On the subject of favorite dualities: Surely the most important are the self-dualities and the most important of these (so important we stop noticing it as we age) is the category of finite-dimensional vector spaces over a given field. Next is Pontryagin's: the category of locally compact groups. The original Pontryagin duality easily generalizes: the category of locally compact modules over a given commutative ring is self-dual. (In the non-commutative case one also obtains a duality but not a self-duality -- unless, of course, the ring is self-dual.) A corollary is that the category of discrete left R-modules is dual to the category of compact right R-modules. (For 50 years I've been trying to turn this into an exercise in abelian categories. There's a nice reduction down to the proposition that R/Z is a cogenerator for the category of compact abelian groups, but that fact seems to require some non-trivial functional analysis.) Strange that two of the most important "dualities" are both Pontryagin's. The other is in algebraic topology theory. Then, of course there's my present favorite: the category of finitely presented group-valued functors from the category of finitely presented modules over a commutative ring.
Next is Pontryagin's: the category of locally compact groups. The original Pontryagin duality easily generalizes: the category of locally compact modules over a given commutative ring is self-dual. (In the non-commutative case one also obtains a duality but not a self-duality -- unless, of course, the ring is self-dual.) A corollary is that the category of discrete left R-modules is dual to the category of compact right R-modules. (For 50 years I've been trying to turn this into an exercise in abelian categories. There's a nice reduction down to the proposition that R/Z is a cogenerator for the category of compact abelian groups, but that fact seems to require some non-trivial functional analysis.)
This reminded me of a long-standing torture: does anybody know an elementary proof at least of the particular case when the base ring - thus the dualizer too - has only two elements? (Demanded by Guram Bezhanishvili; I agreed to try thinking on this one as it seemed somehow close to Boolean algebras, but...)
Then, of course there's my present favorite: the category of finitely presented group-valued functors from the category of finitely presented modules over a commutative ring.
Yes, yes?
In addition to Peter's nice collection of self-dualities there are the Kleisli and Eilenberg-Moore categories of "the" covariant power-set monad (there are really two such monads but either will do), respectively Rel and complete semilattices, both self-dual. One that Mike Barr introduced me to is the subcategory of Rel whose morphisms are the partial injections, those binary relations such that if (x,y) and (x,z) are both present then y = z and likewise for their converses. My personal favorites are finite chains with bottom (showing that \Delta, as the base category of the presheaf category of simplicial sets, comes very close to being self-dual; had \Delta itself been self-dual, Set^{\Delta\op} and Set^\Delta would have been the same thing), and semilattices with a top and all nonempty sups (my candidate for a self-dual system of event/state structures before I replaced it with Chu spaces). One should also mention the topological vector spaces in Barr's book on *-autonomous categories, whose self-duality does for the finite-dimensional vector spaces mentioned by Peter what Pontryagin duality does for finite abelian groups. A feature of Chu spaces I particularly like is that each of the above, as well as finite-dimensional vector spaces and finite abelian groups, can be described as that full subcategory of Chu(Set,K) (K = 2 except for vector spaces and abelian groups) consisting of biextensional Chu spaces whose rows and columns satisfy the same closure conditions. For example the category of finite-dimensional vector spaces over GF(2) (a sneaky way to stick to Chu(Set,2)) embeds in Chu(Set,2) as precisely those finite biextensional Chu spaces whose rows and columns, viewed as bit vectors in the sense a machine-language programmer understands the concept, are both closed under bitwise XOR. This example is given as an exercise at the end of Chapter 2 of http://boole.stanford.edu/pub/coimbra.pdf, my notes for the July 1999 Coimbra School cotaught with John Baez and Cristina Pedicchio. Proposition 2.2 in the same chapter obtains complete semilattices (of any cardinality) as those Chu spaces whose rows and columns are closed under bitwise OR, with the self-duality of CSLat as the immediate Corollary 2.3. This shows that the self-dualities CSLat and Vct_{GF(2)}, at least for finite objects, arise identically except for how they combine their bit-vectors, namely with respectively OR and XOR. Vaughan Peter Freyd wrote:
On the subject of favorite dualities:
Surely the most important are the self-dualities and the most important of these (so important we stop noticing it as we age) is the category of finite-dimensional vector spaces over a given field.
