Re: name for a concept

Re: name for a concept

[Clemens.BERGER]

Let me suggest still another terminology:

For this, call a class S of maps in an arbitrary category *(co)stable*
iff S is closed under composition and under (co)base change. Then call a
commutative square *S-exact * (resp. *S-coexact*) iff the induced map to
the pullback (resp. from the pushout) belongs to S. It is then easy to
check that S-(co)exact squares compose for any (co)stable class S (which
I believe is the minimal condition to impose on any reasonable
distinguished class of commutative squares).

In an abelian category, the class M of monos (resp. the class E of epis)
is not only stable (resp. costable) but also costable (resp. stable).
With this terminology, Hilton's exact squares can either be identified
with the E-exact squares or with the M-coexact squares, which explains
why it is a self-dual concept, cf. the first message of Michael Barr and
the last message of Marco Grandis.

In homotopy theory, there is the important concept of a *homotopy
pullback* which is the ``homotopy invariant'' substitute for an
ordinary pullback. For those who are familiar with Quillen model
categories, it is very useful in practice that if a Quillen model
category is *right proper* (i.e. its class of fibrations is stable),
then a commutative square with two parallel fibrations is a homotopy
pullback *if and only if* the square is exact with respect to the class
of trivial fibrations (those fibrations which are also weak
equivalences). There is of course a dual statement for homotopy pushouts
in a left proper Quillen model category.

With best regards,

Clemens Berger.

Re: name for a concept

[Clemens.BERGER]

In my previous message, one should read:

a commutative square with two parallel fibrations in a right proper
model category is a homotopy pullback if and only if the square is exact
with respect to the class of weak equivalences. These special squares
compose because weak equivalences are stable under base change along
fibrations (this is the definition of right proper).

Clemens Berger.

Re: Name for a concept

[Eduardo Dubuc]

Jean Benabou wrote:

>
> (2)- In most cases the canonical map being epic is not what one really
> wants. Of course Joyal assumes the category where the maps live to be a
> pre-topos, then it's enough, otherwise one cannot "compose" such
> squares.  Do we have to rename the squares where the canonical map is a
> universal epi, or those where its a universal regular epi?
>


very good point

i suggest, since we can live with epics, strict(=regular) epis,
universal such, etc etc,

we should have:

quasi-pullback

strict(=regular) quasi pullback

universal quasi-pullback

of course, the useful concept being: "strict universal quasi-pullback"

e.d.

Re: Name for a concept

[Peter Freyd]

Jean asks:

  Is there a standard name for the squares where the canonical map
  is monic , i.e. the pair of maps A --->B  and  A --->C is jointly
  monic.

In the early 60s at the annual AMS meeting held at Denver, Eilenberg,
Mac Lane and I sat down to "settle" the terminology. ("Denver One" I
called it.) There were just two things we totally agreed on: "weak" is
the operator on definitions that removes uniqueness conditions and
"partial" the operator that removes existence conditions. So the
answer to Jean's question would be "partial pullback".

As for the other side -- when the pair of maps are jointly epic --
I've seen them called "near-pullbacks" in the theoretical computer
science community. Functors between regular categories that preserve
near-pullbacks are precisely those that preserve "weak tabulations"
of (n-ary) relations.

Re: Name for a concept

[Marco Grandis]

I do not know the original problem of M. Barr, may be he is really
interested in getting an epimorphism onto the pullback.

However - I apologise for insisting - I think that the important
notion for such squares should be a natural self-dual generalisation
of pullbacks and pushouts, based on commutative squares and nothing
else - so that, in particular, it cannot depend on the variation of
monos or epis one is interested in.

The one I have proposed in that old paper (under the name of
"semicartesian square") is of this kind:

- the square  (f,g; h,k)  commutes, and for every span  (f',g')
which commutes with the cospan  (h,k)  and every cospan  (h',k')
which commutes with the span  (f,g),  the new span and cospan form a
commutative square.

All this comes from the obvious Galois connection between sets of
spans and cospans (in an arbitrary category), derived from the
commutativity relation.

Explicitly, let us start with two fixed objects A, B.  Let  S  be the
set of spans from A to B:

x = (f: C -> A,  g: C -> B)    (for arbitrary C)

and  C  the set of cospans

y = (h: A -> D, k: B  -> D)   (for arbitrary D).

