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1. Cartesian Closed Category - Wikipedia, The Free Encyclopedia In category theory, a category is cartesian closed if, roughly speaking, any morphism defined on a product of two objects can be naturally identified with a http://en.wikipedia.org/wiki/Cartesian_closed_category

var wgNotice = ""; var wgNoticeLocal = ""; var wgNoticeLang = "en"; var wgNoticeProject = "wikipedia"; Cartesian closed category From Wikipedia, the free encyclopedia Jump to: navigation search In category theory , a category is cartesian closed if, roughly speaking, any morphism defined on a product of two objects can be naturally identified with a morphism defined on one of the factors. These categories are particularly important in mathematical logic and the theory of programming . For generalizations of this notion to monoidal categories , see closed monoidal category Contents Definition Examples Applications Equational theory ... edit Definition The category C is called cartesian closed iff it satisfies the following three properties: it has a terminal object any two objects X and Y of C have a product X Y in C any two objects Y and Z of C have an exponential Z Y in C For the first two conditions above, it is the same to require that any finite (possibly empty) family of objects of C admit a product in C , because of the natural associativity of the categorical product and noticing that the empty product in a category is nothing but the terminal object of that category.

2. Week240 What they really study is the free cartesian closed category on one object x , 15) Joachim Lambek, From lambda calculus to Cartesian closed categories, http://math.ucr.edu/home/baez/week240.html

October 22, 2006 This Week's Finds in Mathematical Physics (Week 240) John Baez I'm back from Shanghai, and classes are well underway now. For the last few weeks I'd been frantically preparing a talk for Stewart Brand's "Seminars About Long-Term Thinking", up in San Francisco. I talked about how we need to "zoom out" of our short-term perspective to understand the history of the earth's climate and what we're doing to it now: 1) John Baez, Zooming out in time, http://math.ucr.edu/home/baez/zoom/ There's a lot of tricky physics in this business. Consider, for example, this graph of cycles governing the Earth's precession, the obliquity of its orbit, and the eccentricity of its orbit: 2) Wikipedia, Milankovitch cycles, http://en.wikipedia.org/wiki/Milankovitch_cycles Here a "kyr" is a thousand years. The yellow curve combines information from all three of these cycles and shows the amount of solar radiation at 65 degrees north latitude. The bottom black curve shows the amount of glaciation. As Milankovitch's theory predicts, you can sort of see a correlation between the yellow and black curves - but it's nothing simple or obvious. One reason is the complex feedback mechanisms within the Earth's climate. Here's a great place to read about this stuff: 3) Barry Saltzman, Dynamical Paleoclimatology: Generalized Theory of Global Climate Change, Academic Press, New York, 2002.

3. Classical Vs Quantum Computation (Week 3) | The N-Category Café In a cartesian closed category, or CCC, we can take products of objects, .. James Dolan, Holodeck strategies and cartesian closed categories. http://golem.ph.utexas.edu/category/2006/10/classical_vs_quantum_computati_3.htm

@import url("/category/styles-site.css"); A group blog on math, physics and philosophy Skip to the Main Content Enough, already! Skip to the content. Note: These pages make extensive use of the latest XHTML and CSS Standards only supported in Mozilla. My best suggestion (and you will thank me when surfing an ever-increasing number of sites on the web which have been crafted to use the new standards) is to upgrade to the latest version of your browser. If that's not possible, consider moving to the Standards-compliant and open-source Mozilla browser. Main October 20, 2006 Classical vs Quantum Computation (Week 3) Posted by John Baez This week we had a guest lecture in our course on Classical versus Quantum Computation Week 3 (Oct. 19) - Guest lecture by James Dolan: Holodeck strategies and cartesian closed categories. James Dolan, Holodeck strategies and the lambda-calculus (a continuation of the above lecture). Todd Trimble, Holodeck games and CCCs (rigorous definitions and proofs). chess go here here In a cartesian closed category, or CCC, we can take products of objects, and also take exponentials. For example, in the category of sets, the product A B , or A B for short, is the usual cartesian product of the sets

4. Cartesian Closed Category@Everything2.com cartesian closed category. printable version chaos category theory Mathematical uniqueness terminal object initial object http://everything2.com/index.pl?node_id=1311841

5. Closed Categories « The Unapologetic Mathematician Here’s an example, though, of a cartesian closed category that looks rather different. It requires the notion of a “predicate calculus”, but not very much http://unapologetic.wordpress.com/2007/08/01/closed-categories/

The Unapologetic Mathematician Mathematics for the interested outsider Home About this weblog Closed Categories A closed category is a symmetric monoidal category where each functor . By what we said yesterday , this means that there is a unique way to sew all these adjoints parametrized by together into a functor. The canonical example of such a category is the category of sets with the cartesian product as its monoidal structure. For each set we need an adjunction and one from should be in bijection with functions from to . And indeed we have such an adjunction: is the set of all functions from to we have the function , which takes two numbers and gives back a third: if we stick in and we get back . But what if we just stick in the first number of the pair? Then what we get back is a function that will take the second number and give us the sum: if we just feed in we get back the function . That is, we can see addition either as a function taking a pair of numbers to a number, or we can see it as a function , taking a number to a function from numbers to numbers.

