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 [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
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
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
On Wed, Sep 1, 2010 at 2:42 PM, Greg Meredith <lgreg.meredith@biosimilarity.com> wrote:
i have always taken this situation
There are *several* definitions that are almost surely "right"
to be a pretty clear critique of category theory's basic formulation.
On the contrary, I think that one of the things category theory contributes to mathematics is a precise theory of many different levels at which two (or more) definitions can be "the same". It happens all over mathematics that different definitions give rise to "the same" object; category theory gives general contexts in which to discuss such things. Moreover it often happens that some subtler sorts of "sameness" are difficult to characterize and prove. Thus it is perhaps not really surprising that in category theory itself, one encounters some of the subtlest and most difficult sorts of "sameness" to deal with.
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.
Indeed -- but one has to be careful (hence the need for operads in Batanin's definition) because in general, it is not the case that "all diagrams built from the canonical isomorphisms commute" in the naive sense. What is true is that all diagrams built *in a canonincal way* from the canonical diagrams should commute -- but then one has to make precise how we are allowed to build canonical diagrams. Mike [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
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 [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
Dear John, Ok! i'm interested. i know you've probably given this (on this list even) dozens of times before, but is there a self-contained account of Leinster's refinement of Batanin's approach? For example, arXiv has a link to the promisingly titled Higher Operads, Higher Categories. Would this be a place to begin? i note the Eugenia Cheng has a survey paper on weak n-categories. Is this a place to begin? Best wishes, --greg On Thu, Sep 2, 2010 at 9:04 PM, John Baez <baez@math.ucr.edu> wrote:
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
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
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 [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
participants (4)
-
Greg Meredith -
John Baez -
Michael Shulman -
Ronnie Brown