Take their set of parts,  PS  and  PC,  ordered by inclusion, and the
following (contravariant) Galois connection between them (X in PS, Y
in PC):

R(X) = set of cospans which commute with all the spans in X,
L(Y) = set of spans which commute with all the cospans in Y.

Now, a square  (x, y)  (span/cospan) commutes iff  {x} is contained
in  L({y})  iff  {y} is contained in R({x}).
A square  (x, y)  is "semicartesian" (or "exact") iff it satisfies
the stronger, equivalent conditions:

1.  R{x} = RL({y})
2.  L{y} = LR({x}).

Marco Grandis

-------------------------
On 1 Dec 2005, at 02:48, Michael Barr wrote:

> Is there a standard name for a square
> A ----> B
> |       |
> |       |
> |       |
> v       v
> C ----> D
> in which the canonical map A ---> B x_D C is epic?  I had always
> called it
> a weak pullback, but Peter Freyd claims that that phrase is
> reserved for
> the case that it satisfies the existence, but not necessarily the
> uniqueness of the definition of pullback.  In fact, he claims it means
> that Hom(E,-) converts it to the kind of square I am talking about.
> What is interesting is that in an abelian category, it satisfies
> this condition iff it satisfies the dual condition iff the evident
> sequence A ---> B x C ---> D is exact.  Putting a zero at the left end
> characterizes a genuine pullback and at the other end a pushout.
>
> Michael
>
>
>
>
>

Re: Name for a concept

[Toby Bartels]

jean benabou wrote in part:

>(1) -  Is there a standard name for the squares where the canonical map
>is monic , i.e. the pair of maps A --->B  and  A --->C is jointly
>monic. I propose semi-pullback

How about "sub-pullback"?
since it is a sub-object of the pullback (if there is one).


-- Toby

Re: Name for a concept

[jean benabou]

Continuation of namings

(1) -  Is there a standard name for the squares where the canonical map 
is monic , i.e. the pair of maps A --->B  and  A --->C is jointly  
monic. I propose semi-pullback

(2)- In most cases the canonical map being epic is not what one really 
wants. Of course Joyal assumes the category where the maps live to be a 
pre-topos, then it's enough, otherwise one cannot "compose" such 
squares.  Do we have to rename the squares where the canonical map is a 
universal epi, or those where its a universal regular epi?

In view of (1), one would like to say that a square is a pullback iff 
it is both a quasi and semi pullback


Début du message réexpédié :

> De: Eduardo Dubuc <edubuc@dm.uba.ar>
> Date: Lun 5 déc 2005  17:16:13 Europe/Paris
> À: categories@mta.ca (Categories list)
> Objet: categories: Re: Name for a concept
>
>>
>> Is there a standard name for a square
>> A ----> B
>> |       |
>> |       |
>> |       |
>> v       v
>> C ----> D
>> in which the canonical map A ---> B x_D C is epic?
>
>
> These are called "quasi-pullbacks" by Joyal, and they form a class of
> "open maps" in the category of squares. The pullbacks form the
> corresponding class of etal maps. These two classes are essential for 
> the
> development of the theory (etal class and open class in the sense of
> Joyal). There are published articles by Joyal and Moerdijk on the
> subject.
>
>
>
>
>

Re: Name for a concept

[Eduardo Dubuc]

>
> Is there a standard name for a square
> A ----> B
> |       |
> |       |
> |       |
> v       v
> C ----> D
> in which the canonical map A ---> B x_D C is epic?


These are called "quasi-pullbacks" by Joyal, and they form a class of
"open maps" in the category of squares. The pullbacks form the
corresponding class of etal maps. These two classes are essential for the
development of the theory (etal class and open class in the sense of
Joyal). There are published articles by Joyal and Moerdijk on the
subject.

Re: Name for a concept

[Marco Grandis]

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

Re: Name for a concept

[Marco Grandis]

[-- Attachment #1: Type: text/plain, Size: 2328 bytes --]

1. In an abelian category, I would prefer "exact", or "Hilton-exact".

Hilton considered such squares (for abelian categories), and proved =20
that an equivalent condition is that this square (of proper =20
morphisms) is "bicommutative" in the category of relations (i.e. it =20
commutes and stays commutative when you reverse two "parallel" arrows =20=

- as relations).

Plainly:   bicartesian square  =3D>  pullback  =3D>  exact;  and dually.

