Fwd: Internalizing n-cells to (n-1)-cells

Fwd: Internalizing n-cells to (n-1)-cells

[Mike Stay]

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Dear moderator: Could you post this message for Greg Meredith, please?


---------- Forwarded message ----------
From: Meredith Gregory <lgreg.meredith@gmail.com>
Date: Sun, Dec 28, 2014 at 1:01 PM
Subject: Fwd: Internalizing n-cells to (n-1)-cells
To: Mike Stay <metaweta@gmail.com>


Dear Mike,

i'm not sure why, but my email to the categories list is not getting
through. Do you think you might be able to post this question on my
behalf?

i'm writing to ask a question about higher categories motivated from
the computer
science perspective. A colleague and i have been looking at an analogue of the
internalization of morphisms typically associated with Currying. In our setting
we're modeling rewrites in various calculi as 2-morphisms, but to
prevent rewrites
from happening too freely we have to reify certain contexts as
1-morphisms to mark
which rewrites are permitted. Essentially, it's a kind of
internalization process
and is closely connected with work by Leifer, Milner, and Sewell.
Now, though, i'm wondering if there has been a more general study of
internalization operators
taking n-cells to (n-1)-cells. Is there essentially only one kind of
internalization process generalizing the exponential object case? Does
anyone have any references?

Best wishes,

--greg

---------- Forwarded message ----------
From: Meredith Gregory <lgreg.meredith@gmail.com>
Date: Sun, Dec 28, 2014 at 12:40 PM
Subject: Re: Internalizing n-cells to (n-1)-cells
To: Tom Leinster <Tom.Leinster@ed.ac.uk>


Dear Tom,

No worries. Thanks for getting back to me. It appears that this is not
only an unanswered question, but an unasked one! ;-) However, given
that the 1-cell to 0-cell case, namely exponential objects play such a
pivotal role in computing and fairly important roles even in
mainstream maths it seems like it might be worthwhile for me to ask it
and seek some answers.

Best wishes in the New Year,

--greg


On Sunday, December 28, 2014, Tom Leinster <Tom.Leinster@ed.ac.uk> wrote:
>
> Dear Greg,
>
> I'm sorry, but I don't know anything at all about this, not even any references.  Apologies!
>
> Tom
>
> On Sat, 27 Dec 2014, Meredith Gregory wrote:
>
>> Dear Tom,
>> i'm writing to ask a question about higher categories motivated from the computer
>> science perspective. A colleague and i have been looking at an analogue of the
>> internalization of morphisms typically associated with Currying. In our setting
>> we're modeling rewrites in various calculi as 2-morphisms, but to prevent rewrites
>> from happening too freely we have to reify certain contexts as 1-morphisms to mark
>> which rewrites are permitted. Essentially, it's a kind of internalization process
>> and is closely connected with work by Leifer, Milner, and Sewell.
>> Now, though, i'm wondering if there has been a study of internalization operators
>> taking n-cells to (n-1)-cells. Is there essentially only one kind of
>> internalization process? Do you have any references?
>>
>> Best wishes in the New Year,
>>
>> --greg
>>
>> --
>> L.G. Meredith
>> Managing Partner
>> Biosimilarity LLC
>> 7329 39th Ave SWSeattle, WA 98136
>>
>> +1 206.650.3740
>>
>> http://biosimilarity.blogspot.com
>>
>>
> --
> The University of Edinburgh is a charitable body, registered in
> Scotland, with registration number SC005336.
>


--
L.G. Meredith
Managing Partner
Biosimilarity LLC
7329 39th Ave SW
Seattle, WA 98136

+1 206.650.3740

http://biosimilarity.blogspot.com




--
L.G. Meredith
Managing Partner
Biosimilarity LLC
7329 39th Ave SW
Seattle, WA 98136

+1 206.650.3740

http://biosimilarity.blogspot.com


-- 
Mike Stay - metaweta@gmail.com
http://www.cs.auckland.ac.nz/~mike
http://reperiendi.wordpress.com

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Re: Internalizing n-cells to (n-1)-cells

[Ronnie Brown]

You mention the exponential object, so I am not sure if you already know
the following suggested answer.

Monoidal closed structures are defined on certain kinds of cubical
omega-groupoids with connections in joint papers

48.  (with P.J. HIGGINS), ``Tensor products and homotopies for
$\omega$-groupoids and crossed complexes'', {\em J. Pure Appl.
Alg.} 47 (1987) 1-33.

and for categories by an analogous process in

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

A kind of "decollage", dropping dimension by 1, is given by taking the
path object PC of a cubical object C, and this essentially drops each
cell by 1, as you ask.  Cubical objects are convenient for this
exponential method for the usual reason, that I^m x I^n \cong I^{m+n}.
Equivalences with other kinds of structures then enable the translation,
at least in principle.

Thierry Coquand has used cubical methods in his work on homotopy type
theory.

Best wishes

Ronnie
PS While I am writing I might as well mention

http://education.lms.ac.uk/2014/12/alexander-grothendieck-some-recollections/


R






On 28/12/2014 23:36, Mike Stay wrote:
> ---------- Forwarded message ----------
> From: Meredith Gregory <lgreg.meredith@gmail.com>
> Date: Sun, Dec 28, 2014 at 1:01 PM
> Subject: Fwd: Internalizing n-cells to (n-1)-cells
> To: Mike Stay <metaweta@gmail.com>
>
>
> Dear Mike,
>
> i'm writing to ask a question about higher categories motivated from
> the computer
> science perspective. A colleague and i have been looking at an analogue of the
> internalization of morphisms typically associated with Currying. In our setting
> we're modeling rewrites in various calculi as 2-morphisms, but to
> prevent rewrites
> from happening too freely we have to reify certain contexts as
> 1-morphisms to mark
> which rewrites are permitted. Essentially, it's a kind of
> internalization process
> and is closely connected with work by Leifer, Milner, and Sewell.
> Now, though, i'm wondering if there has been a more general study of
> internalization operators
> taking n-cells to (n-1)-cells. Is there essentially only one kind of
> internalization process generalizing the exponential object case? Does
> anyone have any references?
>
> Best wishes,
>
> --greg
>
> --
> L.G. Meredith
> Managing Partner
> Biosimilarity LLC
> 7329 39th Ave SW
> Seattle, WA 98136
>
> +1 206.650.3740
>
> http://biosimilarity.blogspot.com
>
>
>
>

-- 



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