I would like to add a few thoughts to the "evil" discussion. My 30+ years involvement with indexed categories have led me to the following understanding. There are two kinds of categories, small and large (surprise!). But the difference is not mainly one of size. Rather it's how well we can pin down the objects. The distinction between sets and classes is often thought of in terms of size but Russell's problem with the set of all sets was not one of size but rather of the nature of sets. Once you think you have the set of all sets, you can construct another set which you had missed. I.e. the notion is changing, slippery. There are set theories where you can have a subclass of a set which is not a set (c.f. Vopenka, e.g.) Smallness is more a question of representability: a functor may fail to be representable because it's too big (no solution set) or, more often, because it's badly behaved (doesn't preserve products, say). Subfunctors of representables are not usually representable. In our work on indexed categories, Schumacher and I had tried to treat this question by considering categories equipped with a groupoid of isomorphisms, which we called *canonical*, and then consider functors defined up to canonical isomorphism. In small categories only identities were canonical whereas in large categories, all isomorphisms were canonical. Our ideas were a bit naive and not well developed and earned us some ridicule, so we quietly stopped talking about it. Recently, Makkai developed an extensive theory of functors defined up to isomorphisms, FOLDS, but did not consider the possibility of specifying which isomorphisms ahead of time, so small categories were not included. When I used to teach category theory, before Dalhousie made me chuck my chalk chuck, I would tell students there were two kinds of categories in practice. Large ones which are categories of structures, corresponding to various branches of mathematics we wished to study. These categories supported various universal constructions, all defined up to isomorphism. Two large categories are considered to be the same if they are equivalent. It was considered impolite to ask if two objects were equal. Then there are the small categories which are used to study the large ones. These are syntactic in nature. For these, one can't expect the kinds of universal constructions that large categories have, but now it's okay, even necessary, to consider equality between objects. I went on to say that there were then four kinds of functors. Functors between large categories were to be thought of as constructions of one structure from another, e.g. the group ring. Functors between small categories were interpretations of one theory in another or reindexing or rearranging. Functors from small to large categories were models or diagrams in the large one. These kinds of functors are perhaps the most important of the four, although this may be debatable. The fourth kind, from large to small are rarer. They can be thought of as gradings or partitions of the large category. Well, after these ramblings, perhaps my message is lost. So here it is: Small categories -> equality of objects okay Large categories -> equality of objects not okay Small is beautiful, not evil. Bob [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
Robert Pare wrote:
Then there are the small categories which are used to study the large ones. These are syntactic in nature.
Don't get me started. Oops, too late.
For these, one can't expect the kinds of universal constructions that large categories have,
Not following. FinSet is an essentially small category, what do you mean that it doesn't enjoy universal constructions? It's even a topos. Then there are the categories enriched in small categories, again subject to cardinality restrictions, which too are perfectly capable of enjoying universal constructions.
but now it's okay, even necessary, to consider equality between objects.
For small as opposed to essentially small categories, yes in some cases. But consider the category of ordinals truncated at say beth_2, certainly a small category when the morphisms are the inequalities. Are you comfortable defining equality on the objects of this category? (PTJ would correctly accuse me of being inconsistent on this point.)
Well, after these ramblings, perhaps my message is lost. So here it is: Small categories -> equality of objects okay Large categories -> equality of objects not okay
I hate to seem argumentative, Bob, but this can't possibly be the difference between small and large.
Small is beautiful, not evil.
