November 1992 - Categories - categories.org.au

!
by barr＠triples.Math.McGill.CA 29 Nov '92

29 Nov '92

I would like to add something to what I said about the !. Jim Otto pointed this out to me a year ago. If you take modules over a commutative ring it is closed monoidal category. It is a model for a fragment of linear logic, missing negation. Define !M as the free module generated by the underlying set of M. Then !(M x N) is !M tensor !N and so is ostensibly a cofree functor. The Eilenberg-Moore category is the category of sets and the Kleisli category is the full subcategory whose objects are the sets underlying modules. To me this seems to mean that unless the ! has some sensible relation to the linear logic of the category, there is really no content to the claim that you have a model of classical logic inside the linear logic. The only models that seem to fill that requirement are the cofree coalgebras, assuming they exist. Michael ==============================================================================

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! as comonad or adjunction?
by Vaughan Pratt 28 Nov '92

28 Nov '92

Can anyone give me a good reason for taking ! literally as a comonad when defining the notion of "model of linear logic"? The problem I have with it is that it seems to force either the co-Kleisli or co-Eilenberg-Moore category on you. I have two objections to this. First, correct me if I'm wrong but I was under the impression that co-Eilenberg-Moore categories were hard to determine. Second, again correct me if I've misunderstood this, but if you have a particular adjunction in mind and F is not comonadic, then when you plug FU into the definition it won't give your F and U back. Instead it will go into convulsions trying to find some comonadic adjunction that you didn't want in the first place. The alternative I have in mind is that the notion of model should be defined so as to permit you to specify an adjunction F -| U such that ! = FU. This allows you to use whatever nice adjunction you had in mind in place of the canonical one determined by FU. One might require that U be monadic, but I don't see any compelling reason to require this. I've been doing just this for the past couple of years, but it occurs to me that the notion of "model of linear logic" might need to be standardized at some point, in which case it would be a pity if it got this detail wrong. Vaughan Pratt ==============================================================================

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Object-speak --> Sets ?
by Al Vilcius 28 Nov '92

28 Nov '92

For various reasons, I find myself needing to know something about object-oriented programming and seek help from my learned friends and acquaintances in understanding what it is all about in categorical terms. "Object-speak" is the terminology used in the so called object- oriented technology, implemented in the (now ten year old) programming language C++ and in other languages as well. It uses terms (taken here from David A. Taylor's "Object- Oriented Technology" Addison-Wesley 1990) such as: object: a software packet that contains a collection of related data (in the form of variables) and methods for operating on that data. method: a procedure contained within an object that is made available to other objects. message: a signal from one object to another that requests the receiving object to carry out one of its methods. class: a template for defining the methods and variables for a particular type of object; can contain subclasses which are special cases. instance: refers to an object that belongs to a particular class. inheritance: a mechanism whereby classes can make use of the methods and variables defined in all classes above them on their branch of the class hierarchy. encapsulation: a technique in which data is packaged together with its corresponding procedures. In object- oriented technology, the mechanism for encapsulation is the object. data abstraction: the process of defining new higher-level data types. polymorphism: the ability to hide different implementations behind a common interface to simplify communications among objects. What does all this really mean? Does all this structure lead to (or arise from) a theory? Is there an interpretation of "object-speak" in a category (with what properties ?) so that it might be understood in category theoretic terms? Are there any references that explicitly make such a connection? Al Vilcius, Toronto ==============================================================================

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Some idle questions
by Allen Knutson 28 Nov '92

28 Nov '92

Are there famous results that can be stated as "Every functor from category A to concrete category B is representable"? Or at least "Every functor from this class that would a priori seem to be too large is representable"? I don't have any use for such a result, I just want to have more feeling for representability vs nonrepresentability. A more specialized question: the representing object for a representable functor from k-Alg to Grp gets a natural commutative Hopf algebra structure. This can be generalized in two ways: lose representability, and just have a functor, or lose commutativity, and just have a noncommutative Hopf algebra. Has anyone ever seen an example where it was desirable to lose both, somehow? Allen K. ==============================================================================

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Re: Fibration of categories
by rkieboom＠tena2.vub.ac.be 20 Nov '92

20 Nov '92

I am interested in fibrations of categories.Being a homotopy theoreticist I got acquainted with this subject through Ronnie BROWN's paper on fibrations of groupoids (1970). Can anyone help me to find recent papers on this subject ? Thanks for the trouble. Rudger Kieboom (Brussels). ==============================================================================

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Categories with faithful functors to a group
by D_FELDMAN＠UNHH.UNH.EDU 19 Nov '92

19 Nov '92

Have categories that are equipped with a faithful functor to some group (or perhaps monoid) ever been singled out as objects of study? I am especially interested in the use they might be put to as a categorical framework for quasiperiodicity. Here is the sort of thing I have in mind. Consider the Euclidean plane decorated with a particular Penrose tiling. For definiteness, regard the tiling as the characteristic function of the union of the edges of all the tiles. Consider the category C whose objects are non-empty open subsets of the plane such that given objects U and V, Hom(U,V) consists of functions f:U --> V which are restrictions of rigid motions of the plane and which commute with the tiling. Then there is a forgetful functor from C to the group of rigid motions of the plane. David Feldman Department of Mathematics University of New Hampshire ==============================================================================

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None
by kelly_m＠maths.su.oz.au 18 Nov '92

18 Nov '92

There are many references. See e.g. Foltz, Lair, Kelly, JPAA 17 (1980), 171 - 177. Max Kelly. ==============================================================================

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Sconing and Relators
by andre＠saul.cis.upenn.edu 18 Nov '92

18 Nov '92

NOTES ON SCONING AND RELATORS John Mitchell and Andre Scedrov ABSTRACT. A semantics of typed lambda calculi based on relations is described. The main mathematical tool is a category-theoretic method of sconing, also called glueing or Freyd covers. Its correspondence to logical relations is also examined. In the case of polymorphic types, a central role is played by relators, i.e., maps that take objects to objects and relations to relations. Sconing may be used to express relators as functors, indeed, as certain object maps on an appropriate category. These notes are intended to be accessible to readers with minimal background in category theory. Available using anonymous ftp from host ftp.cis.upenn.edu and the file pub/papers/scedrov/rel.dvi or pub/papers/scedrov/rel.ps . ==============================================================================

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Re: Additional structure on representing objects(twice)
by pjf＠saul.cis.upenn.edu 18 Nov '92

18 Nov '92

Affine spaces over a give field are easily seen to be algebras of a particular equational theory. Hence the answer to Allen Knutsun's question: Is there a general theory on how to give representing objects additional structure, when one is interested in functors to other categories than Set? In the case I'm dealing with in particular, the terminal category has as objects affine spaces. can be found in the ancient work: Algebra-valued functors in general and tensor products in particular, Colloquium Mathematicum, XIV, 1966. ++++++++++++++++++++++++++++++++++++++++++++++++++

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Additional structure on representing objects
by Allen Knutson 17 Nov '92

17 Nov '92

The object representing a functor from a category C to Set is well understood, being unique if it exists up to unique isomorphism and all that. In the only case I'm familiar with, functors from k-Alg to Grp, the representing objects can be given a Hopf algebra structure with which to determine the group structure. Is there a general theory on how to give representing objects additional structure, when one is interested in functors to other categories than Set (of which the above example would be a particular case)? In the case I'm dealing with in particular, the terminal category has as objects affine spaces. If this is well covered in the literature, please forgive the question and just point me to a reference. Thanks very much! Allen K. ==============================================================================

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