Next is Pontryagin's: the category of locally compact groups. The original Pontryagin duality easily generalizes: the category of locally compact modules over a given commutative ring is self-dual. (In the non-commutative case one also obtains a duality but not a self-duality -- unless, of course, the ring is self-dual.) A corollary is that the category of discrete left R-modules is dual to the category of compact right R-modules. (For 50 years I've been trying to turn this into an exercise in abelian categories. There's a nice reduction down to the proposition that R/Z is a cogenerator for the category of compact abelian groups, but that fact seems to require some non-trivial functional analysis.) Strange that two of the most important "dualities" are both Pontryagin's. The other is in algebraic topology theory.
Then, of course there's my present favorite: the category of finitely presented group-valued functors from the category of finitely presented modules over a commutative ring.
Quoting Peter Freyd <pjf@saul.cis.upenn.edu>:
On the subject of favorite dualities:
Surely the most important are the self-dualities and the most important of these (so important we stop noticing it as we age) is the category of finite-dimensional vector spaces over a given field.
Something on this has been done. Duality for vector bundle objects in the category of Lie groupoids was done by Jean Pradines in 1988, and is part of the fundamental work on symplectic groupoids. The cotangent bundle $T^*G$ of any Lie groupoid $G$ has a groupoid structure with base the dual of $AG$, the Lie algebroid of $G$, and Pradines' construction realizes this as the dual of the tangent prolongation $TG$ of $G$. A double vector bundle (in the sense of Ehresmann) is a particular instance of a vector bundle in the category of Lie groupoids. Pradines' duality can be applied to such a structure in two ways, and these do not commute. If $D$ is a double vector bundle over vector bundles $A$ and $B$, each of which is a vector bundle over a manifold $M$, then $D$ can be dualized over $A$ and over $B$. These dualization operations generate the dihedral group of order 6. See `Duality and triple structures', pp455--481 of `The breadth of symplectic and Poisson geometry', (Weinstein Festschrift), Progr. Math., Birkh\"auser Boston, 2005. Alfonso Gracia-Saz and I are preparing a paper on the duality of $n$-fold vector bundles. Details and references for the double case can be found in my `General Theory of Lie groupoids and Lie algebroids', Cambridge, 2005, Chapter 9. Whether categlorification would add anything to this I do not know. Kirill Mackenzie
Quoting Peter Freyd <pjf@saul.cis.upenn.edu>:
On the subject of favorite dualities:
Surely the most important are the self-dualities and the most important of these (so important we stop noticing it as we age) is the category of finite-dimensional vector spaces over a given field.
Something on this has been done. Duality for vector bundle objects in the category of Lie groupoids was done by Jean Pradines in 1988, and is part of the fundamental work on symplectic groupoids. The cotangent bundle $T^*G$ of any Lie groupoid $G$ has a groupoid structure with base the dual of $AG$, the Lie algebroid of $G$, and Pradines' construction realizes this as the dual of the tangent prolongation $TG$ of $G$. A double vector bundle (in the sense of Ehresmann) is a particular instance of a vector bundle in the category of Lie groupoids. Pradines' duality can be applied to such a structure in two ways, and these do not commute. If $D$ is a double vector bundle over vector bundles $A$ and $B$, each of which is a vector bundle over a manifold $M$, then $D$ can be dualized over $A$ and over $B$. These dualization operations generate the dihedral group of order 6. See `Duality and triple structures', pp455--481 of `The breadth of symplectic and Poisson geometry', (Weinstein Festschrift), Progr. Math., Birkh\"auser Boston, 2005. Alfonso Gracia-Saz and I are preparing a paper on the duality of $n$-fold vector bundles. Details and references for the double case can be found in my `General Theory of Lie groupoids and Lie algebroids', Cambridge, 2005, Chapter 9. Whether categlorification would add anything to this I do not know. Kirill Mackenzie