6. IngentaConnect The Largest Cartesian Closed Category Of Stable Domains The largest cartesian closed category of stable domains. Author GuoQiang Z.1. Source Theoretical Computer Science, Volume 166, Number 1, 20 October 1996 http://www.ingentaconnect.com/content/els/03043975/1996/00000166/00000001/art001

var tcdacmd="dt"; Home About Ingenta Ingenta Labs Ingenta Blog ... Check our FAQs Or contact us to report problems with: Subscription access Article delivery Registration Library administrator tasks ... Other problems For Publishers Why go online? Why choose IngentaConnect? Beyond Print: enhancing your service Access and authentication ... Contact us For Researchers For Authors About IngentaConnect Search and browse Publications available ... Register For Librarians Resource Zone Why choose IngentaConnect? Why choose IngentaConnect Complete? Activating Subscriptions ... Register CM8ShowAd("HorizontalBanner"); The largest cartesian closed category of stable domains Author: Guo-Qiang Z. Source: Theoretical Computer Science , Volume 166, Number 1, 20 October 1996 , pp. 203-219(17) Publisher: Elsevier view table of contents Key: - Free Content - New Content - Subscribed Content - Free Trial Content CM8ShowAd("Skyscraper"); Language: English Document Type: Research article DOI: Affiliations: Department of Computer Science, The University of Georgia, Athens, GA 30602, USA

7. Research Laboratory For Logic And Computation, GC CUNY Cartesian closed categories and lambda calculus II. We define the cartesian closed category generated by a typed lambda calculus, http://web.cs.gc.cuny.edu/~rllc/seminar_fall2003.html

Research Laboratory for Logic and Computation HOME PEOPLE PUBLICATIONS DOWNLOADS ... Seminars FALL 2003 Tuesday, 2pm - 4pm, room 4421 December 2 talk Yegor Bryukhov. Type Theory for a practicing mathematician. Abstract: continue December 2 talk Yegor Bryukhov. Type Theory for a practicing mathematician. Abstract: In this talk we will follow R.Constable's paper "Naive Computational Type Theory" which in turn "follows" the book of Paul Halmos "Naive Set Theory". This paper gives some new perspectives in Type Theory, including a new meaning of openness of Type Theory. We'll start from the fundamentals: "what is type", "propositions as types", and then go to type-theoretic analogues of a set, subset, pair, union, intersection, function, relation, etc. We will consider two meanings of this popular statement "Type Theory is open-ended", one is old and the other one is new. They are related but the new one is much deeper. It shows that the Type Theory is VERY different from the Set Theory. Time permitting we'll discuss dependent intersection (a relatively new result by Alexei Kopylov) and records. November 25 talk Walter Dean (GC and Rutgers). From Church's Thesis to Extended Church's Thesis.

8. CJO - Abstract - The Largest Cartesian Closed Category Of Domains, Considered Co The largest cartesian closed category of domains, considered constructively. DIETER SPREEN Mathematical Structures in Computer Science 150202, 299321, http://journals.cambridge.org/abstract_S0960129504004591

@import url("/css/users_abstract.css"); close Cambridge Journals Online Skip to content Volume 15 Issue 02 - Apr 2005 Cited by Articles (CrossRef) ... Mathematical Structures in Computer Science (2005), 15: 299-321 Cambridge University Press doi:10.1017/S0960129504004591 Published online by Cambridge University Press 14Mar2005 Login Subscribe to journal Email abstract Save article ... Link to this abstract Copy and paste this link: http://journals.cambridge.org/action/displayAbstract?aid=289406 Paper The largest Cartesian closed category of domains, considered constructively DIETER SPREEN spreen@informatik.uni-siegen.de Article author query spreen d [ Google Scholar Abstract (Received November 20 2002) (Revised June 5 2004) Footnotes This research was partially supported by joint DFG/RFBR grant 436 RUS 113/638 and INTAS grant 00-499.

9. Stable Domain Theory Separately, Berry 78 constructed a cartesian closed category whose morphisms preserve directed joins and connected meets, whilst Diers 79 considered http://www.monad.me.uk/stable/

Stable Domain Theory Paul Taylor Please note that, although this page was created in June 2003, the abstracts have simply been copied from the papers as they were written at the time. The paper on Quantitative Domains may be relevant to more recent work on "shapes" and "containers". Semantics of System F DVI (61 kb) PDF (227 kb) Compressed PostScript (78 kb) A5 PS booklet (71 kb) ... What are these? [28 May 2007] Appendix A of Proofs and Types X X X X X X X X X X X X X X An Algebraic Approach to Stable Domains DVI (104 kb) PDF (311 kb) Compressed PostScript (114 kb) A5 PS booklet (105 kb) ... What are these? [28 May 2007] Written January-March 1988. Published in the Journal of Pure and Applied Algebra Day [75] showed that the category of continuous lattices and maps which preserve directed joins and arbitrary meets is the category of algebras for a monad over Set Sp or Pos , the free functor being the set of filters of open sets. Separately, Berry [78] constructed a cartesian closed category whose morphisms preserve directed joins and connected meets , whilst Diers [79] considered similar functors independently in a study of categories of models of disjunctive theories . Girard [85] built on Berry's work to build a new and very lean model of polymorphism In this paper we bring these strands together, defining a monad based on

