Ross, Thanks for the information. I'm not surprised to hear the name 'Isbell' connected with this (nor Lawvere) as there are hints of this idea in 'Taking Categories Seriously' where Lawvere talks about 'Isbell conjugation' (though in Isbell conjugation one considers the categories of covariant and contravariant functors as separate). Looking up 'Isbell envelope' on MathSciNet came up with nothing whilst 'Isbell conjugation' only came up with two papers (one reviewed by you, I think!). One is about 'total categories', though I've yet to look at it to see if it is relevant. When you say that you call it 'the Isbell envelope', what do you mean? Is the category the 'Isbell envelope of/on the original category' or are the objects 'Isbell envelopes' and we have the category of Isbell envelopes (in/on/of the original category)? Thanks for the quick reply, Andrew On Fri, Mar 06, 2009 at 04:13:11PM +1100, Ross Street wrote:

Dear Andrew

This is what Lawvere told me about once, long ago. I think he called it the Isbell envelope; that is what I've called it ever since. It has nice properties. Lawvere explained that, applied to finite dimensional vector spaces, it fully contains the category of banach spaces and bounded linear maps. (I think I've got that right; it's awhile since I checked it.)

Ross

On 06/03/2009, at 2:34 AM, Andrew Stacey wrote:

...

Start with an essentially small category, T, and look at the category whose objects are triples (P,F,c) where: P is a contravariant functor T -> Set, F is a covariant functor T -> Set and c is a natural transformation from P x F to the Hom bi-functor. Morphisms are pairs of natural transformations P_1 -> P_2 and F_2 -> F_1 that intertwine the natural transformations c_1 and c_2.

One could also enrich the whole structure.

Has this cropped up anywhere before? If so, what is it called and where can I learn about it? If not, what shall I call it?

If this is something standard then please pardon my ignorance. I'm fairly new to _real_ category theory and am still just learning the basics.

PS The Isbell envelope arises from the Isbell conjugate pair which is the adjoint pair of functors connecting set^(T^op) and (set^T)^op. It is thus a special case of the general construction in my thesis (TAC Reprints) of the total category with two descriptions which objectifies the notion of adjointness, in order to free it from dependence on enrichments in small sets. (Of course the example depends on enrichments in small sets.) ************************************************************ F. William Lawvere, Professor emeritus Mathematics Department, State University of New York 244 Mathematics Building, Buffalo, N.Y. 14260-2900 USA Tel. 716-645-6284 HOMEPAGE: http://www.acsu.buffalo.edu/~wlawvere ************************************************************ On Thu, 5 Mar 2009, Andrew Stacey wrote:

Dear Categorists,

I'm interested in looking at the following type of thing:

Start with an essentially small category, T, and look at the category whose objects are triples (P,F,c) where: P is a contravariant functor T -> Set, F is a covariant functor T -> Set and c is a natural transformation from P x F to the Hom bi-functor. Morphisms are pairs of natural transformations P_1 -> P_2 and F_2 -> F_1 that intertwine the natural transformations c_1 and c_2.

One could also enrich the whole structure.

Has this cropped up anywhere before? If so, what is it called and where can I learn about it? If not, what shall I call it?

If this is something standard then please pardon my ignorance. I'm fairly new to _real_ category theory and am still just learning the basics.

Thanks,

Andrew Stacey

Ross Street wrote:

Given a category A, the Isbell envelope E(A) is the category whose objects are triplets (F_*, F^*, t) where F_* : A --> Set and F^* : A^{op} --> Set are functors and t_{a,b} : F^*(a) x F_*(b) --> A(a,b) is a family natural in a and b in A. The morphisms are as you described. There is a "double Yoneda" A --> E(A) taking c to (A(c,-), A(-,c), composition).