First, let me say I have avoided contributing to this thread because I don't understand what Vaughan is asking. He knows, as well as anyone, since he put them on the map, about Chu categories. He knows about *-autonomous categories as well. So what is the question, really? The simplest answer to Mamuka's question is the duality between vector spaces (over any field, including the 2 element field) and linearly compact vector spaces. In the case of a finite field, linear compactness is the same as the ordinary topological kind. One proof of this fact is that the category of finite dimensional spaces is self-dual and if two categories are dual, the inductive completion of one is dual to the projective completion of the other. For finite dimensional vector spaces, the inductive completion is vector spaces and the projective completion is linearly compact ones. Another example is the obvious duality between finite sets and finite boolean algebras that gives Stone duality on one hand and the duality between Set and CABA on the other, depending which one you complete which way. Most examples I am aware of of self-dualities are Chu categories (or chu categories). And if V_k is the category of k-vector spaces, then Chu(V_k,k) (an object is a pair of spaces and a bilinear pairing into k) is *-autonomous, as is chu(V_k,k) of separated extensional pairs. Peter's example is one of the very few *-autonomous categories I cannot relate to Chu. Complete (say inf) semi-lattices is another. Michael
First, let me say I have avoided contributing to this thread because I don't understand what Vaughan is asking. He knows, as well as anyone, since he put them on the map, about Chu categories. He knows about *-autonomous categories as well. So what is the question, really? The simplest answer to Mamuka's question is the duality between vector spaces (over any field, including the 2 element field) and linearly compact vector spaces. In the case of a finite field, linear compactness is the same as the ordinary topological kind. One proof of this fact is that the category of finite dimensional spaces is self-dual and if two categories are dual, the inductive completion of one is dual to the projective completion of the other. For finite dimensional vector spaces, the inductive completion is vector spaces and the projective completion is linearly compact ones. Another example is the obvious duality between finite sets and finite boolean algebras that gives Stone duality on one hand and the duality between Set and CABA on the other, depending which one you complete which way. Most examples I am aware of of self-dualities are Chu categories (or chu categories). And if V_k is the category of k-vector spaces, then Chu(V_k,k) (an object is a pair of spaces and a bilinear pairing into k) is *-autonomous, as is chu(V_k,k) of separated extensional pairs. Peter's example is one of the very few *-autonomous categories I cannot relate to Chu. Complete (say inf) semi-lattices is another. Michael
Peter Freyd writes on Pontrjagin duality. I would like to mention the generalisation given in (with P.J. HIGGINS and S.A. MORRIS), ``Countable products of lines and circles: their closed subgroups, quotients and duality properties'', {\em Math. Proc. Camb. Phil. Soc.} 78 (1975) 19-32. One point made is that a duality is not necessarily inherited by closed subgroups and Hausdorff quotients. If it is, it is called a strong duality. Then strong duality is inherited by closed subgroups and Hausdorff quotients! I have taught the classification of closed subgroups of R^n in an analysis course. It is a nice result, and the sums you can set use duality in a nice way - treatment borrowed from Bourbaki. Ronnie Brown
Next is Pontryagin's: the category of locally compact groups. The original Pontryagin duality easily generalizes: the category of locally compact modules over a given commutative ring is self-dual. (In the non-commutative case one also obtains a duality but not a self-duality -- unless, of course, the ring is self-dual.) A corollary is that the category of discrete left R-modules is dual to the category of compact right R-modules
Peter Freyd writes on Pontrjagin duality. I would like to mention the generalisation given in (with P.J. HIGGINS and S.A. MORRIS), ``Countable products of lines and circles: their closed subgroups, quotients and duality properties'', {\em Math. Proc. Camb. Phil. Soc.} 78 (1975) 19-32. One point made is that a duality is not necessarily inherited by closed subgroups and Hausdorff quotients. If it is, it is called a strong duality. Then strong duality is inherited by closed subgroups and Hausdorff quotients! I have taught the classification of closed subgroups of R^n in an analysis course. It is a nice result, and the sums you can set use duality in a nice way - treatment borrowed from Bourbaki. Ronnie Brown
Next is Pontryagin's: the category of locally compact groups. The original Pontryagin duality easily generalizes: the category of locally compact modules over a given commutative ring is self-dual. (In the non-commutative case one also obtains a duality but not a self-duality -- unless, of course, the ring is self-dual.) A corollary is that the category of discrete left R-modules is dual to the category of compact right R-modules
Michael Barr wrote:
First, let me say I have avoided contributing to this thread because I don't understand what Vaughan is asking. He knows, as well as anyone, since he put them on the map, about Chu categories. He knows about *-autonomous categories as well. So what is the question, really?
