Date: Sat, 30 Jan 1999 11:48:10 +0100 (MET) From: Lars Lindqvist Subject: categories: Reading advise on bicategory theory Hi, I am looking for advise on suitable litterature for a beginner in bicategory-theory. The litterature I have found so far is: Kelly, Street Review of the elements of 2-categories Borceux Handbook of categorical algebra 1 I have also ordered the following article: Benabou Introduction to bicategories So far I have only read Borceux's book but it contains mainly definitions. In particular I have difficulties understanding the need for (and consequences of) the coherence axioms associated with the natural isomorphisms expressing the associativity and identity 'axioms'. /Lars Lindqvist Date: Sun, 31 Jan 1999 10:20:19 -0500 (EST) From: John R Isbell Subject: categories: Re: Reading advise on bicategory theory Lars, I would guess that you will get more advice than you can use. Here is my nickel's worth. The matter of coherence is wide open. There is a paper of Yanofsky accepted for publication in JPAA which investigates higher dimensional categories with no coherence assumptions at all -- this coming after a lot of work on weakened coherence. Benabou's basic point was that naturally arising 2-dimensional categories are not quite 2-categories and don't seem to suffer from it. Avoid getting knotted in coherence questions, especially in 1999. _____________________________ John R. Isbell ji2@buffalo.edu or ji2@acsu.buffalo.edu Homepage: www.unipissing.ca/topology/z/a/a/a/05.htm _________________________________________________ | | | Der Mensch ist nur da ganz Mensch, wo er spielt. | | | | -- Friedrich Schiller | |_________________________________________________ | On Sat, 30 Jan 1999, Lars Lindqvist wrote: > Hi, > > I am looking for advise on suitable litterature for a beginner in > bicategory-theory. The litterature I have found so far is: > > Kelly, Street Review of the elements of 2-categories > Borceux Handbook of categorical algebra 1 > > I have also ordered the following article: > > Benabou Introduction to bicategories > > So far I have only read Borceux's book but it contains mainly > definitions. In particular I have difficulties understanding > the need for (and consequences of) the coherence axioms > associated with the natural isomorphisms expressing the > associativity and identity 'axioms'. > > /Lars Lindqvist > > > Date: Sun, 31 Jan 1999 17:28:51 -0800 (PST) From: james dolan Subject: categories: Re: Reading advise on bicategory theory john isbell writes: -The matter of coherence is wide open. There is a paper of Yanofsky -accepted for publication in JPAA which investigates higher dimensional -categories with no coherence assumptions at all -- this coming after a -lot of work on weakened coherence. Benabou's basic point was that -naturally arising 2-dimensional categories are not quite 2-categories -and don't seem to suffer from it. Avoid getting knotted in coherence -questions, especially in 1999. what does "no coherence assumptions at all" mean in this context? does it mean that yanofsky is studying what i call "coarse n-categories", defined recursively as categories enriched over the cartesian closed category where the objects are the coarse [n-1]-categories and the morphisms are the enriched natural isomorphism classes of enriched functors? coarse n-categories are interesting but they're obviously not the whole story; for example it's straightforward to define the "fundamental coarse n-groupoid" of a space, but it's indeed a rather coarse invariant of the homotopy type. From: "Prof. J. Lambek" Date: Tue, 2 Feb 1999 14:21:45 EST Subject: categories: Reading advice Concerning the question by Lindquist: The tensor product automatically satisfies all functoriality, associativity and coherence conditions, if it is introduced by a universal property as by Bourbaki. This is shown for monoidal categories (bicategories with one object) e.g. in my paper ``Multicategories revisited'', Contemporary Mathematics 92(1989). The same argument works for arbitrary bicategories provided, in defining a multicategory, one replaces the free monoid generated by a set by the free category generated by a graph. Jim Lambek Subject: categories: Re: Reading advice Date: Tue, 02 Feb 1999 23:33:02 -0800 From: Vaughan Pratt >From: "Prof. J. Lambek" >Subject: categories: Reading advice > >Concerning the question by Lindquist: > >The tensor product automatically satisfies all functoriality, >associativity and coherence conditions, if it is introduced by a >universal property as by Bourbaki. In view of this would it be fair to say that coherence is not a notion intrinsic to category theory, but rather arises from the traditional set theoretic presentation (or at least point of view) of category theory? Much the same can