(For "the morphisms are as you described" to work, unless I'm missing an op somewhere, in the move from (P,F,c) to (F_*,F^*,t), P as the component that "transforms forward" has to be F^*.) I learned about the Isbell envelope from Ross just this January. Ross said he'd learned it from Bill Lawvere, who he said named it that after learning it from Isbell. In the course of some follow-up discussion, Rich Wood pointed out to us the following translation into the language of profunctors of what both Andrew and Ross wrote just now. To avoid having to play favourites by choosing between (P,F,c) and (F_*,F^*,t) I'll pick the neutral (A,X,r), r: A x X --> ? I've been using myself for this stuff, where the early letters A,B,... transform forwards and the late letters X,Y,... transform backwards, just as with Chu spaces. Organize the presheaves A and X as the profunctors A: C^op x 1 --> Set and X: 1^op x C --> Set in the category Prof (Australian: Mod) of profunctors ((bi)modules, distributors) and their natural transformations. In notation more suggestive of how they compose, these become A: 1 -|-> C X: C -|-> 1 r: AX --> 1_C where AX is profunctor composition at 1, and 1_C: C^op x C --> Set is the identity profunctor 1_C: C -|-> C in Prof, aka the homfunctor of C. In general, composition of profunctors entails a left Kan extension, but in this case the composition is taking place at 1, trivializing the composite AX to A x X: C^op x C --> Set (Andrew's P x F). The Isbell envelope being strikingly reminiscent of the Chu construction, it is natural to ask how they're related, which I'll offer an answer to in the rest of this message. A little calculation shows that for the Isbell envelope of the one-morphism category 1 (whose one object I'll denote * in the next paragraph), E(1) is equivalent to Set x Set^op. But this is also Chu(Set,1) where 1 is now the singleton set. What about Chu(Set,2)? The trick here is to take C to be the two-object category 1+1, augmented with two morphisms from what I'll call the positive copy j of 1 to the negative copy \ell, call this C_2 as the category naturally associated to the profunctor 2: 1 -|-> 1 defined by 2(*,*) = 2. I'd been doing this with what I call Dsh(2) below, but Ross pointed out to me (more generally) that Dsh(K) fully embeds in E(C_K) as follows. I'll write it assuming Rich's profunctor typing of A and X as above so that the second argument of A and the first argument of X is always the object * of 1. (In that language Ross's t_{a,b} : F^*(a) x F_*(b) --> A(a,b) becomes t_{a,b} : \int^c F^*(a,c) x F_*(c,b) --> A(a,b) which exposes the left Kan nature of the product, albeit trivial as noted above.) The objects (A,X,r) of E(C_2) now consist of two pairs of sets: A = (A(j,*),A(\ell,*)) and X = (X(*,j),X(*,\ell)). Chu(Set,2) then emerges from E(C) as the full subcategory of E(C_2) for which A(\ell,*) = X(*,j) = {}. In effect A and X behave as though they were single sets A(j,*) and X(*,\ell), the effect we need for (A,X,r) to be an ordinary Chu space. The generalization to Chu(Set,K) simply entails setting C_K(j,\ell) = K, leaving C_K(\ell,j) empty as implicitly assumed above, with |C_K(j,j)| = |C_K(\ell,\ell)| = 1, and with no other change to the description of the full subcategory of E(C_K) constituting Chu(Set,K). In a posting to this list on 9/7/02 with subject line "Presheaves etc. in a uniform way" I described (even more obscurely than my usual postings) a common generalization of presheaves and Chu spaces that can be described far more clearly using the above language. It amounts to a generalization of the Isbell envelope (which as I said I only learned about this January from Ross). Quite independently of Rich and within a day of him, Jeff Egger suggested to me exactly the following typing for A and X in this generalization. In place of the small C parametrizing the Isbell envelope E(C), take two small categories J and L (explaining the j and \ell in the above). Define a *disheaf* (at CT'04 I called these "communes") on a profunctor K: L -|-> J to be a triple (A,X,r) where A: 1 -|-> J and X: L -|-> 1 are profunctors and r: AX --> K: L -|-> J is a natural transformation. By analogy with Chu(Set,K), write Dsh(Set,K) for the category of disheaves on the profunctor K, whose morphisms are just as for the Isbell envelope, which in turn are just as for Chu spaces. The Isbell envelope E(C) = Dsh(Set,1_C). This is the sense in which Dsh generalizes E. (The "Set" parameter is a bad habit arising out of the notation Chu(V,k) adopted around 1992, I forget by whom. It is probably better to write Chu(K) and Dsh(K) and let V be inferred from context as the ambient V in which one is working, which I'll do now.) As pointed out to me by Ross, E(C) generalizes Dsh(K) by taking C to be what Ross called the *cone* C_K of K: J^op x L --> Set (L -|-> J), namely J+L augmented with morphisms from each j to each \ell picked out of the set K(j,\ell) and composed as prescribed by the functoriality of K. (I learned this category representation of profunctors/bimodules the hard way from Robert Seely at CT'04, who kindly said of my "bipartite categories" during my talk, "That's a bimodule!") The generalization is weaker (in some sense) than the other direction, in that Dsh(K) is not E(C_K) itself but only the full subcategory obtained by requiring A(j,*) = X(*,\ell) = 0 for all j in \ob(J) and \ell in \ob(L). I've recently found Dsh(K) very useful as a foundation for ontology (not so much Aristotle as XML, which among other things supplies the Web Ontology Language OWL with its presentation syntax), where it furnishes an extensional conception of the notion of attribute (which the Wikipedia article "Property (philosophy)" helpfully points out doesn't exist), makes sense of C.I. Lewis's highly controversial notion of qualia from the 1920's (philosophers today divide into qualiaphiles and qualiaphobes, see the Wikipedia article on "Qualia") as the elements of K(j,\ell), and offers a mechanism for interaction between the two components in Cartesian dualism, which failed after a century of unconvincing candidates for a meaningful such mechanism (Malebranche's occasionalism, Leibniz's monads, etc.). From a more mathematical standpoint disheaves are even more general than Chu spaces. I can imagine a few raised eyebrows here---after my decade of propaganda on Chu spaces as the ultimate universal framework, the natural question would be, how could anything be more general than a Chu space? It depends on whether you're willing to settle for some full subcategory of some Chu(Set,K) or want your category "on the nose" without the hassle of having to come up with some ad hoc characterization of the desired subcategory (cf. the distinction above between Dsh(K) as a generalization of E(C) and vice versa where only the latter requires the extra step of specifying a full subcategory). Instead one can point to a profunctor K and say that a K-flapdoodle is precisely an object of Dsh(K). For example there is a straightforwardly exhibited K such that Dsh(K) consists essentially of the inelastic acylic graphs, those having invariant path length. No presheaf category consists of these, and Isbell envelopes only contain them as a full subcategory (I don't know if there's a K large enough that Chu(K) does the same job). I have a paper on this in the works, focusing mainly on the ontology application for now and therefore trying (only half successfully I fear) to minimize the deployment of categorical weapons of math destruction in order to make it more accessible to those likely to care at all about ontology, let me know if you're interested. Vaughan Pratt