to start.
>
>
> Start by reading the above book together with Cheng and Lauda's "Higher
> categories: an illustrated guidebook":
>
> http://www.cheng.staff.shef.ac.uk/guidebook/
>
> and Leinster's "A Survey of Definitions of n-Category":
>
> http://arxiv.org/abs/math/0107188
>
> Then try Lurie's "Higher Topos Theory":
>
> http://arxiv.org/abs/math/0608040
>
> They're all free online!
>
> Expect to spend a decade on this stuff.  Or, wait two decades for people to
> polish it up, and then spend half a decade learning the basics and half a
> decade learning what people have done in the next two decades.  That may be
> more efficient.
>
> Is there a definitive definition of omega-categories somewhere in the
>> literature or is it still unknown?
>
>
> There are *several* definitions that are almost surely "right" and likely
> to
> be studied for many years hence.  There is no particular reason to expect
> that one definition will be best for all applications - but there's a lot
> of
> reason to expect that all the "right" definitions will be shown to be
> equivalent (in a rather subtle sense).
>
>
>> Can it be stated in elementary terms (I mean in terms of object, arrows,
>> ... without references to simplicial sets or topology) ?
>>
>
> You should learn to love simplicial sets - they're way too important to
> avoid!
>
> If for some reason you're allergic to simplicial sets, you might like
> Batanin's definition of omega-categories.   But then you need to like
> operads.  You could state it without operads, but then it becomes quite
> long.
>
> The book by Cheng and Lauda takes various definitions and makes them less
> scary by illustrating how they work with lots of pictures.
>
>
>> In the definition of a bicategory, one could replace the coherence
>> axioms by the statement that all diagrams built from the canonical
>> ismorphisms commute. Can it be generalized to n=3, ... , omega.
>>
>
> You could say that's the basic idea behind Batanin's definition.
>
> Best,
> jb
>
>




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Re: Makkai's suggestion

[Greg Meredith]

Dear David,

[This reply never made it to the list and i got a note from Bob
Rosebrugh about text-only messages and no attachments. i've a feeling
that the gmail client did something strange with my reply -- since i
thought everything i typed was predominantly ASCII. The following
version of the original reply should be just plain ASCII.]

>> the fact that it is exceptionally hard to
>> get categorical composition to line up with parallel composition (in the
>> sense of concurrent computation) in a manner that respects Curry-Howard,
>
>I am not aware of this fact. What do you mean? Please explain or give
>references.

Arguably, the two best approaches are Interaction Categories
(Abramsky, et al) and Bi-graphs (Milner, et al). The crucial concern
is to align categorical composition with parallel composition in a
manner that respects Curry-Howard. Thus, processes must be interpreted
as morphisms. The central complication is how to do this in a setting
supporting a natural interpretation of mobility, i.e. the dynamic
reconfiguration of communication topology. As an example, consider the
pi-calculus process,

*(u?(v).(new w)v!(w)) | (new x)u!(x).x?(y).P

Find below one of it's reductions. The process represents a simplified
model of a little mini-protocol we use on the internet every day: on
some well known URL (the channel u above) ask for a unique (and
potentially private) URL (the channel w above) on which to do further
communication.

A categorical interpretation (meeting the desiderata above) needs to
interpret -- for example --

1) *(u?(v).(new w)v!(w)) as a morphism, say [| *(u?(v).(new w)v!(w)) |]

2) (new x)u!(x).x?(y).P as a morphism, say [| (new x)u!(x).x?(y).P |] and

3) *(u?(v).(new w)v!(w)) | (new x)u!(x).x?(y).P as a morphism, say [|
*(u?(v).(new w)v!(w)) | (new x)u!(x).x?(y).P |]

such that

[| *(u?(v).(new w)v!(w)) | (new x)u!(x).x?(y).P |]
=
[| *(u?(v).(new w)v!(w)) |] o [| (new x)u!(x).x?(y).P |]

Sangiorgi, Fiore and others have given fully abstract interpretations
meeting these criteria, by the letter of the law, so to speak. Imho,
one would never use any of these semantic accounts to do calculations
of properties of even these simple protocols. That's the sense in
which i mean that parallel composition does not line up well with
categorical composition. The "semantic" categorical account doesn't
help clarify things. It doesn't aid practical calculation. For
concurrent computation the process calculi still have greater utility
for both practical calculations theoretical framework supplying a
robust notion of equivalence and substitutability (i.e.,
bisimulation).

One reduction of the example process.

*(u?(v).(new w)v!(w)) | (new x)u!(x).x?(y).P
->
*(u?(v).(new w)v!(w)) | u?(v).(new w)v!(w) | (new x)u!(x).x?(y).P
=
*(u?(v).(new w)v!(w)) | (new x)(u?(v).(new w)v!(w) | u!(x).x?(y).P)
->
*(u?(v).(new w)v!(w)) | (new x)((new w)x!(w) | x?(y).P)
=
*(u?(v).(new w)v!(w)) | (new x)((new w)(x!(w) | x?(y).P))
->
*(u?(v).(new w)v!(w)) | (new x)((new w)(P{w/y}))



Best wishes,

--greg

On Sat, Sep 4, 2010 at 3:10 AM, David Leduc <david.leduc6@googlemail.com> wrote:
>
> On 9/1/10, Greg Meredith <lgreg.meredith@biosimilarity.com> wrote:
>> the fact that it is exceptionally hard to
>> get categorical composition to line up with parallel composition (in the
>> sense of concurrent computation) in a manner that respects Curry-Howard,
>
> I am not aware of this fact. What do you mean? Please explain or give
> references.



