Given the private replies I got to yesterday's queries, it is obvious I was not clear enough, and indeed there was an unhelpful typo.
I'm wondering, if anybody has ever described the following monoidal structure on the category of oriented multigraphs, what MacLane calls graphs, the most common kind of graph in category theory (but not
OK Saunders, from now on they're graphs. This what happens when you hang out with combinatorists AND category theorists.
Given graphs X, Y, the set |X-oY| of vertices on X-oY is the set of graph morphisms X --> Y.
So right from the start this is not the usual presheaf CC structure, where the set of vertices is the set of all functions |X| --> |Y| . So in what follows I use categorical notation for vertices, arrows, etc.
Given f,g : X --> Y the set of arrows f-->g is the set of pairs (p_0,p_1) of functions such that
forall x in |X|, p_0(x) : f(x)-->g(x)
forall k: x-->y in X, p_1(k) : f(x)-->g(y)
Now the typo has been corrected. So an arrow f --> g is like a natural transformation, with p_0 the usual familly of arrows indexed by the vertices/objects of X, but since things don't compose, you add the diagonal p_1 as part of the information. There is some kinship to homotopies, as M. Barr has remarked.
This co-contra bifunctor has a tensor left adjoint, which is symmetric and monoidal.
Is this more understandable? Thanks again Francois
On Francois Lamarche's question. If I understand correctly, the tensor product X tensor Y has the obvious objects (x, y) and arrows of three types (a, y): (x, y) --> (x', y), for a: x -> x' in X, y in Y, (x, b): (x, y) --> (x, y'), for x in X, b: y -> y' in Y, (a, b): (x, y) --> (x', y'), for a and b as above. *** Remark 1. We are thus simulating "identities" of X and Y (which are not given). In other words, we are considering the cartesian product X'xY' of the free reflexive graphs over X and Y, and taking out its identities. Would it not be simpler to work with REFLEXIVE GRAPHS and their cartesian closed structure? In my opinion, reflexive graphs are more natural than graphs: reflexive graph = 1-truncated simplicial set = 1-truncated cubical set It is the topos of presheaves over a FULL subcategory of non-empty ordinals (or cardinals as well), actually the initial segment 1, 2. Remark 2. Roughly speaking, a category enriched over reflexive graphs (wrt the cc structure) is a "2-category without vertical composition". It has cells a: f -> g: X -> Y, with a categorical horizontal composition; it also has trivial cells f -> f: X -> Y ("vertical identities"). All this is clearly related to homotopy and its abstract settings in "2-dimensional categories" (in some sense). And indeed topological spaces, with continuous maps and homotopies, form a rather obvious example. The horizontal composition of homotopies a: f -> g: X -> Y, b: h -> k: Y -> Z is b(a(x, t), t) t in [0, 1], which is indeed categorical. Remark 3. [The sequel is relevant for homotopy; I do not know if it may be relevant in CS, but I always had the impression that abstract homotopy should be of use there, eg with respect to deformations of processes, in some sense.] I do not think that the latter is the "right" 2-dimensional categorical setting for abstract homotopy (even as a starting point). The previous horizontal composition of homotopies is rather artificial; it is what you get from the "double homotopy" b(a(x, t), t') (t, t') in [0, 1]^2 through the diagonal t = t' of the square. (The "double homotopy" itself is quite natural, as produced by the cubical enrichment due to the cylinder functor; it is also important in homotopy.) When the diagonal of the "standard interval" is missing (eg for chain complexes of abelian groups), there is no canonical horizontal composition of homotopies (working with the vertical composition, you get two of them; the middle four interchange does not hold). But there still are canonical horizontal compositions of "maps with homotopies" and "homotopies with maps". This is why I think that the basic 2-dimensional categorical setting for abstract homotopy should only treat such "reduced horizontal composition": arrows with cells, cells with arrows, but NOT cells with cells. Formally, it is again a category enriched over reflexive graphs, BUT wrt the following monoidal closed structure: X tensor Y: - the subgraph of XxY whose arrows are pairs (a, b), where a or b is an identity; [X, Y]: - vertices: the graph morhisms; - arrows: their transformations (without "diagonals"). References: a) The last enrichment (with further developments) has been used for abstract homotopy in: M. Grandis, On the categorical foundations of homological and homotopical algebra, Cahiers Top. Geom. Diff. Categ. 33 (1992), 135-175. [sketch] - , Homotopical algebra in homotopical categories, Appl. Categ. Structures 2 (1994), 351-406. b) A notion equivalent to a category enriched in the same sense had already been studied in: K.H. Kamps, Ueber einige formale Eigenschaften von Faserungen und h-Faserungen, Manuscripta Math. 3 (1970), 237-255. c) For homotopy in groupoid-enriched categories, see Gabriel-Zisman's text (1967). *** With best regards Marco Grandis Dipartimento di Matematica Universita' di Genova via Dodecaneso 35 16146 GENOVA, Italy e-mail: grandis@dima.unige.it tel: +39.010.353 6805 fax: +39.010.353 6752 http://www.dima.unige.it/STAFF/GRANDIS/ ftp://pitagora.dima.unige.it/WWW/FTP/GRANDIS/