REFERENCE:
P. Hilton, Correspondences and exact squares, in: Proc. Conf. on =20
Categorical Algebra, La Jolla 1965, Springer, pp. 254-271.

2. Studying more general categories of relations, I considered =20
"semicartesian squares"  (f,g, h,k),  defined - in any category - by =20
the following self-dual property (after being commutative, of course):

  Whenever  (f',g', h,k)  and  (f,g, h',k')  commute, also the =20
external 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,  f: A --> B,   etc).

- Again: bicartesian square  =3D>  pullback  =3D>  semicartesian,  and =20=

dually.

- If pb's  and/or  po's exist, one can give a lot of equivalent =20
properties; eg:

--  (f,g)  and the pb of  (h,k) have the same po (or the same =20
commutative squares out of them).

- In an abelian category, semicartesian amounts to the previous notion.
- In Set, it characterises again those squares which are =20
bicommutative in Rel.

REFERENCE:
M. Grandis, Sym=E9trisations de categories et factorisations =20
quaternaires, Atti Accad. Naz. Lincei Mem. Cl. Sci. Fis. Mat. Natur. =20
14 sez. 1 (1977), 133-207.

3. A 2-dimensional version of this property (actually a STRUCTURE on =20
2-cells), was introduced by Guitart, and called "H-exact", if I =20
remember well (H for Hilton)

REFERENCES:

- R. Guitart, Carr=E9s exacts et carr=E9s deductifs, Diagrammes 6 =
(1981), =20
G1-G17.
- R. Guitart and L. Van den Bril, Calcul des satellites et =20
pr=E9sentations des bimodules =E0 l'aide des carr=E9s exacts, Cahiers =20=

Topologie G=E9om. Diff=E9rentielle 24 (1983), no. 3, 299-330.
(and some other papers by the same authors).

Best regards

Marco Grandis


[-- Attachment #2: Type: text/html, Size: 14070 bytes --]

Re: Name for a concept

[Ronald Brown]

This kind of condition occurs in topology as a fibrant square - but where
all the maps are fibrations as is the map A ---> B x_D C . This can be
generalised to cubes. See a paper by R. Steiner on

`Resolutions of spaces by n-cubes of fibrations',  J. London Math. Soc.(2),
34, 169-176, 1986

used to build a complete (strict) algebraic model of homotopy n-types which
allows some computations.

This raises the spectre in algebra of

Resolutions of A by free  crossed n-cubes of A.

to give a more `nonabelian' homological algebra. Of course crossed n-cubes
of A should be equivalent to n-fold groupoids in A. This would presumably
bring in higher versions of nonabelian tensor products in A; a bibliography
of such a tensor, mainly for n=2, has 90 items.

This probably does not help to answer Mike's question on the name!

Ronnie
www.bangor.ac.uk/r.brown/nonabtens.html



----- Original Message -----
From: "Michael Barr" <mbarr@math.mcgill.ca>
To: "Categories list" <categories@mta.ca>
Sent: Thursday, December 01, 2005 1:48 AM
Subject: categories: Name for a concept


> Is there a standard name for a square
> A ----> B
> |       |
> |       |
> |       |
> v       v
> C ----> D
> in which the canonical map A ---> B x_D C is epic?  I had always called it
> a weak pullback, but Peter Freyd claims that that phrase is reserved for
> the case that it satisfies the existence, but not necessarily the
> uniqueness of the definition of pullback.  In fact, he claims it means
> that Hom(E,-) converts it to the kind of square I am talking about.
> What is interesting is that in an abelian category, it satisfies
> this condition iff it satisfies the dual condition iff the evident
> sequence A ---> B x C ---> D is exact.  Putting a zero at the left end
> characterizes a genuine pullback and at the other end a pushout.
>
> Michael
>
>
>

Name for a concept

[Michael Barr]

Is there a standard name for a square
A ----> B
|       |
|       |
|       |
v       v
C ----> D
in which the canonical map A ---> B x_D C is epic?  I had always called it
a weak pullback, but Peter Freyd claims that that phrase is reserved for
the case that it satisfies the existence, but not necessarily the
uniqueness of the definition of pullback.  In fact, he claims it means
that Hom(E,-) converts it to the kind of square I am talking about.
What is interesting is that in an abelian category, it satisfies
this condition iff it satisfies the dual condition iff the evident
sequence A ---> B x C ---> D is exact.  Putting a zero at the left end
characterizes a genuine pullback and at the other end a pushout.

Michael