Re: Question on factorization systems

["Dr. Cyrus F Nourani" <projectm2@lycos.com>]

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
>>

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