10. [Giuseppe.Longo@THEORY.CS.CMU.EDU: Re: The Cmu Workshop] These form a cartesian closed category which has fixed points for domain equations. It is shown that a ``universal domain exists. http://www.cis.upenn.edu/~bcpierce/types/archives/1988/msg00043.html

[Prev] [Next] [Index] [Thread] [Giuseppe.Longo@THEORY.CS.CMU.EDU: Re: the cmu workshop] To types@THEORY.LCS.MIT.EDU Subject : [Giuseppe.Longo@THEORY.CS.CMU.EDU: Re: the cmu workshop] From meyer@THEORY.LCS.MIT.EDU Date : Tue, 19 Apr 88 16:30:51 ast Prev: No Subject Next: Preliminary Program for 4th MFPS Workshop Index(es): Main Thread

11. Theses From Uppsala University : 5883 - Effective Domains And Admissible Domain In Paper I we define a cartesian closed category of effective bifinite domains. We also show that there is a natural cartesian closed category of http://publications.uu.se/theses/abstract.xsql?dbid=5883

12. Constructing A Category Text - Physics Forums Library Once I ve found C , I would like to construct a cartesian closed category C , again with there being a full embedding of C into C , and universal amongst http://physicsforums.com/archive/index.php/t-80719.html

Physics Help and Math Help - Physics Forums Mathematics PDA View Full Version : Constructing a category Hurkyl Suppose I have some arbitrary category C. I would like to construct a cartesian category C' with C embedded in it. If at all possible, the embedding would be full, and C' would be universal amongst all such constructions. What would be a good way to go about doing that? Can I even do that in general? Once I've found C', I would like to construct a cartesian closed category C'', again with there being a full embedding of C into C'', and universal amongst all such constructions. Once I have that, I what I really want is some topos E in which C is embedded, preferably fully. It would be nice, too, if E was universal amongst all such topoi, or at least being minimal amongst extensions. Actually, the first category I want to do this with is already cartesian, and has a subobject classifier. (It would be cool if it was still the subobject classifier when extended to a topos) But I'm still curious about the more general case too! mathwonk yukkk. ok what is a cartesian category?

13. Category Theory limits and colimits, adjoint functors, cartesian closed categories and typed lambdacalculus, the cartesian closed category of Scott domains. http://www.fit.vutbr.cz/study/course-l.php.en.iso-8859-2?id=5799

14. Ccard V2.0 - Mantras JS5b In cartesian closed categories, coproducts distribute over products. BP34 A cartesian closed category is a category with a terminal object, http://www.verify-it.de/sub/ccard/mantras_v20.html

Oliver Möller Last modified: Wed Jun 30 16:47:31 1999

15. Equilogical Spaces We show that this category in contradistinction to Top0 - is a cartesian closed category. The direct proof outlined here uses the equivalence of the http://www.cs.cmu.edu/Groups/LTC/abstracts/equ.html

Equilogical Spaces Submitted A. Bauer, L. Birkedal, D.S. Scott Abstract It is well known that one can build models of full higher-order dependent type theory (also called the calculus of constructions) using partial equivalence relations (PERs) and assemblies over a partial combinatory algebra (PCA). But the idea of categories of PERs and ERs (total equivalence relations) can be applied to other structures as well. In particular, we can easily define the category of ERs and equivalence-preserving continuous mappings over the standard category Top of topological T -spaces; we call these spaces (a topological space together with an ER) equilogical spaces and the resulting category Equ . We show that this category - in contradistinction to Top - is a cartesian closed category. The direct proof outlined here uses the equivalence of the category Equ to the category PEqu of PERs over algebraic lattices (a full subcategory of Top that is well known to be cartesian closed from domain theory). In another paper with Carboni and Rosolini (cited herein) a more abstract categorical generalization shows why many such categories are cartesian closed. The category Equ obviously contains Top as a full subcategory, and it naturally contains many other well known subcategories. In particular, we show why, as a consequence of work of Ershov, Berger, and others, the Kleene-Kreisel hierarchy of countable functionals of finite types can be naturally constructed in

16. Deliverables: A Categorical Approach To Program Development In Type Theory In fact the combining operations (excluding iteration) are exactly those of a cartesian closed category whose objects are the pre and postconditions and http://www.lfcs.inf.ed.ac.uk/reports/92/ECS-LFCS-92-242/

People Events Research Home Deliverables: A categorical approach to program development in type theory Rod Burstall and James McKinna Introduction: This paper outlines a method for constructing a program together with a proof of its correctness with respect to a given specification. The technology used is type theory, in fact an extended version of Calculus of Constructions Luo's ECC as implemented in Pollack's `Lego' system ; here program means a primitive recursive description of a function using simply typed lambda calculus with higher order functions and data types such as numbers and lists. Given a precondition and a postcondition we consider pairs consisting of a program a proof that the program satisfies the postcondition given the precondition. We call such a pair together with the pre- and postconditions a deliverable , since it is what a software house should ideally deliver to its client instead of just a program. Now we observe that various operations, composition, pairing, iteration and so on, can be used to combine deliverables, and that these operations can be implemented in ECC. Consider for example composition. If (f, p)

17. A Note On Connectedness In Cartesian Closed Categories Primaxily working in the category of limit spaces and continuous maps we suggest a new concept of connectivity with application in all categories where http://www.hindawi.com/GetArticle.aspx?doi=10.1155/S0161171297000148