Agreed, so long as this is not at the expense of large. Nice to be able to close on a note of consensus. :) Vaughan [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
I find Bob Pare posting on large versus small super interesting, and the first contribution since Russell (at the origin of Grothendieck Universes) with really new and radical considerations. Of course, Bob's posting is rather misterious, makes you think, but it is impossible to analyze technicaly. It will be impossible also to explain it more by writting. Needs personal disscussion. Bob, what do you mean by "this and that ?", after the answer: Well, then it is so !! ... but still do not understand what you really say ..., and etc etc ... Cheers Bob Eduardo. [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
Thanks to all who replied to my posting either privately or on Categories. I'd like to clarify a few points. Ross's interpretation of what I meant, and which Vaughan agrees with, is not what I was trying to say. It is part of the picture but not the one I was promoting. Small categories play a different role in category theory than large categories. Small categories are used for indexing things. Large categories consist of the things we are indexing. Okay, syntactic was not the right word. Combinatorial might be better although that has finiteness overtones. Strict? Well, I'll just stick with small. Small is not the same as essentially small. As Jeff Egger pointed out, the category of finite sets is not small. It's not even clear what this category is. The ZFCists would say that for each set A we get a finite set {A} and another {{A}}, and Barwise might even wonder if these last two are distinct. But I'm digressing. Of course every small category can be considered as a large one (perhaps large isn't the right word either). Then two equivalent ones would be considered "the same". I don't think that this is how the working categorician works. Sometimes equivalent categories are "the same" and sometimes not. An equivalence relation is a category but we would lose something (everything?) if we identified it with equality in the quotient set. So I'm saying that, as a matter of "categorical hygiene", we could be a bit more explicit about what kind "equivalence" we are allowing. And without changing mathematical practice, I'm advocating that "small" should imply that it's okay to talk of equality of objects. As usual it helps put things in relief to generalize them. Consider category theory in a world based on a topos S. A small category is a category object in S. A large category like S or Group(S) is an indexed category, given by a pseudo-functor S^op -> Cat, or a fibration over S if you prefer. A small category C gives a large one by homming. But something is lost in the process. The object functor Cat -> Set is not a 2-functor so if you compose it with a pseudo-functor S^op -> Cat you get an assignment that doesn't preserve composition in any sense. So a general indexed category doesn't have a discrete category of objects. What you can do is take the groupoid of isomorphisms 2-functor Cat -> Gpd, so that a large has a groupoid of objects, thinking of a groupoid as a set with a 2-equality on it. An indexed category gotten from a small category gives an actual functor S^op -> Cat, so now you can compose with Ob : Cat -> Set so that a small category has a discrete groupoid of objects. And that's where the idea of considering categories with a specified "equality groupoid" of isomorphisms, and equivalences defined to have the isomorphisms from the corresponding groupoids. Small categories have only identities, so equivalence means isomorphism, and large categories have all isos. At the time there were no good examples of intermediate groupoids (just products of small and large categories and the like) but Hilbert spaces with unitary isos is a good one. Well this is my second posting in a week bringing my life-time total to three! Seems a good place to stop. Best wishes for the new year. May your happiness be large and your disappointments small. Bob [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
This is in reply to a posting by Bob Pare. Dear Bob: You wrote: “As usual it helps put things in relief to generalize them. Consider category theory in a world based on a topos S. A small category is a category object in S. A large category like S or Group(S) is an indexed category, given by a pseudo-functor S^op -> Cat, or a fibration over S if you prefer. A small category C gives a large one by homming. But something is lost in the process.” That is precisely how I always thought of small versus large relative to a topos S (the universe of discourse). Further, a large category A (fibered over S) "is" small if and only if the corresponding pseudo-functor A: S^op--> Cat is representable. As you say, a small category C (internal to the topos S) may always be considered as a large one (via its externalization [C]: S^op ->Cat). These considerations came up in my and our joint work on stacks (Cahiers, 1979), and more recently in my work with Claudio Hermida on 2-stacks. The notions of a stack and of a 2-stack are taken to be intrinsic to a topos S, that is, relative to the topology of its (regular) epimorphisms, as introduced by Lawvere 1974. Dimension 1. A small category C (internal to a topos S) always has a stack completion when regarded as a large category via its externalization [C]. The stack completion of C is given by yon: [C] -> [C]*= LocRep(S^(C^op)), a weak equivalence functor. Applied to a groupoid G, this gives the classification theorem for G-torsors (Diaconescu 1975). An axiom of stack completions (ASC) in its