Duality is of necessity between categories, and involves associating an object (say an algebra or space) of one category with its dual in another, or in the same category in the self-dual case. Downstairs, i.e. in 2-CAT. By "categorifying duality" I meant a duality of 2-categories in which one associates an object (this time a category rather than an algebra) of one 2-category with its dual in another, with the functors being reversed (op) as opposed to the natural transformations (co). Upstairs, i.e. in 3-CAT. Regarding Chu, I was going to respond that the Chu construction works downstairs with categories (from my usual V=Set perspective), or at most V-categories, categories enriched in V, as objects of the 2-category V-CAT. However if the enriched Chu construction can be organized to allow V to be a 2-category, with Chu(V,k) then being a 3-category, maybe Mike is on to a promising approach (though it's not clear that's what he actually meant). It's an aspect of Chu spaces I know next to nothing about however. My first guess would be that it (moving Chu up into 3-CAT) ought to work fine, with the caveat that the simple notion of Stone topology as a totally disconnected compact Hausdorff topology would turn into the proverbial thousand flowers---there's far more room for such stuff in 3-CAT than 2-CAT. (Actually there's also a lot of unexplored such territory even just in ordinary Chu(Set,3).) Along those lines, Peter Johnstone's mention of quasi-injective toposes dual to continuous categories in his 1982 paper with Joyal is surely just scratching the surface of the possible permutations and combinations up there in 3-CAT. Peter's
example is one of the very few *-autonomous categories I cannot relate to Chu. Complete (say inf) semi-lattices is another.
Oh, but complete inf semilattices are one of the most elegant self-dualities of chupology. They embed in Chu(Set,2) as those biextensional Chu spaces (biextensional = no repeated rows or columns), of any cardinality, such that the set of rows is closed under arbitrary AND (think of them as bit vectors) and likewise for the set of columns. No other conditions. Using OR instead of AND for both rows and columns gives sup semilattices. But that was in my previous post, where I also mentioned that XOR in place of OR, at least for finite Chu spaces, embeds FinVct_{GF(2)}. In either case the symmetry of the conditions makes the self-duality immediate. All that is in the 1999 Coimbra notes I cited. Vaughan
Michael Barr wrote:
First, let me say I have avoided contributing to this thread because I don't understand what Vaughan is asking. He knows, as well as anyone, since he put them on the map, about Chu categories. He knows about *-autonomous categories as well. So what is the question, really?
Duality is of necessity between categories, and involves associating an object (say an algebra or space) of one category with its dual in another, or in the same category in the self-dual case. Downstairs, i.e. in 2-CAT. By "categorifying duality" I meant a duality of 2-categories in which one associates an object (this time a category rather than an algebra) of one 2-category with its dual in another, with the functors being reversed (op) as opposed to the natural transformations (co). Upstairs, i.e. in 3-CAT. Regarding Chu, I was going to respond that the Chu construction works downstairs with categories (from my usual V=Set perspective), or at most V-categories, categories enriched in V, as objects of the 2-category V-CAT. However if the enriched Chu construction can be organized to allow V to be a 2-category, with Chu(V,k) then being a 3-category, maybe Mike is on to a promising approach (though it's not clear that's what he actually meant). It's an aspect of Chu spaces I know next to nothing about however. My first guess would be that it (moving Chu up into 3-CAT) ought to work fine, with the caveat that the simple notion of Stone topology as a totally disconnected compact Hausdorff topology would turn into the proverbial thousand flowers---there's far more room for such stuff in 3-CAT than 2-CAT. (Actually there's also a lot of unexplored such territory even just in ordinary Chu(Set,3).) Along those lines, Peter Johnstone's mention of quasi-injective toposes dual to continuous categories in his 1982 paper with Joyal is surely just scratching the surface of the possible permutations and combinations up there in 3-CAT. Peter's
example is one of the very few *-autonomous categories I cannot relate to Chu. Complete (say inf) semi-lattices is another.
