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
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
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
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.
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.
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
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.
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.
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.
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
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.
participants (8)
-
Clemens.BERGER -
Eduardo Dubuc -
jean benabou -
Marco Grandis -
Michael Barr -
Peter Freyd -
Ronald Brown -
Toby Bartels