--
L.G. Meredith
Managing Partner
Biosimilarity LLC
1219 NW 83rd St
Seattle, WA 98117

+1 206.650.3740

http://biosimilarity.blogspot.com


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Makkai's suggestion

[John Baez]

Greg writes:


> Of course, after following such a path of least resistance, a journeyman
> categoryist might look at a variety of alternatives and consider their
> trade-offs. In this effort, the very quality you point out is of great
> utility in developing a genuine understanding of the design space. The
> initial presentation, this path of least resistance formulation, however,
> ought to have a precise sense in which it is *initial*, like an initial
> algebra.


Leinster's refinement of Batanin's approach defines weak infinity-categories
as algebras of an "initial globular operad with contractions".

Here "globular" means we're doing infinity-categories in the obvious way,
where given two n-morphisms f,g: x -> y we can talk about (n+1)-morphisms
from f to g.

"Algebra of a globular operad with contractions" means we can compose these
n-morphisms in all the pictorially obvious ways, and every pictorially
plausible law holds *up to a higher morphism*.

"Initial" means we're doing this in exactly the right way: for example,
there aren't any *extra* ways of composing morphisms, and we're not sticking
in *too many* of these higher morphisms.

I am sure people will eventually come up with better ways to do
infinity-category theory.  Eventually most math majors will learn it in
college (unless our current civilization collapses in less than, say, 150
years).    But the approaches we've got right now are already pretty good.
Learn 'em!

Best,
jb


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Re: Makkai's suggestion

[Ronnie Brown]

John Baez writes on 03/09/2010:
----------------------------------

Leinster's refinement of Batanin's approach defines weak infinity-categories
as algebras of an "initial globular operad with contractions".

Here "globular" means we're doing infinity-categories in the obvious way,
where given two n-morphisms f,g: x ->  y we can talk about (n+1)-morphisms
from f to g.

"Algebra of a globular operad with contractions" means we can compose these
n-morphisms in all the pictorially obvious ways, and every pictorially
plausible law holds *up to a higher morphism*.

"Initial" means we're doing this in exactly the right way: for example,
there aren't any *extra* ways of composing morphisms, and we're not sticking
in *too many* of these higher morphisms.

I am sure people will eventually come up with better ways to do
infinity-category theory.  Eventually most math majors will learn it in
college (unless our current civilization collapses in less than, say, 150
years).    But the approaches we've got right now are already pretty good.
Learn 'em!

---------------------------------------------------------
I would like to put in a plea for analogous studies of cubical
approaches to weak higher categories. I started looking for higher
groupoids in homotopy theory in the 1960s but all the time was seeking
cubical structures, as these were clearly relevant to my starting point,
the van Kampen Theorem for the fundamental group and groupoid. It was
easy, even nicely childish,  to draw pictures of a square subdivided
into little squares by lines parallel to the sides, and so to seek for
some maths which expressed the big square as the composite of the little
squares. (The further idea needed was `commutative cube', which required
extra structure.) The cubical singular complex of a space was obviously
some kind of `lax or weak infinity-fold groupoid', but I am not clear if
this idea is captured by any of the current expositions. For the notion
of higher groupoid with a van Kampen Theorem yielding  specific and
precise calculations one wanted colimits rather than `lax colimits (??)'
and so needed to take some kind of homotopy classes to get a strict
structure. Over a period of 11 years it was found with Philip Higgins
that this could be done usefully, but non trivially, for filtered spaces.

The argument of the new book on `Nonabelian algebraic topology'
(downloadable from my web page,
www.bangor.ac.uk/r.brown/nonab-a-t.html, planned to appear in 2010 with
the EMS) is that filtered spaces can be taken as a satisfactory starting
point for algebraic topology, at the border between homotopy and
homology, avoiding a direct use of singular homology and simplicial
approximation, and getting for example nonabelian results in dimension
2. The hope is that the book will allow convenient evaluation and
hopefully extension of these methods.

In all this work we noted various globular ideas but could never use
them in our context to get any new results in homotopy theory. So we
could define `algebraic inverse to subdivision' cubically, but not
globularly. We could define classifying spaces simplicially or
cubically, but there seems no evidence that this works globularly.
Monoidal closed structures worked fine cubically, and could be
translated to other contexts, using equivalences of categories. This is
done for example for strict globular omega categories in

(F.A. AL-AGL, R. BROWN and R. STEINER), `Multiple categories: the
equivalence between a globular and cubical approach', Advances in
Mathematics, 170 (2002) 71-118.

following an earlier approach for the groupoid case by Philip Higgins
and me (JPAA, 1987). The paper by Higgins

`Thin elements and commutative shells in cubical {$\omega$}-categories'.
Theory Appl. Categ. 14 (2005) No. 4, 60--74

shows that the globular case is needed to define commutative shells in
this context, so I am not saying one can get away without globular notions.

Is it possible to capture the properties of the cubical singular complex
of a space (which has been around since Kan's work in 1955, and possible
earlier) in terms of these operadic ideas, and make this useful? Even
more naively, does this relate to `little cubes operads'? How to relate
the putative cubical and globular notions?

Ronnie Brown




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