Question on factorization systems

Question on factorization systems

[Michael Barr]

First let me explain that our math dept email system has been down for ten
days and there is no indication when it will be back, although our sysop
has been working on it day and night.  I will circulate an announcement
when it is running again.  Meantime, use this address.

Is there one place that develops all the properties of factorization
systems?  We are especially interested in the non-strict case, that is in
which the right factor needn't be epic, nor the left factor be  monic, but
the unique diagonal fill-in condition holds.

Michael


[For admin and other information see: http://www.mta.ca/~cat-dist/ ]

Re: Question on factorization systems

[Till Mossakowski]

The general non-strict case is covered in some detail in chapter 4 of
Ad??mek, Herrlich, Strecker:
Abstract and Concrete Categories - The Joy of Cats
katmat.math.uni-bremen.de/acc/acc.pdf???

Best, Till

Am 20.04.2014 14:24, schrieb Michael Barr:
> First let me explain that our math dept email system has been down for ten
> days and there is no indication when it will be back, although our sysop
> has been working on it day and night.  I will circulate an announcement
> when it is running again.  Meantime, use this address.
>
> Is there one place that develops all the properties of factorization
> systems?  We are especially interested in the non-strict case, that is in
> which the right factor needn't be epic, nor the left factor be  monic, but
> the unique diagonal fill-in condition holds.
>
> Michael
>



[For admin and other information see: http://www.mta.ca/~cat-dist/ ]

Re: Question on factorization systems

[Richard Garner]

Dear Mike,

A good recent account is in:

A. Carboni, G. Janelidze, G. M. Kelly, and R. Par\'e, On localization
and stabilization for factorization systems, Applied Categorical
Structures (1997), no. 1, 1–58.

Richard

On Sun, Apr 20, 2014, at 10:24 PM, Michael Barr wrote:
> First let me explain that our math dept email system has been down for
> ten
> days and there is no indication when it will be back, although our sysop
> has been working on it day and night.  I will circulate an announcement
> when it is running again.  Meantime, use this address.
> 
> Is there one place that develops all the properties of factorization
> systems?  We are especially interested in the non-strict case, that is in
> which the right factor needn't be epic, nor the left factor be  monic,
> but
> the unique diagonal fill-in condition holds.
> 
> Michael
> 

[For admin and other information see: http://www.mta.ca/~cat-dist/ ]

Re: Question on factorization systems

[Till Mossakowski]

P.S.

the correct URL is

katmat.math.uni-bremen.de/acc/acc.pdf

(without the "???", I do not know where these came from...)


[ Note from moderator: the book is also available at
http://www.tac.mta.ca/tac/reprints/articles/17/tr17.pdf ]


Am 21.04.2014 08:23, schrieb Till Mossakowski:
> The general non-strict case is covered in some detail in chapter 4 of
> Ad??mek, Herrlich, Strecker:
> Abstract and Concrete Categories - The Joy of Cats
> katmat.math.uni-bremen.de/acc/acc.pdf???
>
> Best, Till
>
> Am 20.04.2014 14:24, schrieb Michael Barr:
>> First let me explain that our math dept email system has been down for ten
>> days and there is no indication when it will be back, although our sysop
>> has been working on it day and night.  I will circulate an announcement
>> when it is running again.  Meantime, use this address.
>>
>> Is there one place that develops all the properties of factorization
>> systems?  We are especially interested in the non-strict case, that is in
>> which the right factor needn't be epic, nor the left factor be  monic, but
>> the unique diagonal fill-in condition holds.
>>
>> Michael
>>
>
>
> [For admin and other information see: http://www.mta.ca/~cat-dist/ ]



[For admin and other information see: http://www.mta.ca/~cat-dist/ ]

Re: Question on factorization systems

[tholen]

Michael,

The formally correct answer to your "one place" question is certainly
"No", but here are a few suggestions.

As a primer on orthogonal factorization systems (E,M) without epi-mono
constraints in a published book I would recommend Section 14 of the
Adamek-Herrlich-Strecker book, simply because (unlike many other
accounts of the topic) it is free of redundant requirements.

In my view, that section is, however, not the best in terms of
discussing closure of M (and E) under limits (colimits). In a paper
with John MacDonald (LNM 962,Springer 1982, pp 175-1982) we showed the
equivalence of:

i. (E,M) orth f.s. in C;
ii. E (considered as a full subcat of C^2) is coreflective in C^2, and
E is closed under composition;
iii. M is reflective in C^2, and M is closed under composition.

