General notions of equivalence and exactness
Hello. In a category with pullbacks, say that a binary relation X <- R -> Y is "z-closed" if it satisfies the following axiom (interpreted as usual): If x R y and x' R y and x' R y' then x R y'. (The "z" in "z-closed" refers to the pattern of variables in the premise.) Z-closedness seems to be a sensible generalization of "equivalence" to relations between two different objects. (e.g. In computer science, it is common to relate the state spaces of two different systems.) Note that an endorelation is an equivalence relation if and only if it is z-closed and reflexive. Also note that, in an abelian category, every relation is z-closed. The [z-closed v. equivalence] connection seems to extend to [pullbacks v. kernel pairs]. Every span that arises from a pullback is a z-closed relation. Say that a category is "z-effective" if every z-closed relation arises as a pullback. - every abelian category is straightforwardly z-effective; - in a topos, every z-closed relation arises as a pullback span. Indeed, an extensive regular category has effective equivalence relations if and only if it is z-effective. These notions and ideas seem quite elementary, even fundamental, and I would be surprised if no-one had thought of them before. I borrowed the terminology "z-closed" from a paper by Erik de Vink and Jan Rutten (Theoret Comput Sci, 221:271-293, 1999) but I couldn't find any other references. Have I missed something? I'd be grateful for any observations or suggestions. Sam PS. I'd like to take the opportunity to acknowledge the helpful replies (public and private) to my question about W-types, a few months ago.
Hi Sam, These "zigzag closed" relations are called "difunctional". Some old references are: [1] J Riguet. Relations binaries, fermetures, correspondances de Galois (1948) [2] J Riguet. Quelques proprieties des relations difonctionelles (1950) [3] Katuzi Ono. On some properties of binary relations (1957). and once one knows what to search for, it turns out they're well-known. An interesting characterization (which is how we(*) discovered them) is that in sets, they're the TT-closed relations, where if x,x'\in A, x(R^TT)x' if forall k k' : A->2, (forall y y', yRy' -> k y = k y') -> k x = k x' ([4] M Abadi. TT-closed relations and admissibility (2000) considers the situation in cpos). Nick (*) Martin Hofmann, Andrew Kennedy, Lennart Beringer and I. Martin initially called these Quasi-PERs. -----Original Message----- From: cat-dist@mta.ca [mailto:cat-dist@mta.ca] On Behalf Of Sam Staton Sent: 29 May 2008 10:02 To: categories@mta.ca Subject: categories: General notions of equivalence and exactness Hello. In a category with pullbacks, say that a binary relation X <- R -> Y is "z-closed" if it satisfies the following axiom (interpreted as usual): If x R y and x' R y and x' R y' then x R y'. (The "z" in "z-closed" refers to the pattern of variables in the premise.) Z-closedness seems to be a sensible generalization of "equivalence" to relations between two different objects. (e.g. In computer science, it is common to relate the state spaces of two different systems.) Note that an endorelation is an equivalence relation if and only if it is z-closed and reflexive. Also note that, in an abelian category, every relation is z-closed. The [z-closed v. equivalence] connection seems to extend to [pullbacks v. kernel pairs]. Every span that arises from a pullback is a z-closed relation. Say that a category is "z-effective" if every z-closed relation arises as a pullback. - every abelian category is straightforwardly z-effective; - in a topos, every z-closed relation arises as a pullback span. Indeed, an extensive regular category has effective equivalence relations if and only if it is z-effective. These notions and ideas seem quite elementary, even fundamental, and I would be surprised if no-one had thought of them before. I borrowed the terminology "z-closed" from a paper by Erik de Vink and Jan Rutten (Theoret Comput Sci, 221:271-293, 1999) but I couldn't find any other references. Have I missed something? I'd be grateful for any observations or suggestions. Sam PS. I'd like to take the opportunity to acknowledge the helpful replies (public and private) to my question about W-types, a few months ago.