18. Cartesian Closed Stable Categories The aim of this paper is to establish some cartesian closed categories which are between the two cartesian closed categories SLP (the category of Ldomains http://portal.acm.org/citation.cfm?id=1063574.1063583

19. FLoC '02 - DOMAIN Sunday July 21st Invited talk Some open problems concerning cartesian closed categories Finally, we will show that it category of domain is cartesian closed. http://floc02.diku.dk/DOMAIN/Sunday.html

DOMAIN on Sunday Detailed program Sunday July 21st, 2002 See also the unified by-slot program All sessions take place in auditorium 3. Session 5 Bill Lawvere, State U of New York-Buffalo, USA Invited talk: Some open problems concerning cartesian closed categories Dana Scott showed in 1970 that there are many cartesian closed categories containing arbitrarily large objects which are isomorphic to their own self-exponential. Tarski's high school identities suggest a more general question: (1) Which exponential rigs can be objectively realized? Hurewicz's homotopy category is also cartesian-closed, but has the unusual property that its points functor is the left adjoint of its own left adjoint. (2) Can any CCC be mapped to one with that property? That property also applies to infinitesimals which moreover enjoy further right adjoints, so that functionals (such as differential forms) are actually functions with a bigger codomain. (3) Can the lambda-calculus be expanded so as to provide an efficient means of presenting CCC's with such additional right adjoints? A topos has exponential types as a consequence of the power types. The power types also permit, via Dedekind's infinite intersection method, the construction of a "natural" number object given only the geometrical phenomenon of a monomorphic endo-map which is not an isomorphism.

20. [0710.5202] The Category Of 3-computads Is Not Cartesian Closed As a corollary we get that neither the category of all computads nor the category of $n$computads, for $n 2$, do form locally cartesian closed categories, http://arxiv.org/abs/0710.5202

arXiv.org math Search or Article-id Help Advanced search All papers Titles Authors Abstracts Full text Help pages Full-text links: Download: PostScript PDF Other formats Citations CiteBase p revious n ... ext Mathematics > Category Theory Title: The category of 3-computads is not cartesian closed Authors: Mihaly Makkai Marek Zawadowski (Submitted on 27 Oct 2007) Abstract: Comments: 3 pages Subjects: Category Theory (math.CT) MSC classes: Cite as: arXiv:0710.5202v1 [math.CT] Submission history From: Marek Zawadowski [ view email Sat, 27 Oct 2007 01:05:44 GMT (8kb) Which authors of this paper are endorsers? Link back to: arXiv form interface contact

21. Springer Online Reference Works cartesianclosed category. A category such that the following axioms are satisfied. A1) there exists a terminal object ; http://eom.springer.de/c/c130030.htm

Encyclopaedia of Mathematics C Article refers to Cartesian-closed category A category such that the following axioms are satisfied: A1) there exists a terminal object A2) for any pair of objects of there exist a product and given projections A3) for any pair of objects of there exist an object and an evaluation arrow such that for any arrow there is a unique arrow with These conditions are equivalent to the following: is a category with given products such that the functors have each a specified right-adjoint, written respectively as: Some examples of Cartesian-closed categories are: E1) any Heyting algebra E2) the category for any small category with itself; E3) the category of sheaves over a topological space, and more generally a (Grothendieck) topos E4) any elementary topos E5) the category of all (small) categories; E6) the category of graphs and their homomorphisms; E7) the category of -CPOs. These definitions can all be put into a purely equational form. References M. Eytan This text originally appeared in Encyclopaedia of Mathematics - ISBN 1402006098

22. Atlas: Some Cartesian Closed Topological Hulls In Approach Theory. By Mark Nauwe For example, in 2, G. Bourdaud indicated the existence of a family of cartesian closed topological constructs in CONV, the category of convergence http://atlas-conferences.com/c/a/e/q/12.htm

Atlas home Conferences Abstracts about Atlas Categorical Methods in Algebra and Topology (CatMAT 2000) August 21-25, 2000 University of Bremen Bremen, Germany Organizers Hans-E. Porst, Horst Herrlich View Abstracts Conference Homepage Some cartesian closed topological hulls in approach theory. by Mark Nauwelaerts University of Antwerp Although being topological is a nice property for a construct, it may be desirable to have more convenient properties, such as being cartesian closed topological (CCT), which means that it is possible to equip the set of functions between two objects with some appropriate structure in such a way that one obtains canonical function spaces having some nice properties. Considering a category A that lacks this property, one can look for its CCT hull B , that is, the smallest possible extension of A that has this required property. For example, in [2], G. Bourdaud indicated the existence of a "family" of cartesian closed topological constructs in CONV , the category of convergence spaces and continuous maps , such that the CCT hull of TOP and the CCT hull of CREG are specific instances of this family.

