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- By Rose Teteki Abbey, K. C. Abraham, David Tuesday Adamo, LeRoy H. Aden, Efrain Agosto, Victor Aguilan, Gillian T. W. Ahlgren, Charanjit Kaur AjitSingh, Dorothy B E A Akoto, Giuseppe Alberigo, Daniel E. Albrecht, Ruth Albrecht, Daniel O. Aleshire, Urs Altermatt, Anand Amaladass, Michael Amaladoss, James N. Amanze, Lesley G. Anderson, Thomas C. Anderson, Victor Anderson, Hope S. Antone, María Pilar Aquino, Paula Arai, Victorio Araya Guillén, S. Wesley Ariarajah, Ellen T. Armour, Brett Gregory Armstrong, Atsuhiro Asano, Naim Stifan Ateek, Mahmoud Ayoub, John Alembillah Azumah, Mercedes L. García Bachmann, Irena Backus, J. Wayne Baker, Mieke Bal, Lewis V. Baldwin, William Barbieri, António Barbosa da Silva, David Basinger, Bolaji Olukemi Bateye, Oswald Bayer, Daniel H. Bays, Rosalie Beck, Nancy Elizabeth Bedford, Guy-Thomas Bedouelle, Chorbishop Seely Beggiani, Wolfgang Behringer, Christopher M. Bellitto, Byard Bennett, Harold V. Bennett, Teresa Berger, Miguel A. Bernad, Henley Bernard, Alan E. Bernstein, Jon L. Berquist, Johannes Beutler, Ana María Bidegain, Matthew P. Binkewicz, Jennifer Bird, Joseph Blenkinsopp, Dmytro Bondarenko, Paulo Bonfatti, Riet en Pim Bons-Storm, Jessica A. Boon, Marcus J. Borg, Mark Bosco, Peter C. Bouteneff, François Bovon, William D. Bowman, Paul S. Boyer, David Brakke, Richard E. Brantley, Marcus Braybrooke, Ian Breward, Ênio José da Costa Brito, Jewel Spears Brooker, Johannes Brosseder, Nicholas Canfield Read Brown, Robert F. Brown, Pamela K. Brubaker, Walter Brueggemann, Bishop Colin O. Buchanan, Stanley M. Burgess, Amy Nelson Burnett, J. Patout Burns, David B. Burrell, David Buttrick, James P. Byrd, Lavinia Byrne, Gerado Caetano, Marcos Caldas, Alkiviadis Calivas, William J. Callahan, Salvatore Calomino, Euan K. Cameron, William S. Campbell, Marcelo Ayres Camurça, Daniel F. Caner, Paul E. Capetz, Carlos F. Cardoza-Orlandi, Patrick W. Carey, Barbara Carvill, Hal Cauthron, Subhadra Mitra Channa, Mark D. 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Deichmann Edwards, Miguel A. De La Torre, George E. Demacopoulos, Thomas de Mayo, Leah DeVun, Beatriz de Vasconcellos Dias, Dennis C. Dickerson, John M. Dillon, Luis Miguel Donatello, Igor Dorfmann-Lazarev, Susanna Drake, Jonathan A. Draper, N. Dreher Martin, Otto Dreydoppel, Angelyn Dries, A. J. Droge, Francis X. D'Sa, Marilyn Dunn, Nicole Wilkinson Duran, Rifaat Ebied, Mark J. Edwards, William H. Edwards, Leonard H. Ehrlich, Nancy L. Eiesland, Martin Elbel, J. Harold Ellens, Stephen Ellingson, Marvin M. Ellison, Robert Ellsberg, Jean Bethke Elshtain, Eldon Jay Epp, Peter C. Erb, Tassilo Erhardt, Maria Erling, Noel Leo Erskine, Gillian R. Evans, Virginia Fabella, Michael A. Fahey, Edward Farley, Margaret A. Farley, Wendy Farley, Robert Fastiggi, Seena Fazel, Duncan S. Ferguson, Helwar Figueroa, Paul Corby Finney, Kyriaki Karidoyanes FitzGerald, Thomas E. FitzGerald, John R. Fitzmier, Marie Therese Flanagan, Sabina Flanagan, Claude Flipo, Ronald B. Flowers, Carole Fontaine, David Ford, Mary Ford, Stephanie A. Ford, Jim Forest, William Franke, Robert M. Franklin, Ruth