Has anyone worked out the details of "higher Lawvere theories" so that one can say "the free bicategory on this object, these morphisms, these 2-morphisms, modulo these equations of 2-morphisms"? -- Mike Stay - metaweta@gmail.com http://www.cs.auckland.ac.nz/~mike http://reperiendi.wordpress.com [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
The strict case has been studied quite a bit using n-polygraphs (aka computads), which give a way of presenting an n-category by generators and relations (where relations are given by n+1 dimensional cells). Often the concern is decidablility of the (higher-dimensional) word problem, using rewriting. They originally came out of work by Street and Burroni. More recently, Yves Guiraud has written about them quite extensively. See e.g.: http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.144.3106&rep=rep1&type=pdf Best, Aleks Kissinger On 2 August 2013 17:35, Mike Stay <metaweta@gmail.com> wrote:
Has anyone worked out the details of "higher Lawvere theories" so that one can say "the free bicategory on this object, these morphisms, these 2-morphisms, modulo these equations of 2-morphisms"? -- Mike Stay - metaweta@gmail.com http://www.cs.auckland.ac.nz/~mike http://reperiendi.wordpress.com
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
Hi, In a paper: Coherence, Homotopy and 2-Theories K-Theory 23: Pgs 203 - 235. (2001). I worked out much of 2-Theories and their relationship with coherence theory. Abstract: 2-Theories are a canonical way of describing categories with extra structure. 2-theory-morphisms are used when discussing how one structure can be replaced with another structure. This is central to categorical coherence theory. We place a Quillen model category structure on the category of 2-theories and 2-theory-morphisms where the weak equivalences are biequivalences of 2-theories. A biequivalence of 2-theories (Morita equivalence) induces and is induced by a biequivalence of 2-categories of algebras. This model category structure allows one to talk of the homotopy of 2-theories and discuss the universal properties of coherence. There is also a version on the arXiv: http://xxx.lanl.gov/abs/math.CT/0007033 All the best, Noson Yanofsky -----Original Message----- From: Mike Stay [mailto:metaweta@gmail.com] Sent: Friday, August 02, 2013 12:35 PM To: categories Subject: categories: Higher Lawvere theories? Has anyone worked out the details of "higher Lawvere theories" so that one can say "the free bicategory on this object, these morphisms, these 2-morphisms, modulo these equations of 2-morphisms"? -- Mike Stay - metaweta@gmail.com http://www.cs.auckland.ac.nz/~mike http://reperiendi.wordpress.com [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
On Fri, Aug 2, 2013 at 10:35 AM, Mike Stay <metaweta@gmail.com> wrote:
Has anyone worked out the details of "higher Lawvere theories" so that one can say "the free bicategory on this object, these morphisms, these 2-morphisms, modulo these equations of 2-morphisms"?
Thanks for all the helpful responses, everyone! -- Mike Stay - metaweta@gmail.com http://www.cs.auckland.ac.nz/~mike http://reperiendi.wordpress.com [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
participants (3)
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Aleks Kissinger -
Mike Stay -
Noson S. Yanofsky