As Im and Kelly (J. Korean Math. Soc. 23 (1986) 1-18) pointed out,
reflectivity in C^2 leads to all the desired limit stability properties
of the class M in C. This approach to orth. f.s. is taken in the first
chapter of my book with Dikranjan on Closure Operators (Kluwer 1995).
(There, however, we assume for "convenience" M to be a class of monos,
but, as is pointed out there, the essential proofs all work in
generality.)

There is also the important aspect of considering an orth. f.s. (E,M)
as an Eilenberg-Moore (!) structure with respect to the monad C |-->
C^2 in CAT, for which I would refer you to my paper with Korostenski
(JPAA 85 (1993) 57-72) and with George Janelidze (JPAA 142 (1999)
99-130).

Sorry, certainly not just one place, especially since the above
references don't do justice to tons of other contributions. And things
get even more complicated if we talk historical firsts, which would
start with Mac Lane (Bull. AMS 56 (1950))...

Regards,

Walter


Quoting Michael Barr <akapbarr@gmail.com>:

> First let me explain that our math dept email system has been down for ten
> days and there is no indication when it will be back, although our sysop
> has been working on it day and night.  I will circulate an announcement
> when it is running again.  Meantime, use this address.
>
> Is there one place that develops all the properties of factorization
> systems?  We are especially interested in the non-strict case, that is in
> which the right factor needn't be epic, nor the left factor be  monic, but
> the unique diagonal fill-in condition holds.
>
> Michael
>




[For admin and other information see: http://www.mta.ca/~cat-dist/ ]

Re: Question on factorization systems

[Dr. Cyrus F Nourani]

Orthogonal areas: Factorizations on product topologies on fields with 
adjunctions can have interesting applications to explore. There are a 
chapter or two at A Functorial Model Theroy that can be applicable. 
http://appleacademicpress.com/title.php?id=9781926895925 Best regards. 
Akdmkrd.tripod.com DE cyrusfn@alum.mit.edu

Apr 21, 2014 09:00:21 AM, tholen@mathstat.yorku.ca wrote:
Michael,
>
>The formally correct answer to your "one place" question is certainly
>"No", but here are a few suggestions.
>
>As a primer on orthogonal factorization systems (E,M) without epi-mono
>constraints in a published book I would recommend Section 14 of the
>Adamek-Herrlich-Strecker book, simply because (unlike many other
>accounts of the topic) it is free of redundant requirements.
>
>In my view, that section is, however, not the best in terms of
>discussing closure of M (and E) under limits (colimits). In a paper
>with John MacDonald (LNM 962,Springer 1982, pp 175-1982) we showed the
>equivalence of:
>
>i. (E,M) orth f.s. in C;
>ii. E (considered as a full subcat of C^2) is coreflective in C^2, and
>E is closed under composition;
>iii. M is reflective in C^2, and M is closed under composition.
>
>As Im and Kelly (J. Korean Math. Soc. 23 (1986) 1-18) pointed out,
>reflectivity in C^2 leads to all the desired limit stability properties
>of the class M in C. This approach to orth. f.s. is taken in the first
>chapter of my book with Dikranjan on Closure Operators (Kluwer 1995).
>(There, however, we assume for "convenience" M to be a class of monos,
>but, as is pointed out there, the essential proofs all work in
>generality.)
>
>There is also the important aspect of considering an orth. f.s. (E,M)
>as an Eilenberg-Moore (!) structure with respect to the monad C |-->
>C^2 in CAT, for which I would refer you to my paper with Korostenski
>(JPAA 85 (1993) 57-72) and with George Janelidze (JPAA 142 (1999)
>99-130).
>
>Sorry, certainly not just one place, especially since the above
>references don't do justice to tons of other contributions. And things
>get even more complicated if we talk historical firsts, which would
>start with Mac Lane (Bull. AMS 56 (1950))...
>
>Regards,
>
>Walter
>
>
>Quoting Michael Barr akapbarr@gmail.com>:
>
>> First let me explain that our math dept email system has been down for ten
>> days and there is no indication when it will be back, although our sysop
>> has been working on it day and night.  I will circulate an announcement
>> when it is running again.  Meantime, use this address.
>>
>> Is there one place that develops all the properties of factorization
>> systems?  We are especially interested in the non-strict case, that is in
>> which the right factor needn't be epic, nor the left factor be  monic, but
>> the unique diagonal fill-in condition holds.
>>
>> Michael
>>

[For admin and other information see: http://www.mta.ca/~cat-dist/ ]