I did not quite understand Francois' construction. However, my first reaction to a question like that is that it ought to be a homotopy. So I will say what a homotopy reduces to in this case and leave it to Francois to decide if this is what he has. I suspect that rather few people know what a simplicial homotopy is and, of those, rather few have ever actually verified one. I am in that minority^2, so perhaps I tend to see them where they are not the most natural, but I think it quite remarkable that they can arise where no real topology (but some combinatorics) is present. I have to begin by saying how a graph becomes a simplicial set. Actually, that is a lie, since unless you are dealing with reflexive graphs--that are equipped with a selected loop at each vertex--you will only get a face complex. But homotopies are still definable. A category is a simplicial set by taking for n-simplexes composable n-tuples of arrows. This doesn't work for graphs, since the "interior faces" (all except the lowest and highest numbered) all depend on composition. But there is a face complex in which an n-simplex is simply an n-simplex in the graph. So a 2-simplex is a triangle--obviously non-commutative and a 3-simplex is a tetrahedron and so on. You can describe a composable n-tuple in a category as commutative n-simplex, so this isn't so different. Now given this, if f,g: X --> Y are graph morphisms, what is a homotopy? Well, write X as d^0,d^1: X_1 --> X_0 and similarly for Y. Then f consists of f_0: X_0 --> Y_0 and f_1: X_1 --> Y_1 giving a serially commutative square. Just a homomtopy between functors turns out to be simply a natural transformation, a homotopy in this case turns out to consist of a function p_0: X_0 --> Y_1 and a function p_1: X_1 --> Y_1 such that there is a diagram (not, of course commutative; what a diagram does is specify source and target) as follows. In this diagram I assume x: x^0 --> x^1 in X, and f(x): y^0 --> y^1 and g(x): z^0 --> z^1 in Y. f(x) y^0 -----------> y^1 | \ | | \ | | \ | | \ | | \ | | \ | | \ | p_0(x^0)| p_1(x)\ |p_0(x^1) | \ | | \ | | \ | | \ | | \ | | \ | v g(x) vv z^0 -----------> z^1 So if this is what Francois was saying, then the answer is it a homotopy of face complexes. Of course, if you replaced X_1 by X_1 + X_0, you would have a reflexive graph and I assume (I have not checked this) you would then get a simplicial homotopy. BTW, homotopies do not generally compose--and the ones described here do not appear to either. Categories are special because of their internal composition. It makes me wonder if the well-known failure of dinats to compose could be related to this in some way. Having seen Francois' clarification, I think this is exactly what he had. Michael ------------------------------------------------------------------- History shows that the human mind, fed by constant accessions of knowledge, periodically grows too large for its theoretical coverings, and bursts them asunder to appear in new habiliments, as the feeding and growing grub, at intervals, casts its too narrow skin and assumes another... Truly the imago state of Man seems to be terribly distant, but every moult is a step gained. - Charles Darwin, from "The Origin of Species"
Various replies to my query during the day (my thanks to Ronnie Brown and Lutz Schroeder) made me realize that my SMC structure was actually the lifting of the CC structure on reflexive graphs (those with a choice of loop) to ordinary non-reflexive graphs. Here is a bit more detail: We all know these two categories are categories of presheaves. So let G and R be the categories such that Set^G is graphs and Set^R is reflexive graphs. There is an embedding G --> R, which generates the usual triple of functors between the presheaf categories. So it seems this functorial machinery allows to transform the CC structure in Set^R into an SMC structure in Set^G, preserving the forgetful functor. There must be general conditions that allow this. I'm not pursuing this any more right now, because it has to have been done before, and in a much more general setting. Now Michael's comment is also (among other things) about the tension between graphs and reflexive graphs, and naturally there is Lawvere's "Qualitative distinctions between some toposes of generalized graphs" that says a lot about that tension. there may be more in there than we suspect Francois
participants (3)
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Francois Lamarche -
grandis＠dima.unige.it -
Michael Barr