Oh, but complete inf semilattices are one of the most elegant self-dualities of chupology. They embed in Chu(Set,2) as those biextensional Chu spaces (biextensional = no repeated rows or columns), of any cardinality, such that the set of rows is closed under arbitrary AND (think of them as bit vectors) and likewise for the set of columns. No other conditions. Using OR instead of AND for both rows and columns gives sup semilattices. But that was in my previous post, where I also mentioned that XOR in place of OR, at least for finite Chu spaces, embeds FinVct_{GF(2)}. In either case the symmetry of the conditions makes the self-duality immediate. All that is in the 1999 Coimbra notes I cited. Vaughan
Hi -
First, let me say I have avoided contributing to this thread because I don't understand what Vaughan is asking.
He asked if dualities categorify. I guess he meant something like this: There are lots of interesting examples of a pair of categories C,D together with an object c in C and an object d in D such that hom(-,c): C -> D and hom(-,d): D -> C are part of an equivalence of categories. In the nicest examples, c and d are in some sense the same mathematical entity regarded as living in two different categories - a "schizophrenic object", in the words of Harold Simmons. So, can we find equally nice examples where C and D are instead 2-categories? In particular, can we find examples where C and D are 2-categorical generalizations of the 1-categorical examples we already know? In particular, he suggested taking the example where C is the category of finite distributive lattices and finding an analogous example where C is the 2-category of (maybe finite, in some sense?) distributive categories. For more on "schizophrenic objects", Peter Johnstone's review of Clark and Davies' "Natural dualities for the working algebraist" makes good reading: http://north.ecc.edu/alsani/ct99-00(8-12)/msg00116.html
Hi -
First, let me say I have avoided contributing to this thread because I don't understand what Vaughan is asking.
He asked if dualities categorify. I guess he meant something like this: There are lots of interesting examples of a pair of categories C,D together with an object c in C and an object d in D such that hom(-,c): C -> D and hom(-,d): D -> C are part of an equivalence of categories. In the nicest examples, c and d are in some sense the same mathematical entity regarded as living in two different categories - a "schizophrenic object", in the words of Harold Simmons. So, can we find equally nice examples where C and D are instead 2-categories? In particular, can we find examples where C and D are 2-categorical generalizations of the 1-categorical examples we already know? In particular, he suggested taking the example where C is the category of finite distributive lattices and finding an analogous example where C is the 2-category of (maybe finite, in some sense?) distributive categories. For more on "schizophrenic objects", Peter Johnstone's review of Clark and Davies' "Natural dualities for the working algebraist" makes good reading: http://north.ecc.edu/alsani/ct99-00(8-12)/msg00116.html
John Baez wrote:
[...] So, can we find equally nice examples [of representable dualities] where C and D are instead 2-categories? In particular, can we find examples where C and D are 2-categorical generalizations of the 1-categorical examples we already know?
In particular, he suggested taking the example where C is the category of finite distributive lattices and finding an analogous example where C is the 2-category of (maybe finite, in some sense?) distributive categories.