23. Category Theory For Computer Science polymorphic functions seen as natural transformations; the category of graphs seen as a functor category; examples of cartesian closed categories Set, http://www.brics.dk/~nygaard/CTfCS/

Category Theory for Computer Science Autumn 2002 - Department of Computer Science University of Aarhus News Lectures ... People News The course has terminated. Merry Christmas! Lectures Mondays, 12-14 in the r3 meeting room. December 9th A coalgebraic treatment of automata. Presented by Saurabh Agarwal. December 2nd Infinite data structures, coalgebra, and coinduction. Presented by Michael Westergaard ( slides November 25th Finite data structures, algebra, and induction. Presented by Henning Korsholm Rohde ( slides November 18th Transition systems generalised into presheaves. Presented by Marco Carbone. November 11th Exercises related to last week's lecture. Designing subtyping disciplines in programming languages. Presented by Branimir Lambov. November 4th Exercise related to last week's lecture. Effects in functional programming handled by monads. Presented by Karl Kristian Krukow ( slides October 28th Modelling recursive types. Presented by Mads Sig Ager (

24. Anubis Language The CAM, which is based on the Theory of cartesian closed Categories, Indeed, it is well known by category theoretists that the free cartesian closed http://www.anubis-language.com/?s=pr7pWYkG0ixt05FEjwpQgVtVYBo&a=go_philo

25. Cartesian Closed Topological Hull Of The Construct Of Closure Spaces Secondly, within this extension L the cartesian closed topological hull L* of we produce a concrete functor to the category of power closed collections http://www.tac.mta.ca/tac/volumes/8/n18/8-18abs.html

Cartesian closed topological hull of the construct of closure spaces V. Claes, E. Lowen-Colebunders and G. Sonck A cartesian closed topological hull of the construct CLS of closure spaces and continuous maps is constructed. The construction is performed in two steps. First a cartesian closed extension L of CLS is obtained. We apply a method worked out by J. Adamek and J. Reiterman for constructing extensions of constructs that in some sense ``resemble'' the construct of uniform spaces. Secondly, within this extension L the cartesian closed topological hull L* of CLS is characterized as a full subconstruct. In order to find the internal characterization of the objects of L* we produce a concrete functor to the category of power closed collections based on CLS as introduced by J. Adamek, J. Reiterman and G.E. Strecker. Keywords: closure space, cartesian closedness, function space, cartesian closed topological hull. 2000 MSC: 54A05, 18D15, 54C35. Theory and Applications of Categories , Vol. 8, 2001, No. 18, pp 481-489. http://www.tac.mta.ca/tac/volumes/8/n18/n18.dvi

26. Cartesian Closed Double Categories, Their Lambda-Notation, And The Pi-Calculus We introduce the notion of cartesian closed double category to provide mobile calculi for communicating systems with specific semantic models One dimension http://csdl.computer.org/comp/proceedings/lics/1999/0158/00/01580246abs.htm

var sc_project=2763585; var sc_invisible=0; var sc_partition=28; var sc_security="3c678009"; var addtoLayout=0; var addtoMethod=0; var AddURL = escape(document.location.href); // this is the page's URL var AddTitle = escape(document.title); // this is the page title Advanced Search CS Search Google Search Home Digital Library Podcasts Site Map ... Table of Contents Abstract 14th Annual IEEE Symposium on Logic in Computer Science (LICS'99) p. 246 Cartesian Closed Double Categories, Their Lambda-Notation, and the Pi-Calculus Roberto Bruni , University of Pisa Ugo Montanari , University of Pisa Full Article Text: DOI Bookmark: http://doi.ieeecomputersociety.org/10.1109/LICS.1999.782620 Send link to a friend Abstract We introduce the notion of cartesian closed double category to provide mobile calculi for communicating systems with specific semantic models: One dimension is dedicated to compose systems and the other to compose their computations and their observations. Also, inspired by the connection between simply typed lambda-calculus and cartesian closed categories, we define a new typed framework, called double lambda-notation, which is able to express the abstraction/application and pairing/projection operations in all dimensions. In this development, we take the categorical presentation as a guidance in the interpretation of the formalism. A case study of the pi-calculus, where the double lambda-notation straightforwardly handles name passing and creation, concludes the presentation.

27. Diagonal Arguments And Cartesian Closed Categories Diagonal arguments and cartesian closed categories with author commentary. F. William Lawvere. Originally published in Diagonal arguments and cartesian http://emis.library.cornell.edu/journals/TAC/reprints/articles/15/tr15abs.html

Diagonal arguments and cartesian closed categories with author commentary F. William Lawvere Originally published in: Diagonal arguments and cartesian closed categories, Lecture Notes in Mathematics , 92 (1969), 134-145, used by permission. 2000 MSC: 08-10, 02-00 Republished in: Reprints in Theory and Applications of Categories , No. 15 (2006), pp 1-13 http://www.tac.mta.ca/tac/reprints/articles/15/tr15.dvi http://www.tac.mta.ca/tac/reprints/articles/15/tr15.ps http://www.tac.mta.ca/tac/reprints/articles/15/tr15.pdf TAC Reprints Home

28. Practical Foundations Of Mathematics The raw calculus Interpretation The b and h-rules The universal property cartesian closed categories of domains. 4.8 NATURAL TRANSFORMATIONS http://www.cs.man.ac.uk/~pt/Practical_Foundations/html/c4.html

Practical Foundations of Mathematics Paul Taylor IV. Cartesian Closed Categories Practical Foundations of Mathematics Paul Taylor Cambridge University Press Introduction ... IV