rough form says that S satisfies it if for ever small category C in S, the fibration LocRep(S^{C^op}) is representable by a category C* so that [C]* and [C*] are equivalent as fibrations. As shown by Joyal and Tierney (1991) by means of Quillen model structures, but also by a general argument involving the existence of a set of generators, (Duskin 1980), Grothendieck toposes satisfy (ASC). Dimension 2. The 2-dimensional analogue of the above set-up was discussed in my lecture at CT 2008 (joint work with Claudio Hermida). Our main result is that, for a topos S satisfying (ASC), any 2-category 1-stack C in S, regarded as a 2-fibration, has a 2-stack completion, to wit yon: [C] -> [C]*=LocRep(Stack^(C^op)), a weak 2-equivalence 2-functor. Applied to a 2-gerbe G and suitably interpreted, this gives a classification theorem for G-2-torsors. The validity of an appropriately formulated (ASC)^2 for a Grothendieck topos S (see slides for my lecture at CT 2008) is true by a general argument involving the existence of a set of generators. What is still missing, however, is a construction of a small 2-category 1-stack C*representing the 2-fibration [C]*= LocRep(Stack^(C^op)) in the case of a Grothendieck topos S. The Quillen model structure on 2-Cat given by Lack 2002 is not suitable for this purpose. Dimension n. Analogue results in higher dimensions are less tractable but a pattern emerges from the passage from dimension 1 to dimension 2. Remark (concerning a previous posting of yours): Although using the S-indexed versions of fibrations over S is useful, just as presentations of groups are useful, the entire discussion of stacks can be carried out at the level of fibrations (ditto 2-fibrations). I fail to understand what is all the fuss about the use of S-indexed categories if taken in that spirit. Certainly not deserving ridicule! You also wrote: “Well this is my second posting in a week bringing my life-time total to three! Seems a good place to stop.” To me, that is a poor reason to give. This forum could certainly profit from your interventions, but I understand your qualms, as I myself quite often abstain from an urge to intervene. Happy New Year to you and everyone, Marta ************************************************ Marta Bunge Professor Emerita Dept of Mathematics and Statistics McGill UniversityBurnside Hall, Office 1005 805 Sherbrooke St. West Montreal, QC, Canada H3A 2K6 Office: (514) 398-3810/3800 Home: (514) 935-3618 marta.bunge@mcgill.ca http://www.math.mcgill.ca/~bunge/ ************************************************ [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
On 03/01/2010, at 6:57 PM, Vaughan Pratt wrote:
For these, one can't expect the kinds of universal constructions that large categories have,
Not following. FinSet is an essentially small category, what do you mean that it doesn't enjoy universal constructions? It's even a topos.
Dear Vaughan Part of what Bob Paré was arguing, I believe, was that we should be flexible (pun intended) about what "small" means. If "small" means "finite" then FinSet is not "essentially small". Also, "small" could mean "no more than one element". Ross [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
Ross Street wrote:
Part of what Bob Paré was arguing, I believe, was that we should be flexible (pun intended) about what "small" means. If "small" means "finite" then FinSet is not "essentially small". Also, "small" could mean "no more than one element".
Thanks, Ross. Hopefully Bob will phrase it that way next time. ;) If 2 is the usual symmetric monoidal closed category with objects 0 and 1 and only non-identity morphism 0 --> 1, then Chu(2,1) has four objects while Chu(2,0) has only three, but both are self-dual. The CEO of search engine company Cuil (Old Irish for knowledge) had finite categories of this kind in her 1997 Ph.D. thesis. What got me started on my previous message was that Bob was calling these "syntactic" when to me they were semantic. If by "syntactic" he meant "finite," or more generally less than some specified ordinal, then I have no problem with that, other than that I'd prefer he be specific about the ordinal rather than vaguely saying "syntactic." Vaughan [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
Bob Pare' made the excellent point that not only size but quality is relevant. I definitely agree with the spirit of his remarks. Bob happens to have used in passing the term 'syntactic'. For clarity, the use of that term needs to be sharpened to avoid misunderstanding. Actually, the term 'syntax' refers NOT to small categories such as algebraic theories or rings, but rather to their PRESENTATION by signatures or by polynomial generators, et cetera. The process of presentation is an adjoint pair quite distinct from the semantical adjoint pair: both adjoint pairs have a category of theories or of rings in common but are otherwise quite independent. In particular, syntax is NOT the adjoint of semantics. Cratylus, Chomsky, and their 21st century followers can be refuted by looking soberly at the actual practice of mathematics (wherein the construction of sequences of words and of diagrams is pursued with great care for the purpose of communication. That syntax is only remotely dependent on the structure of the content that is to be communicated). Both of the functors ?--------------> theories -------------->Large categories Syntax Semantics are needed. The domain category of the first can be chosen in various useful ways: sketches or diagrams of signatures et cetera. Happy new year! Bill On Fri 01/01/10 9:48 AM , pare@mathstat.dal.ca (Robert Pare) sent:
I would like to add a few thoughts to the "evil" discussion.