Enrico Vitale just sent me the answer for that one: C = the 2-category of idempotent-closed categories, D = the 2-category of presheaf categories. This categorifies C = Pos, D = StoneDLat by passing from 2 to Set as the enriching autonomous category (so in that sense one could say we were in 3-CAT all along, though presumably only trivially so by virtue of only having identity modifications when V = 2, I think). Although I'd heard the phrase "Morita equivalence" many times over the years, it meant nothing to me until recently when Bill Lawvere was talking about graphs as presheaves on the monoid consisting of the three monotone functions on the ordinal 2 and I finally woke up to the connection between splitting the two idempotents and ME (the equivalence, not the condition). The idempotent closure of that monoid, meaning the result of splitting the idempotents, is just the initial segment of Delta of length 2, aka the ordinals 1 and 2 and their monotone functions. The impact on the models, here graphs, is that splitting the idempotents results in giving the self-loops that were playing the role of vertices their own datatype V, as coded by the ordinal 1. This new category of graphs is not the old one as its objects now have vertices in their own right, but it is equivalent to the old one. {2} and {1,2}, each made a category with respectively 3 and 7 monotone functions, are Morita equivalent: they have equivalent idempotent closures, and homming into Set maps them to equivalent categories, the iff that makes Morita equivalence important. ME is the kernel of idempotent closure, which is a categorification, with Set in place of 2, of the functor Ord --> Pos (Ord the category of preordered sets, Pos of posets) that collapses the cliques. The reason there is no representable duality between Ord and a suitable cousin of StoneDLat (FinOrd and FinDLat for the Stonaphobes) is that preorders are equivalent to posets and the Yoneda embedding taking elements of P to primes in 2^P, while fully faithful, is only good up to equivalence. (The homfunctor being transposed here is the order <= : P\op x P --> 2.) The categorification of this, meaning in this case not the passage from 2-CAT to 3-CAT but from enrichment in 2 to enrichment in Set, still has to deal with equivalence in the same way (though here it goes with the territory and so is less noticeable than back down at Ord vs. Pos where we tend to think isomorphism rather than equivalence). But Hom: C\op x C --> Set is not itself an equivalence but only a "retract that retracts retracts", the essence of Morita equivalence (a dual of Freyd's "trivial for a trivial reason"?). In order to take the "log to the base Set" we can't really "retract all the retracts" because we may need to keep some of them around but then which ones (like picking a dense subset of a continuum: which subset?). We can however put them all in, which is to say, split all the idempotents, so we do that in order to get a normal form. The rest of this duality is then the triviality that the internal hom of CAT is contravariant in its first argument. Morita equivalence is the only thing to be worried about. Proposition 5.28 of Kelly's "Basic Concepts of Enriched Category Theory", namely that Cauchy completion (Kelly's name for the enriched counterpart of idempotent closure) permits taking logs to any autonomous base V, then produces a proper class of dualities, one for every autonomous V. In particular we can recover Pos\op ~ StoneDLat by taking V = 2. (Pos and Ord, preordered sets, while not equivalent any more than CAT and its subcategory of idempotent-closed categories are equivalent, have equivalent objects which is all we need ask of a duality.) There are two "good" 3-object V's, the non-Heyting one of which enriches the "prossets" that Haim Gaifman and I wrote about in LICS'87, so these have their dual objects in the same way, by homming into 3, a construct I talked about incomprehensibly at the Newton Institute meeting on geometry in computation some years ago, not recognizing that it was a duality. Metric spaces, another duality there. And so on. But then every such duality has its subdualities, for example Set\op ~ CABA as a subduality of Pos\op ~ StoneDLat, so a great many more dualities there. Enrico also mentioned the Gabriel-Ulmer duality for locally finitely presentable categories, and the Adamek-Lawvere-Rosicky duality for varieties. Are these in addition to the above or can they be recovered from them? Likewise for the duality Peter Johnstone mentioned? Vaughan Pratt
John Baez wrote:
[...] So, can we find equally nice examples [of representable dualities] where C and D are instead 2-categories? In particular, can we find examples where C and D are 2-categorical generalizations of the 1-categorical examples we already know?
In particular, he suggested taking the example where C is the category of finite distributive lattices and finding an analogous example where C is the 2-category of (maybe finite, in some sense?) distributive categories.