29. Categories: Cartesian Closed Categories Of Internal Categories To categories@mta.ca; Subject categories cartesian closed categories of internal categories; From Andree Ehresmann Andree.Ehresmann@upicardie.fr http://north.ecc.edu/alsani/ct01(9-12)/msg00087.html

Date Prev Date Next Thread Prev Thread Next ... Thread Index categories: Cartesian closed categories of internal categories To categories@mta.ca Subject : categories: Cartesian closed categories of internal categories From Andree.Ehresmann@u-picardie.fr Date : Fri, 23 Nov 2001 17:35:48 +0100 Sender cat-dist@mta.ca http://perso.wanadoo.fr/vbm-ehr Prev by Date: categories: re: Exact adjunctions Next by Date: categories: Characterizing FinSet up to equivalence Previous by thread: categories: k-spaces and Hurewicz Next by thread: categories: Re: Characterizing FinSet up to equivalence Index(es): Date Thread

30. JSTOR On The Unification Problem For Cartesian Closed Categories cartesian closed categories (CCCs) have played and continue to play an important role in the study of the semantics of programming languages. http://links.jstor.org/sici?sici=0022-4812(199706)62:2<636:OTUPFC>2.0.CO;2-H

31. C[omp]UTE: Games And Cartesian Closed Categories Date Sat, 18 Nov 2006 214109 0300 (CLST) Subject Games and cartesian closed Categories From andrew cooke andrew@ Something interesting from Baez http://www.acooke.org/cute/GamesandCa0.html

Contents Front Page (latest) Previous Next Games and Cartesian Closed Categories http://math.ucr.edu/home/baez/week240.html (scroll down to point 4 and read on, including subsequent points). Andrew Comment on this post Contents Front Page (latest) Previous ... Next

32. TAC Abstracs These results subsume earlier ones using cartesian closed categories, as well as those employing socalled Henkin and Kripke lambda-models. http://www.disi.unige.it/eventsandseminars/tac/abstracts98.html

Abstracts The topos of 3-colored graphs The problem of 3 coloring finite undirected graphs is NP-complete and therefore there is interest in the structure of graphs which are and are not 3 colorable. The category of undirected graphs U is a topos. A graph with a given 3 coloring is said to be 3-colored. The category C of 3-colored graphs is is a cocomplete topos, hence also complete. There is an essential geometric morphism between U and C, and the three functors involved have considerable natural appeal. These functors justify two of the heuristics used to quicken algorithms which determine whether or not a 3 coloring exists for argument finite undirected graphs. Indeed, one of these heuristics deserves to be better known. ... Finally, we observe that instead of B we might take any pca of the form L/T where T is a theory containing the (obvious) conversion theory for L and itself being containEd in Th(A). This may be considered as a solution of (a variant of) the Longley-Phoa Conjecture claiming that realisability over a term model of untyped lambda-calculus gives rise to a fully abstract model of PCF.

33. Roberto Di Cosmo's Abstracts By the close relation between closed cartesian categories and models of these calculi, we also produce a characterization of those isomorphisms which hold http://www.pps.jussieu.fr/~dicosmo/Publications/Abstracts.html

Abstracts Here are the abstracts of most of the works you can find referenced in my publications page , where you will find the pointers to the online available documents. Provable isomorphisms of types: from Lambda Calculus to Information Retrieval and Language Design. This book (see here for a more detailed overview) is devoted to the study of the notion of type-isomorphisms in functional languages, both from a theoretical and a practical point of view. Based on my PhD dissertation, it has been extensively revised, updated and provided with a completely new introduction to the topic, that makes it accessible to a wide spectrum of readers. It tries hard to provide a complete reference and discussion of all research done in this area, from the definition of confluent rewriting systems for typed lambda calculi equipped with various extensionality rules, to the characterization of isomorphisms of types in different typed calculi, to the applications to extensions of ML-style type-inference algorithms and the design of library search tools based on types (you can test a sample implementation right now: english or french Provable isomorphisms of types A constructive characterization is given of the isomorphisms which must hold in all models of the typed lambda calculus with surjective pairing. By the close relation between closed Cartesian categories and models of these calculi, we also produce a characterization of those isomorphisms which hold in all CCC's. By the correspondence between these calculi and proofs in intuitionistic positive propositional logic, we thus provide a characterization of equivalent formulae of this logic, where the definition of equivalence of terms depends on having ``invertible'' proofs between the two terms. Rittri (1989), on types as search keys in program libraries, provides an interesting example of use of these characterizations.

34. Date Sun, 1 May 1994 163756 +0500 (GMT+400) Subject Michael Date Tue, 3 May 1994 155358 +0500 (GMT+400) Subject Confused about cartesian closed categories. Date Tue, 03 May 94 152132 +0100 From http://www.mta.ca/~cat-dist/archive/1994/94-5