In Genoa, in the late 60's and after, our group was studying categories of relations. I was interested in relations on abelian categories, for homological algebra, while Gabriele Darbo, Franco Parodi and others were more interested - after the general construction - in relations on sets, and even more in "corelations on sets" (relations on Set^op), called "transductors". (It is the dual construction, based on equivalence classes of cospans of sets, i.e. quotients of the sum of domain and codomain; used to simulate electric connections between two sets of terminals [see how they compose], and as a formal basis for a general "theory of devices".) At that time, a category with involution u |--> u* (typically, a category of relations in some sense) was called "von Neumann regular" if the condition u.u*.u = u holds for every arrow (plainly related to von Neumann regularity of semigroups and rings). The category of relations of sets is not vN-regular, the simplest counterexample being likely the Z-shaped relation which transgresses your condition: R: {x, y} --> {x', y'} x R y, x' R y, x' R y' This relation was precisely called "Z" in the paper [Pa] F. Parodi, Simmetrizzazioni di una categoria II, Sem. Mat. Univ. Padova, 44 (1970), 223-262. http://archive.numdam.org/ARCHIVE/RSMUP/RSMUP_1970__44_/ RSMUP_1970__44__223_0/RSMUP_1970__44__223_0.pdf On the other hand, as you say, category of relations on abelian categories are von Neumann regular (which is a crucial fact in studying subquotients, see Mac Lane's text on Homology). But, interestingly, the category of CORELATIONS on sets is also von Neumann regular, see the paper above [Pa]. Best regards Marco Grandis On 29 May 2008, at 11:01, Sam Staton wrote:
Hello. In a category with pullbacks, say that a binary relation X <- R -> Y is "z-closed" if it satisfies the following axiom (interpreted as usual):
If x R y and x' R y and x' R y' then x R y'.
(The "z" in "z-closed" refers to the pattern of variables in the premise.)
Z-closedness seems to be a sensible generalization of "equivalence" to relations between two different objects. (e.g. In computer science, it is common to relate the state spaces of two different systems.) Note that an endorelation is an equivalence relation if and only if it is z-closed and reflexive. Also note that, in an abelian category, every relation is z-closed.
The [z-closed v. equivalence] connection seems to extend to [pullbacks v. kernel pairs]. Every span that arises from a pullback is a z-closed relation. Say that a category is "z-effective" if every z-closed relation arises as a pullback.
- every abelian category is straightforwardly z-effective; - in a topos, every z-closed relation arises as a pullback span. Indeed, an extensive regular category has effective equivalence relations if and only if it is z-effective.
These notions and ideas seem quite elementary, even fundamental, and I would be surprised if no-one had thought of them before. I borrowed the terminology "z-closed" from a paper by Erik de Vink and Jan Rutten (Theoret Comput Sci, 221:271-293, 1999) but I couldn't find any other references.
Have I missed something? I'd be grateful for any observations or suggestions.
Sam
PS. I'd like to take the opportunity to acknowledge the helpful replies (public and private) to my question about W-types, a few months ago.
Many thanks to all who replied to my message, in private and publicly. I notice that these concerns also arose on this mailing list back in 1992: http://www.mta.ca/~cat-dist/archive/1992/92-06.txt At that time, Michael Barr was asking, amongst other things, about the exactness property that "every Mal'cev [=difunctional=z-closed] relation is a pullback", which holds both in toposes and in abelian categories (as I mentioned below). I wonder if anything more came out of that. He mentioned a possible connection with "effective unions", but I haven't been able to get anything to work there. By the way, following the comments about Mal'cev operators, and Peter Freyd's "Mal'cev allegories", I note that (exact) categories in which every relation is difunctional have been called "Mal'cev categories" [see e.g. the book by Bourn and Borceux on the topic (pointed out by Peter Lumsdaine), or Carboni, Lambek, Pedicchio, Diagram chasing in Mal'cev categories, JPAA 69]. Sam On 29 May 2008, at 10:01, Sam Staton wrote:
Hello. In a category with pullbacks, say that a binary relation X <- R -> Y is "z-closed" if it satisfies the following axiom (interpreted as usual):
If x R y and x' R y and x' R y' then x R y'.
(The "z" in "z-closed" refers to the pattern of variables in the premise.)
Z-closedness seems to be a sensible generalization of "equivalence" to relations between two different objects. (e.g. In computer science, it is common to relate the state spaces of two different systems.) Note that an endorelation is an equivalence relation if and only if it is z-closed and reflexive. Also note that, in an abelian category, every relation is z-closed.
The [z-closed v. equivalence] connection seems to extend to [pullbacks v. kernel pairs]. Every span that arises from a pullback is a z-closed relation. Say that a category is "z-effective" if every z-closed relation arises as a pullback.
- every abelian category is straightforwardly z-effective; - in a topos, every z-closed relation arises as a pullback span. Indeed, an extensive regular category has effective equivalence relations if and only if it is z-effective.
These notions and ideas seem quite elementary, even fundamental, and I would be surprised if no-one had thought of them before. I borrowed the terminology "z-closed" from a paper by Erik de Vink and Jan Rutten (Theoret Comput Sci, 221:271-293, 1999) but I couldn't find any other references.
Have I missed something? I'd be grateful for any observations or suggestions.
Sam
PS. I'd like to take the opportunity to acknowledge the helpful replies (public and private) to my question about W-types, a few months ago.
participants (4)
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Marco Grandis -
Nick Benton -
Sam Staton -
Sam Staton