My 30+ years involvement with indexed categories have led me to the following understanding. There are two kinds of categories, small and large (surprise!). But the difference is not mainly one of size. Rather it's how well we can pin down the objects. The distinction between sets and classes is often thought of in terms of size but Russell's problem with the set of all sets was not one of size but rather of the nature of sets. Once you think you have the set of all sets, you can construct another set which you had missed. I.e. the notion is changing, slippery. There are set theories where you can have a subclass of a set which is not a set (c.f. Vopenka, e.g.)Smallness is more a question of representability: a functor may fail to be representable because it's too big (no solution set) or, more often, because it's badly behaved (doesn't preserve products, say). Subfunctorsof representables are not usually representable.
In our work on indexed categories, Schumacher and I had tried to treat this question by considering categories equipped with a groupoid of isomorphisms, which we called *canonical*, and then consider functors defined up to canonical isomorphism. In small categories only identitieswere canonical whereas in large categories, all isomorphisms were canonical.Our ideas were a bit naive and not well developed and earned us some ridicule,so we quietly stopped talking about it. Recently, Makkai developed an extensive theory of functors defined up to isomorphisms, FOLDS, but did not consider the possibility of specifying which isomorphisms ahead of time, so small categories were not included.
When I used to teach category theory, before Dalhousie made me chuck my chalk chuck, I would tell students there were two kinds of categories inpractice. Large ones which are categories of structures, corresponding tovarious branches of mathematics we wished to study. These categories supported various universal constructions, all defined up to isomorphism.Two large categories are considered to be the same if they are equivalent.It was considered impolite to ask if two objects were equal. Then there are the small categories which are used to study the large ones. These are syntactic in nature. For these, one can't expect the kinds of universal constructions that large categories have, but now it's okay, even necessary, to consider equality between objects. I went on to say that there were then four kinds of functors. Functors between large categorieswere to be thought of as constructions of one structure from another, e.g.the group ring. Functors between small categories were interpretations ofone theory in another or reindexing or rearranging. Functors from small to large categories were models or diagrams in the large one. These kindsof functors are perhaps the most important of the four, although this maybe debatable. The fourth kind, from large to small are rarer. They can be thought of as gradings or partitions of the large category.
Well, after these ramblings, perhaps my message is lost. So here it is: Small categories -> equality of objects okay Large categories -> equality of objects not okay Small is beautiful, not evil.
Bob
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
In addition, a key property of semantic categories (as opposed to syntactic) is their concreteness, i.e., their objects have carriers. Commutativity with the forgetful functor is essential for results stating equivalence of syntactic and semantic constructs. Zinovy On Tue, Jan 5, 2010 at 12:31 PM, F William Lawvere <wlawvere@buffalo.edu> wrote:
Bob Pare' made the excellent point that not only size but quality is relevant. I definitely agree with the spirit of his remarks.
Bob happens to have used in passing the term 'syntactic'. For clarity, the use of that term needs to be sharpened to avoid misunderstanding.
Actually, the term 'syntax' refers NOT to small categories such as algebraic theories or rings, but rather to their PRESENTATION by signatures or by polynomial generators, et cetera. The process of presentation is an adjoint pair quite distinct from the semantical adjoint pair: both adjoint pairs have a category of theories or of rings in common but are otherwise quite independent.
In particular, syntax is NOT the adjoint of semantics. Cratylus, Chomsky, and their 21st century followers can be refuted by looking soberly at the actual practice of mathematics (wherein the construction of sequences of words and of diagrams is pursued with great care for the purpose of communication. That syntax is only remotely dependent on the structure of the content that is to be communicated).
Both of the functors
?--------------> theories -------------->Large categories Syntax Semantics
are needed. The domain category of the first can be chosen in various useful ways: sketches or diagrams of signatures et cetera.
Happy new year!
Bill
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
Dear categorists, The question for me is not : small or large categories, but small or large structures. The Bourbaki's time was the time of small structures (groups, monoïds, rings, spaces, etc), the "categorical time" is the time of structures of structures,i.e. large structures (groupoids, complete categories, abelian catégories, topos, etc.). The bridge beetwen the firsts and the seconds is the Yoneda lemma, that is to say the introduction of logic, thus of the only true evil category : the category of sets. All the other large categorical stuctures introduced by categorists are deduced from category Set, by abstraction, generalisations, constructions or restrictions. In fact "category theory" is an (wonderfull but) inappropriate name (here the word category is only an important and historical keyword): the reason is that a lot of data (limits, classifiant object, etc) appear as properties because of their universal properties (unicity up to isomorphism). But "category theory" is an illusion, nobody studies seriouly the categorical structures (I don't know any structure theorem on the categorical structures without additional properties). That is my starting point of reflexion on this subject. I think there is a true theory of the small categories, but it is not yet born, if it must ever exist. This theory should be, not only for the 1-categories, but also for the n and omega-categories (strict or not possibly). And, for me, the Yoneda lemma is an important tool (but only a tool). Such a theory may be eventually important for the computer science (particularly for the formal languages). My best wishes for the new year, Albert burroni [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
I'm not sure I understand this
In particular, syntax is NOT the adjoint of semantics. Cratylus, Chomsky, and their 21st century followers can be refuted by looking soberly at the actual practice of mathematics (wherein the construction of sequences of words and of diagrams is pursued with great care for the purpose of communication. That syntax is only remotely dependent on the structure of the content that is to be communicated).