Date: Sun, 1 May 1994 16:37:56 +0500 (GMT+4:00) Subject: translation traps Date: Sat, 30 Apr 94 20:37:54 EDT From: Michael Barr This is all very simple stuff, which I should have worked out a long time ago. What is the correct definition of a Cartesian closed category? Some books seem to say that a CCC has a finite limits plus the relevant conditions to give you exponitails. While other books seem to define it as finite products plus exponitails. Am I reading all these books wrong (I always thought it was all finite limits so I might of systimacitally misread lots of books since). Or can you get finite limits from finite products (but where do you get the equalisers form?) Regards Justin Pearson Date: Wed, 4 May 1994 10:18:01 +0500 (GMT+4:00) Subject: Re: Confused about Cartesian closed categories Date: Tue, 3 May 94 17:16:17 EDT From: Michael Barr I am afraid that different authors make different asssumptions. I guess the minimum you want is finite products (except that a C-monoid is sometimes defined as a CCC with one object, so it has only finite non-empty products), while others will want finite limits. They are definitely different, no doubt of that and the complexity of the computations in the initial CCC of those types is different too, so the difference is important. There is an open question of some interest: can an FP CCC with NNO be embedded into an FL CCC with NNO in a way that preserves the NNO. I conjecture that it cannot, but if it could, that would have some interesting consequences. Michael Date: Wed, 4 May 1994 10:20:58 +0500 (GMT+4:00) Subject: Re: Confused about Cartesian closed categories Date: Wed, 4 May 1994 09:40:27 +0100 (BST) From: Roy Crole

35. Good Math, Bad Math : From Lambda Calculus To Cartesian Closed Categories What I m going to do in these two posts is show the correspondence between lambda calculus and the cartesian closed categories. If you re not familiar with http://scienceblogs.com/goodmath/2006/08/from_lambda_calculus_to_cartes.php

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36. Chronological List Of Publications Diagonal Arguments and cartesian closed Categories with Author Commentary Reprints in Theory and Applications of Categories, No. 15, 2006, 113. (link) http://www.buffalo.edu/~wlawvere/list.html

F. William Lawvere Chronological list of publications Home Subject Classification Bottom of page (most recent) 1. Functorial Semantics of Algebraic Theories Proceedings of the National Academy of Science 50 , No. 5 (November 1963), 869-872. 2. Elementary Theory of the Category of Sets Proceedings of the National Academy of Science 52 , No. 6 (December 1964), 1506-1511. Algebraic Theories, Algebraic Categories, and Algebraic Functors, Theory of Models ; North-Holland, Amsterdam (1965), 413-418. Functorial Semantics of Elementary Theories Journal of Symbolic Logic , Abstract, Vol. 31 (1966), 294-295. The Category of Categories as a Foundation for Mathematics La Jolla Conference on Categorical Algebra , Springer-Verlag (1966), 1-20. Some Algebraic Problems in the Context of Functorial Semantics of Algebraic Theories Springer Lecture Notes in Mathematics No. 61 , Springer-Verlag (1968), 41-61. Ordinal Sums and Equational Doctrines Springer Lecture Notes in Mathematics No. 80 , Springer-Verlag (1969), 141-155. Diagonal Arguments and Cartesian Closed Categories Springer Lecture Notes in Mathematics No. 92

37. List For KWIC List Of MSC2000 Phrases closed categories (closed monoidal and cartesian closed 18D15 categories, foundations of homotopy theory topological 55U40 categories, functors 46M15 http://www.math.unipd.it/~biblio/kwic/msc/m-kl_11_07.htm

calculus # integral transforms, operational 44-XX calculus # one-variable calculus # quantum stochastic calculus # stochastic calculus of variations and the Malliavin calculus # umbral calculus (e.g.: curve sketching, extremum problems) # differential calculus and other discrete methods # lattice gravity, Regge calculus and related topics # $q$- calculus and the teaching of calculus # comprehensive works on calculus in topological algebras # functional calculus of functions on infinite-dimensional spaces calculus of functions taking values in infinite-dimensional spaces calculus of Mikusinski and other operational calculi calculus of variations and optimal control; optimization 49-XX calculus of variations and the Malliavin calculus # stochastic calculus of vector functions calculus on manifolds; nonlinear operators calculus, BCK and BCI logics) # substructural logics (including relevance, entailment, linear logic, Lambek calculus, etc. # analytical theory: series, transformations, transforms, operational calculus, etc.) # hermitian and normal operators (spectral measures, functional

38. Search Results Diagonal arguments and cartesian closed categories Diagonal arguments and cartesian closed categories with author commentary. F. William Lawvere. http://astech.library.cornell.edu/ast/search_results.cfm?btnG=go&gsa_client=defa

39. Categorical Logic In this connection the lambdacalculus is treated via the theory of cartesian closed categories. Similarly higher-order logic is modelled by the categorical http://www.andrew.cmu.edu/user/awodey/catlog/

Categorical Logic Fall 2006 Course Information Place: PH 226B Time: TTh 3 - 4:20 Instructor: Steve Awodey Office: Baker 152 (mail: Baker 135) Office Hour: Friday 1-2, or by appointment Phone: 8947 Email: awodey@andrew Secretary: Baker 135 Overview This course focuses on applications of category theory in logic and computer science. A leading idea is functorial semantics, according to which a model of a logical theory is a set-valued functor on a structured category determined by the theory. This gives rise to a syntax-invariant notion of a theory and introduces many algebraic methods into logic, leading naturally to the universal and other general models that distinguish functorial from classical semantics. Such categorical models occur, for example, in denotational semantics. In this connection the lambda-calculus is treated via the theory of cartesian closed categories. Similarly higher-order logic is modelled by the categorical notion of a topos. Using sheaves, topos theory also subsumes Kripke semantics for intuitionistic logic. Prerequisites 80-413/713 Category Theory, or equivalent.

