Higher Lawvere theories?

Higher Lawvere theories?

[Mike Stay]

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


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Re: Higher Lawvere theories?

[Aleks Kissinger]

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
>

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Re: Higher Lawvere theories?

[Thomas Fiore]

Hello All,
Related work in this direction is in my thesis
http://arxiv.org/abs/math.CT/0408298
which appeared as an AMS Memoir, though the notion of 2-theory used there
is *not* the same as a theory enriched in Cat.

Thomas M. Fiore. Pseudo Limits, Biadjoints, and Pseudo Algebras:
Categorical Foundations of Conformal Field Theory.
Memoirs of the American Mathematical Society 182 (2006), no. 860. 171 pages.

and related work is in the follow up article

Fiore, Hu, Kriz. Laplaza Sets, or How to Select Coherence Diagrams for
Pseudo Algebras<http://www-personal.umd.umich.edu/~tmfiore/1/FioreHuKrizLaplazaSets.pdf>.
Advances
in Mathematics, Volume 218, Number 6, pages 1705-1722, 2008. 17 pages.

best regards,
Tom Fiore


>
> On Fri, Aug 2, 2013 at 12:35 PM, 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
>>

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Re: Higher Lawvere theories?

[Mike Stay]

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


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Re: Higher Lawvere theories?

[Noson S. Yanofsky]

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



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