Both of the functors
?--------------> theories -------------->Large categories Syntax Semantics
are needed. The domain category of the first can be chosen in various useful ways: sketches or diagrams of signatures et cetera.
Do you mean that if we choose some kind of sketches for the domain category then theories are a reflective subcategory, more or less the 'definitionally closed' sketches? Then a presentation of a theory T would be (up to isomorphism) any unit arrow of the adjunction with T as codomain? best, Colin [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
Isn't there something "evil" in the definition of dual category itself? Claudio [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
Dear Colleagues, I have not quite absorbed all the email on this yet, so may be repeating something already said. But perhaps it would be helpful to mention that, in regard to questions like this, I have found enriched categories helpful: consider either 1 the functor category [->,Set] (an object is a pair of sets X and Y and a function from X to Y) or 2 the category Sub(Set) (an object is a set X together with a subset X', and a map from (X,X') to (Y,Y') is a function from X to Y for which the image of X' lies in Y' These categories, especially the first, both have the properties one typically seeks for a V in studying V-categories. Spelling out what a V-category is in the second case yields a category C with a subcategory for which the inclusion is the identity on objects. Happy New Year to all, John. Quoting Robert Pare <pare@mathstat.dal.ca>:
I would like to add a few thoughts to the "evil" discussion.
My 30+ years involvement with indexed categories have led me to the following understanding. There are two kinds of categories, small and large (surprise!). But the difference is not mainly one of size. Rather it's how well we can pin down the objects. The distinction between sets and classes is often thought of in terms of size but Russell's problem with the set of all sets was not one of size but rather of the nature of sets. Once you think you have the set of all sets, you can construct another set which you had missed. I.e. the notion is changing, slippery. There are set theories where you can have a subclass of a set which is not a set (c.f. Vopenka, e.g.) Smallness is more a question of representability: a functor may fail to be representable because it's too big (no solution set) or, more often, because it's badly behaved (doesn't preserve products, say). Subfunctors of representables are not usually representable.
... [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
A few little comments on "small is beautiful" 1) The nice thing about small cats is that externalizing them gives rise to a split fibration and that allows one to speak about equality of objects. But if we have got a split fibration P : XX -> BB it may be considered as a small cat over \widehat{BB} = Set^{BB^op} (for Set big enough). To my knowledge this observation is due to Jean B'enabou and also found its way into Bart Jacob's book. 2) Identifying small with representable seems to be an idea going back to Grothendieck already (as told to me by Jean B'enabou and taken up by him). Namely Grothendieck's notion of representable morphism in \widehat{BB} captures the notion of a family of small things indexed by a possibly large index object (arbitrary presheaf). The identification of small as representable lies at the heart of B'enabou's definitions of the properties locally small and well powered for fibrations. In this sense the definition of n elementary topos also amounts to a smallness condition: namely as a category EE with finite limits such that its fundamental fibration EE^2 -> E is well powered. 3) Use of the idea "small is representable" has been made in Algebraic Set Theory (AST) in the formulation of Awodey, Simpson and collaborators (see www.phil.cmu.edu/projects/ast/ for more information). There starting from a topos EE or (when working "predicatively") from a locally cartesian closed pretopos EE one considers the topos Sh(EE) of sheaves over EE w.r.t. the coherent, i.e. finite cover topology. Sh(EE) is thought of a category of classes and the full subcat of representables as the full subcat of sets. But Sh(EE) is a bit too large because for objects X in Sh(EE) the diagonal \delta_X (equality on X) need not be a representable morphism (and well behaved predicates should be since otherwise separation would lead out of sets). Thus instead of Sh(EE) one considers the full subcategory Idl(EE) of Sh(EE) on those separated objects, i.e. those where the diagonal is a representable mono). Notice that separated for a presheaf over EE (a split discrete fibration over EE) means that equality is definable in the sense of B'enabou. It