Subject: categories: Re: Name for a concept

From: Marco Grandis <[email-masked]>
Date: Mon, 5 Dec 2005 15:44:26 +0100

In reply to M. Barr's posting.

Richard Wood tells me that my posting on this subject, dated 2  
December 2005, was unreadable with 'elm' and nearly so with 'pine',  
due to rich-text marks.
I am reposting it in plain text (hopefully), with a few small  
additions - and apologies

MG
_____

I think that such squares should be called "exact" or  
"semicartesian" (where cartesian square = pb, cocartesian = po).
They should be viewed as the natural self-dual generalisation of  
pullback and pushout (and their name should be "self-dual", in some  
way). They appear whenever one studies categories of relations.

1. In an abelian category (where they are chracterised by the exact  
sequence you have mentioned), I would prefer "exact", or "Hilton-exact".
Hilton considered such squares (for abelian categories), and proved  
that an equivalent condition is that this square (of proper  
morphisms) is "bicommutative" in the category of relations (i.e. it  
commutes and stays commutative when you reverse two "parallel" arrows  
- as relations).

Plainly:   bicartesian square  =>  pullback  =>  exact;  and dually.

REFERENCE:
P. Hilton, Correspondences and exact squares, in: Proc. Conf. on  
Categorical Algebra, La Jolla 1965, Springer, pp. 254-271.

2. Studying more general categories of relations, I considered  
"semicartesian squares"  (f,g, h,k),  defined - in any category - as  
the commutative squares satisfying the following self-dual property:

  Whenever  (f',g', h,k)  and  (f,g, h',k')  commute, also the outer  
square  (f',g', h',k')  commutes

                          B
         f'           f        h          h'
   A'          A                 D          D'
         g'          g        k          k'
                         C

(add slanting arrows  f': A' --> B,  g': A --> C,  f: A --> B,  etc).

- Again: bicartesian square  =>  pullback  =>  semicartesian,  and  
dually.

- If pb's  and/or  po's exist, there are a lot of equivalent  
properties; eg:

--  (f,g)  and the pb of  (h,k)  have the same po (or the same  
commutative squares out of them).

- In an abelian category, semicartesian amounts to the previous notion.
- In Set, it characterises again those squares which are  
bicommutative in Rel.

REFERENCE:
M. Grandis, Symétrisations de categories et factorisations  
quaternaires, Atti Accad. Naz. Lincei Mem. Cl. Sci. Fis. Mat. Natur.  
14 sez. 1 (1977), 133-207.

3. A 2-dimensional version of this property (actually a STRUCTURE on  
2-cells), was introduced by Guitart, and called "H-exact", if I  
remember well (H for Hilton)

REFERENCES:
- R. Guitart, Carrés exacts et carrés deductifs, Diagrammes 6 (1981),  
G1-G17.
- R. Guitart and L. Van den Bril, Calcul des satellites et  
présentations des bimodules à l'aide des carrés exacts, Cahiers  
Topologie Géom. Différentielle 24 (1983), no. 3, 299-330.
(and some other papers by the same authors).

Best regards

Marco Grandis