Enrico Vitale just sent me the answer for that one: C = the 2-category of idempotent-closed categories, D = the 2-category of presheaf categories. This categorifies C = Pos, D = StoneDLat by passing from 2 to Set as the enriching autonomous category (so in that sense one could say we were in 3-CAT all along, though presumably only trivially so by virtue of only having identity modifications when V = 2, I think). Although I'd heard the phrase "Morita equivalence" many times over the years, it meant nothing to me until recently when Bill Lawvere was talking about graphs as presheaves on the monoid consisting of the three monotone functions on the ordinal 2 and I finally woke up to the connection between splitting the two idempotents and ME (the equivalence, not the condition). The idempotent closure of that monoid, meaning the result of splitting the idempotents, is just the initial segment of Delta of length 2, aka the ordinals 1 and 2 and their monotone functions. The impact on the models, here graphs, is that splitting the idempotents results in giving the self-loops that were playing the role of vertices their own datatype V, as coded by the ordinal 1. This new category of graphs is not the old one as its objects now have vertices in their own right, but it is equivalent to the old one. {2} and {1,2}, each made a category with respectively 3 and 7 monotone functions, are Morita equivalent: they have equivalent idempotent closures, and homming into Set maps them to equivalent categories, the iff that makes Morita equivalence important. ME is the kernel of idempotent closure, which is a categorification, with Set in place of 2, of the functor Ord --> Pos (Ord the category of preordered sets, Pos of posets) that collapses the cliques. The reason there is no representable duality between Ord and a suitable cousin of StoneDLat (FinOrd and FinDLat for the Stonaphobes) is that preorders are equivalent to posets and the Yoneda embedding taking elements of P to primes in 2^P, while fully faithful, is only good up to equivalence. (The homfunctor being transposed here is the order <= : P\op x P --> 2.) The categorification of this, meaning in this case not the passage from 2-CAT to 3-CAT but from enrichment in 2 to enrichment in Set, still has to deal with equivalence in the same way (though here it goes with the territory and so is less noticeable than back down at Ord vs. Pos where we tend to think isomorphism rather than equivalence). But Hom: C\op x C --> Set is not itself an equivalence but only a "retract that retracts retracts", the essence of Morita equivalence (a dual of Freyd's "trivial for a trivial reason"?). In order to take the "log to the base Set" we can't really "retract all the retracts" because we may need to keep some of them around but then which ones (like picking a dense subset of a continuum: which subset?). We can however put them all in, which is to say, split all the idempotents, so we do that in order to get a normal form. The rest of this duality is then the triviality that the internal hom of CAT is contravariant in its first argument. Morita equivalence is the only thing to be worried about. Proposition 5.28 of Kelly's "Basic Concepts of Enriched Category Theory", namely that Cauchy completion (Kelly's name for the enriched counterpart of idempotent closure) permits taking logs to any autonomous base V, then produces a proper class of dualities, one for every autonomous V. In particular we can recover Pos\op ~ StoneDLat by taking V = 2. (Pos and Ord, preordered sets, while not equivalent any more than CAT and its subcategory of idempotent-closed categories are equivalent, have equivalent objects which is all we need ask of a duality.) There are two "good" 3-object V's, the non-Heyting one of which enriches the "prossets" that Haim Gaifman and I wrote about in LICS'87, so these have their dual objects in the same way, by homming into 3, a construct I talked about incomprehensibly at the Newton Institute meeting on geometry in computation some years ago, not recognizing that it was a duality. Metric spaces, another duality there. And so on. But then every such duality has its subdualities, for example Set\op ~ CABA as a subduality of Pos\op ~ StoneDLat, so a great many more dualities there. Enrico also mentioned the Gabriel-Ulmer duality for locally finitely presentable categories, and the Adamek-Lawvere-Rosicky duality for varieties. Are these in addition to the above or can they be recovered from them? Likewise for the duality Peter Johnstone mentioned? Vaughan Pratt
participants (7)
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John Baez -
K C H Mackenzie -
Mamuka Jibladze -
Michael Barr -
Peter Freyd -
Ronnie Brown -
Vaughan Pratt