was suggested to Awodey et.al. by Joyal that the separated objects in Sh(EE) can be characterized as those presheaves over EE which can be obtained as an "ideal colimit" of representable objects ("ideal" meaning directed diagram of monos). A further nice characterization of X being in Idl(EE) is that the image of a map y(A) -> X (taken in Sh(EE)) is again representable. Now working in Idl(EE) one can define for X in Sh(EE) its "class of subsets" P(X) as follows: P(X)(Y) is the collection of subobjects of y(I) x X whose source is representable, i.e. monos of the form y(J) >--> y(I) x X. By iterating P one obtains fixpoints (not representable) of P which serve as universes for interpreting appropriately weak set theories. As already mentioned by Bob the set theorist Vopenka wrote a lot about set theories where subclasses of sets needn't be sets again. He called "semiset" a subclass of a set which is not a set itself. Although Vopenka doesn't emphasize this point he is working in an ultra power extension of V_\omega (because he wants the negation of the Infinity axiom to hold) and there subclasses of a set need not be in the ultra power extension. As I have heard (and seen some notes of talks by him) B'enabou quite some time ago worked on a Nonstandard Theory of Classes which relates to Nelson's Internal Set Theory like GBN to ZFC. I am vaguely aware of extensive work by NSA people on nonstandard class and set theories motivated by similar ideas (but this was much later) but they have a somewhat richer ontology. There are 2 books to mention in this context Anatoly G. Kusraev, E. I. Gordon, S. S. Kutateladze "Infinitesimal Analysis" Kluwer Academic Pub (2002) V. Kanovei, M. Reeken "Nonstandard Analysis, Axiomatically" Springer 2004 Both views have in common that "set" has nothing to do with size but rather with "being definable in a reasonable sense" (the collection of standard elements of a set is typically not a set because "standard" is not a clear cut notion). This was concealed by early axiomatizations of class theory. I wonder now whether these two notions of "smallness" (better called "sethood") can be reconciled more precisely. -- Thomas [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
Dear Bob, This reminds me of a distinction that arises topologically. In a discrete space, equality is "OK" in the the sense that the diagonal is an open subspace of the square. This works fine also for point-free spaces, by a result in the Joyal--Tierney monograph. (A space X is discrete iff all finite diagonals X -> X^n are open maps.) That suggests working more generally with (point-free) topological categories: the collections of objects and morphisms are spaces. Then the ones with object equality OK are the small ones, where the spaces of objects and morphisms are discrete, i.e. sets. At first sight this doesn't help us with large categories. But actually we go a long way if we generalize spaces a la Grothendieck. For example, the "topologized version of the class of sets" is then the object classifier S[U], a Grothendieck topos whose points are sets. That may look like a clumsy way of replacing something unbeautiful (the large category of sets) by something even worse. But in fact S[U] can be presented in a way that doesn't presuppose knowledge of all of Set, by using a site on a small category of finite sets, whose objects are the natural numbers and whose morphisms correspond to functions between the finite cardinals. Many large categories, including categories of structures such as Group, Ring etc., can be replaced in this way by topical categories, whose collections of objects and morphisms are toposes and whose domain, codomain etc. functors are geometric morphisms. Topical functors, again, are made from geometric morphisms, which imposes continuity conditions on the functors (e.g. preservation of filtered colimits, analogous to Scott continuity). In effect this revises the notion of "class", replacing formulae in set theory by theories in geometric logic. Example: In the topical category of groups, the (generalized) space of objects is the group classifier S[Gp] while the space of morphisms is S[GpHom], the classifier for pairs of groups with homomorphism between them. Similarly for rings we have S[Rg] and S[RgHom]. The group ring construction is then given by geometric morphisms S[Gp] -> S[Rg] and S[GpHom] -> S[RgHom], satisfying the functoriality conditions. (Actually, in this example the morphism parts are given canonically once we have S[Gp] -> S[Rg].) Best wishes, Steve Vickers. Robert Pare wrote:
... Well, after these ramblings, perhaps my message is lost. So here it is: Small categories -> equality of objects okay Large categories -> equality of objects not okay Small is beautiful, not evil.