surely be said of naturality, whose abstract essence is that of 2-cells but which is standardly presented concretely, where the interchange axiom becomes a not entirely trivial theorem. Vaughan Pratt From: john baez Subject: categories: coherence Date: Wed, 3 Feb 1999 18:11:46 -0800 (PST) J. Lambek writes: >The tensor product automatically satisfies all functoriality, >associativity and coherence conditions, if it is introduced by a >universal property as by Bourbaki. Vaughan Pratt writes: >In view of this would it be fair to say that coherence is not a notion >intrinsic to category theory, but rather arises from the traditional >set theoretic presentation (or at least point of view) of category theory? I think it's fair to say that operations automatically satisfy all the right coherence laws if you define them using universal properties. This is the idea behind Jim Dolan's and my definition of weak n-categories: all the ways of composing cells are defined by means of universal properties, so one doesn't need to explicitly list coherence laws - they're automatic. Indeed, if you ask what are the "right" coherence laws, perhaps the easiest answer is: the coherence laws automatically satisfied by universal constructions! (There are also some answers coming from homotopy theory but probably deep down they are the same answer.) We pound these points home with great rhetorical flourishes in the following paper: Categorification, in Higher Category Theory, eds. Ezra Getzler and Mikhail Kapranov, American Mathematical Society, Providence, 1998, pp. 1-36. Also available electronically at http://math.ucr.edu/home/baez/cat.ps Date: Fri, 5 Feb 1999 14:03:37 -0500 (EST) From: Noson Yanofsky Subject: categories: no coherence assumptions James Dolan wrote: >what does "no coherence assumptions at all" mean in this context? >does it mean that yanofsky is studying what i call "coarse >n-categories", defined recursively as categories enriched over the >cartesian closed category where the objects are the coarse >[n-1]-categories and the morphisms are the enriched natural >isomorphism classes of enriched functors? I am not sure whether his "course 2-categories" is my associative categories. Here is the abstract of my paper available at http://xxx.lanl.gov/abs/math.QA/9804106 Obstructions to Coherence: Natural Noncoherent Associativity Abstract We study what happens when coherence fails. Categories with a tensor product and a natural associativity isomorphism that does not necessarily satisfy the pentagon coherence requirements (called associative categories) are considered. Categorical versions of associahedra where naturality squares commute and pentagons do not are constructed (called Catalan groupoids, $\An$). These groupoids are used in the construction of the free associative category. They are also used in the construction of the theory of associative categories (given as a 2-sketch). Generators and relations are given for the fundamental group, $\pi(\An)$, of the Catalan groupoids -- thought of as a simplicial complex. These groups are shown to be more than just free groups. Each associative category, $\bf B$, has related fundamental groups $\pi(\bf B_n)$ and homomorphisms $\pi(P_n):\pi(\An) \longrightarrow \pi(\bf B_n)$. If the images of the $\pi(P_n)$ are trivial, i.e. there is only one associativity path between any two objects, then the category is coherent. Otherwise the images of $\pi(P_n)$ are obstructions to coherence. Some progress is made in classifying noncoherence of associative categories. Prof. J. Lambek wrote: >The tensor product automatically satisfies all functoriality, >associativity and coherence conditions, if it is introduced by a >universal property as by Bourbaki. Yes, but Bourbaki's universal properties required for the isomorphism A(BC) ---> (AB)C is that a(bc) |---> (ab)c (see Algebre II.64). Mac Lane's classical counterexample a(bc)|--->(-1)(ab)c surely does not satisfy this coherence condition (page 163 of CWM). By the way, the fundamental group that corresponds to this associative category is Z_2 since going around once does not give the identity but going around twice does. John Baez wrote: >Indeed, if you ask what are the "right" coherence laws, perhaps the >easiest answer is: the coherence laws automatically satisfied by universal >constructions! The whole point of my paper is that there are no "right" coherence laws. Each structure has interesting theorems that can be proven about it. Finding the "right" coherence axioms is like finding the "right" axioms to study symmetry. Which one is "right": semi-groups, groups, Abelian groups Lie groups etc.? If coherence theory is to be thought of as some type of higher-dimensional algebra, then there are many structures that are of interest. Noson Yanofsky