In my Vancouver talk at CT04 I defined a bipartite category to be a bipolar category with no morphisms from negative to positive objects. (By "bipolar" I mean having two types of objects, positive and negative, or black and white, or manic and depressive.) Robert Seely immediately remarked that this notion already had a name, namely "bimodule," which prompted me to apologize for not being familiar with that abstract usage of the notion. Since I've always associated modules (and hence bimodules) with rings (and semimodules with semirings), at the very least I'd expect this notion to be called something like "abstract bimodule" to indicate independence of any additive or even semiadditive structure. Is there an established terminology for this notion? And if not, does anyone have any strong preference between "bipartite category" and "abstract bimodule?" Note that I'll be writing initially for a CS audience, for whom bipartite graphs are a staple. Vaughan Pratt 14-Oct-2004 16:44:29 -0300,1854;000000000000-00000000
On Thu, 14 Oct 2004, Vaughan Pratt wrote:
Since I've always associated modules (and hence bimodules) with rings (and semimodules with semirings), at the very least I'd expect this notion to be called something like "abstract bimodule" to indicate independence of any additive or even semiadditive structure.
I think that category-theorists long ago sold the pass on the idea that the term "module" implies the presence of additive structure; so I am happy to accept "bimodule" in this context. The trouble with "bipartite" is that bipartite graph people are generally non-directed, so that there is a complete symmetry between the two parts -- which doesn't exist in Vaughan's case. Peter Johnstone 18-Oct-2004 09:42:44 -0300,2424;000000000000-00000000
Based on the reactions so far it looks like I'll be going with one of "module" or "bimodule" rather than "bipartite category." A decisive point here was Steve Lack's remark about V-modules, making the point that "abstract module" was less a matter of abstracting the additive structure away than of putting it in by taking V to be additive (whence R-module). So presumably a Set-module would count as an ordinary module, or just a module, by analogy with "an ordinary category, or just a category". (Who'd have guessed the deference paid set theory by the movie "Ordinary People?") (As a digression, this got me to wondering whether abstraction and enrichment could be presented as a suitable adjunction in such cases. The case at hand seems to be a special kind of abstraction: what would such an adjoint be for the notion of 2-cell as an abstract natural transformation, for example? I.e. how would one enrich 2-categories so that the 1-cells became functors and the 2-cells natural transformations?) Peter J. pointed out that bipartite graphs are normally understood to be undirected. However the polarization of the objects makes proscribing negative-to-positive edges entirely equivalent to undirectedness, obvious from the identical matrix formulations of the two kinds of graph. Any difference between bipartite graphs and (Set-)modules would surely come rather from the former disallowing homogeneous edges, those from positive to positive or negative to negative; admitting homogeneous edges breaks the above equivalence. Vaughan Pratt 18-Oct-2004 09:45:40 -0300,1581;000000000000-00000000
On 15 Oct, Vaughan Pratt wrote:
Based on the reactions so far it looks like I'll be going with one of "module" or "bimodule" rather than "bipartite category."
Perhaps it is worth mentioning that we (Robert Seely and I) have used the term "polarized category" to describe the presentation of a module between categories as a category over 2 (the arrow category). This because the categorical semantics of polarized logic turns out to be in polarized categories (aka modules) ... it is worth remembering that simply because polarized categories are something else as well does not mean that they do not have a theory which derives more directly from that special presentation! At any rate bipartite I must agree is the wrong word. -robin (Robin Cockett) 23-Oct-2004 10:16:08 -0300,8448;000000000001-00000000
Dear Vaughan, I think it would be a mistake to introduce a new term for these, since there are already several including "profunctor" and "distributor". The additive notions (ring, module, bimodule) generalize well to enriched categories, where it is standard practice to keep the terms module and bimodule without apology. You seem to have rediscovered the set enriched case. Regards, Steve Vickers. Vaughan Pratt wrote:
In my Vancouver talk at CT04 I defined a bipartite category to be a bipolar category with no morphisms from negative to positive objects. (By "bipolar" I mean having two types of objects, positive and negative, or black and white, or manic and depressive.) Robert Seely immediately remarked that this notion already had a name, namely "bimodule," which prompted me to apologize for not being familiar with that abstract usage of the notion.
Since I've always associated modules (and hence bimodules) with rings (and semimodules with semirings), at the very least I'd expect this notion to be called something like "abstract bimodule" to indicate independence of any additive or even semiadditive structure.
Is there an established terminology for this notion? And if not, does anyone have any strong preference between "bipartite category" and "abstract bimodule?" Note that I'll be writing initially for a CS audience, for whom bipartite graphs are a staple.
Vaughan Pratt
18-Oct-2004 09:42:44 -0300,2236;000000000000-00000000
Vaughan Pratt wrote:
In my Vancouver talk at CT04 I defined a bipartite category to be a bipolar category with no morphisms from negative to positive objects. (By "bipolar" ...
Is there an established terminology for this notion? And if not, does anyone have any strong preference between "bipartite category" and "abstract bimodule?" Note that I'll be writing initially for a CS audience, for whom bipartite graphs are a staple.
Vaughan Pratt
As non-CS and only marginally categorical I would vote for bipartite as bimodule in the alg context brings to mind two objects with one acting on the other two different ways jim 18-Oct-2004 09:47:03 -0300,1562;000000000000-00000000
participants (5)
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jim stasheff -
Prof. Peter Johnstone -
robin＠cpsc.ucalgary.ca -
Steve Vickers -
Vaughan Pratt