Bob
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
I largely agree with Bob Pare's New Year's Day posting, and disagree with those who disagree with him, although Albert Burroni is right to generalise from categories to structures.
... there were two kinds of categories in practice. [Large and small.]
I went on to say that there were then four kinds of functors. - Functors between large categories were to be thought of as constructions of one structure from another, e.g. the group ring. - Functors between small categories were interpretations of one theory in another or reindexing or rearranging. - Functors from small to large categories were models or diagrams in the large one. These kinds of functors are perhaps the most important of the four, although this may be debatable. - The fourth kind, from large to small are rarer. They can be thought of as gradings or partitions of the large category.
Small categories -> equality of objects okay Large categories -> equality of objects not okay Small is beautiful, not evil.
I would, however, like to substitute INTERNAL for SMALL. Tradionally, a "small" category is one with a "set" of objects and morphisms. However, since set theory is the problem and not the solution to the foundations of mathematics, we can avoid this by translating "set" into "object of a topos" and more generally a lex category (one with pullbacks and a terminal object") or an arithmetic universe. (I believe that there was a development of "locally internal" instead of "locally small" categories by some French categorists in the 1970s -- Burroni again maybe?.) Having done this, I would like to rescue the word "set" from the set theorists, and use it to mean "object of a topos", lex category or arithmetic universe. Vaughan Pratt's remark that "FinSet is an essentially small category" is entirely consistent with what Bob said. FinSet is a LARGE category for which there is a SMALL category "finset" (whose set of objects is N) and a WEAK EQUIVALENCE, ie a full and faithful functor finset->FinSet that is essentially surjective. This is one of the "most important" functors, according to Bob. An internal category (and more generally internal structure) of course inherits equality from its carrier, which is by definition a set (object of a topos or lex category). A large category or external structure has no carrier, and therefore no notion of equality, as Bob said. Another way of seeing the large/small or external/internal distinction is that a small or internal structure gives a NAME to the external one, which we may conversely call the SEMANTICS (meaning) of the name. So, again, I agree with Bob in identifing semantics/syntax with large/small. You might object that this is a syntax without an alphabet of symbols. Again we need an internal notion, this time an INTERNAL LANGUAGE. Unfortunately, a lot of people have muddled up the terminology here. What *I* mean by an internal language is an internal structure of a suitable kind for investigating formal grammar. For example, its set of "words" is the internal free monoid on its set of "letters", which is why we need an arithmetic universe. Regrettably, other people have used the phrase "internal language" to mean a language that is equivalent to a structure, which I call a PROPER LANGUAGE, where I have anglicised French "propre" or Italian "proprio". Given, for example, an internal CCC, it has a proper language that is an INTERNAL SIMPLY TYPED LAMBDA CALCULUS. From this we may construct the internal CATEGORY OF CONTEXTS AND SUBSTITUTIONS, which is the internal CLASSIFYING CATEGORY for the lambda calculus, in particular having an internal stucture-preserving functor to the given internal CCC, and this is an interval equivalence. (I'm not going to go into whether it is a weak or a strong equivalence -- see Section 7.6 of "Practical Foundations" for this.) Besides the book, please see also mathoverflow.net/questions/8731/categorical-foundations-without-set- theory Paul Taylor [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
Dear Paul,
(I believe that there was a development of "locally internal" instead of "locally small" categories by some French categorists in the 1970s -- Burroni again maybe?.)
I must mention that it is not me, but Jacques Penon who has worked on "locally internal" notions. I would like to add (to my yesterday mail) some indications on what I have called "small categories theory". In fact, this theory begins to exist. It is, for instance, what is called the "higher dimensionnal words problem", but also a developpement on higher automata theory (I have made many talks on this subjet, but not published --- I can send a manuscript to anybody interested). I have for example proved that finite and finitary Lawvere theory are finitely presentable 2-monoïds (it is my motivation for introducing the notion of polygraphs, previously introduced by Ross Street under the name of computads). It is perhaps not well-known by the categorists because it is published in a computer sciences revue : http://people.math.jussieu.fr/~burroni/mapage/highwordpb.pdf Best, Albert [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
participants (14)
-
burroni@math.jussieu.fr -
claudio pisani -
Colin McLarty -
Eduardo J. Dubuc -
F William Lawvere -
John Power -
Marta Bunge -
pare@mathstat.dal.ca -
Paul Taylor -
Ross Street -
Steve Vickers -
Thomas Streicher -
Vaughan Pratt -
Zinovy Diskin