40. CCS1a: Categories, Proofs And Processes | Mathematical Institute - University Of Categories, functors, natural transformations. Isomorphisms. monics and epics. Products and coproducts. Universal constructions. cartesian closed categories http://www.maths.ox.ac.uk/courses/part-c/ccs1a-categories-proofs-and-processes

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41. Peter Selinger Papers We introduce the class of control categories, which combine a cartesianclosed structure with a premonoidal structure in the sense of Power and Robinson. http://www.mscs.dal.ca/~selinger/papers.html

Papers This is a list of Peter Selinger 's papers, along with abstracts and hyperlinks. Linear-non-linear model for a computational call-by-value lambda calculus (with B. Valiron Idempotents in dagger categories On a fully abstract model for a quantum linear functional language (with B. Valiron Tree checking for sparse complexes (with M. Caboara and S. Faridi Simplicial cycles and the computation of simplicial trees (with M. Caboara and S. Faridi A lambda calculus for quantum computation with classical control (with B. Valiron Dagger compact closed categories and completely positive maps Towards a semantics for higher-order quantum computation A brief survey of quantum programming languages ... has nonempty products See also: Proceedings of QPL 2006 (editor). Proceedings of QPL 2005 (editor). Proceedings of QPL 2004 (editor). Proceedings of CTCS 2002 (editor, with R. Blute). My compiler for the core join calculus My finite lambda model applet My compiler for the lambda-mu-nu calculus A linear-non-linear model for a computational call-by-value lambda calculus Proceedings of the Eleventh International Conference on Foundations of Software Science and Computation Structures (FOSSACS 2008), Budapest

42. LaTeX Project: An Introduction To LaTeX Its title is cartesian closed categories and the price of eggs. Its author is Jane Doe. It was written in September 1994. The document consists of a title http://www.latex-project.org/intro.html

An introduction to LaTeX You are here: LaTeX project site LaTeX is a document preparation system for high-quality typesetting. It is most often used for medium-to-large technical or scientific documents but it can be used for almost any form of publishing. LaTeX is not a word processor! Instead, LaTeX encourages authors not to worry too much about the appearance of their documents but to concentrate on getting the right content. For example, consider this document: Cartesian closed categories and the price of eggs Jane Doe September 1994 Hello world! To produce this in most typesetting or word-processing systems, the author would have to decide what layout to use, so would select (say) 18pt Times Roman for the title, 12pt Times Italic for the name, and so on. This has two results: authors wasting their time with designs; and a lot of badly designed documents! LaTeX is based on the idea that it is better to leave document design to document designers, and to let authors get on with writing documents. So, in LaTeX you would input this document as: Or, in English:

43. Domains And Lambda Calculi (book Announcement) Chapter 5 gives a complete presentation of the problem of classifying the largest cartesian closed categories of algebraic directed complete partial orders http://www.seas.upenn.edu/~sweirich/types/archive/1997-98/msg00317.html

[Prev] [Next] [Index] [Thread] Domains and Lambda Calculi (book announcement) To types@cs.indiana.edu Subject : Domains and Lambda Calculi (book announcement) From amadio@gyptis.univ-mrs.fr Date : Fri, 27 Mar 1998 17:43:21 GMT CC amadio@protis.univ-mrs.fr Delivery-Date : Fri, 27 Mar 1998 11:44:50 -0500 Reply-to amadio@gyptis.univ-mrs.fr Book announcement: DOMAINS AND LAMBDA CALCULI by Roberto AMADIO and Pierre-Louis CURIEN Cambridge Tracts in Theoretical Computer Science 46, ISBN 0-521-62277-8, Cambridge University Press We are pleased to announce that the book is now available. Order informations can be found at http://www.cup.cam.ac.uk/Scripts/webbook.asp?isbn=0521622778 and for North-America at http://www.cup.org/Titles/62/0521622778.html Prev: postdoc position at INRIA Sophia Antipolis - pls forward Next: AMAST'98 first CFP Index(es): Main Thread

44. Selected Papers And Notes By Andrej Bauer These two categories are both locally cartesian closed extensions of countably based T0spaces. A natural question to ask is how they are related. http://andrej.com/papers/

Andrej Bauer selected papers and notes "Two Constructive Embedding-Extension Theorems with Applications to Continuity Principles and to Banach-Mazur Computability " with Alex Simpson December 19, 2003 We prove two embedding and extension theorems in the context of the constructive theory of metric spaces. The first states that Cantor space embeds in any inhabited complete separable metric space (CSM) without isolated points, X, in such a way that every sequentially continuous function from Cantor space to Z extends to a sequentially continuous function from X to R. The second asserts an analogous property for Baire space relative to any inhabited locally non-compact CSM. Both results rely on having careful constructive formulations of the concepts involved. As a first application, we derive new relationships between "continuity principles" asserting that all functions between specified metric spaces are pointwise continuous. In particular, we give conditions that imply the failure of the continuity principle "all functions from X to R are continuous", when X is an inhabited CSM without isolated points, and when X is an inhabited locally non-compact CSM. One situation in which the latter case applies is in models based on ``domain realizability'', in which the failure of the continuity principle for any inhabited locally non-compact CSM, X, generalizes a result previously obtained by Escardo and Streicher in the special case X = C[0,1].