Franzén, Edward H. Friedman, Samuel Frouisou, Lorelei F. Fuchs, Jojo M. Fung, Inger Furseth, Richard R. Gaillardetz, Brandon Gallaher, China Galland, Mark Galli, Ismael García, Tharscisse Gatwa, Jean-Marie Gaudeul, Luis María Gavilanes del Castillo, Pavel L. Gavrilyuk, Volney P. Gay, Metropolitan Athanasios Geevargis, Kondothra M. George, Mary Gerhart, Simon Gikandi, Maurice Gilbert, Michael J. Gillgannon, Verónica Giménez Beliveau, Terryl Givens, Beth Glazier-McDonald, Philip Gleason, Menghun Goh, Brian Golding, Bishop Hilario M. Gomez, Michelle A. Gonzalez, Donald K. Gorrell, Roy Gottfried, Tamara Grdzelidze, Joel B. Green, Niels Henrik Gregersen, Cristina Grenholm, Herbert Griffiths, Eric W. Gritsch, Erich S. Gruen, Christoffer H. Grundmann, Paul H. Gundani, Jon P. Gunnemann, Petre Guran, Vidar L. Haanes, Jeremiah M. Hackett, Getatchew Haile, Douglas John Hall, Nicholas Hammond, Daphne Hampson, Jehu J. Hanciles, Barry Hankins, Jennifer Haraguchi, Stanley S. Harakas, Anthony John Harding, Conrad L. Harkins, J. William Harmless, Marjory Harper, Amir Harrak, Joel F. Harrington, Mark W. Harris, Susan Ashbrook Harvey, Van A. Harvey, R. Chris Hassel, Jione Havea, Daniel Hawk, Diana L. Hayes, Leslie Hayes, Priscilla Hayner, S. Mark Heim, Simo Heininen, Richard P. Heitzenrater, Eila Helander, David Hempton, Scott H. Hendrix, Jan-Olav Henriksen, Gina Hens-Piazza, Carter Heyward, Nicholas J. Higham, David Hilliard, Norman A. Hjelm, Peter C. Hodgson, Arthur Holder, M. Jan Holton, Dwight N. Hopkins, Ronnie Po-chia Hsia, Po-Ho Huang, James Hudnut-Beumler, Jennifer S. Hughes, Leonard M. Hummel, Mary E. Hunt, Laennec Hurbon, Mark Hutchinson, Susan E. Hylen, Mary Beth Ingham, H. Larry Ingle, Dale T. Irvin, Jon Isaak, Paul John Isaak, Ada María Isasi-Díaz, Hans Raun Iversen, Margaret C. 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Yee, Viktor Yelensky, Yeo Khiok-Khng, Gustav K. K. Yeung, Angela Yiu, Amos Yong, Yong Ting Jin, You Bin, Youhanna Nessim Youssef, Eliana Yunes, Robert Michael Zaller, Valarie H. Ziegler, Barbara Brown Zikmund, Joyce Ann Zimmerman, Aurora Zlotnik, Zhuo Xinping
- Edited by Daniel Patte, Vanderbilt University, Tennessee
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- The Cambridge Dictionary of Christianity
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- 05 August 2012
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- 20 September 2010, pp xi-xliv
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! and ? – Storage as tensorial strength
- R. F. Blute, J. R. B. Cockett, R. A. G. Seely
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- Mathematical Structures in Computer Science / Volume 6 / Issue 4 / August 1996
- Published online by Cambridge University Press:
- 04 March 2009, pp. 313-351
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We continue our study of the negation-free structure of multiplicative linear logic, as represented by the structure of weakly distributive categories, to consider the ‘exponentials’! and ? in the weakly distributive context. In addition to the usual triple and cotriple structure that one would expect on each of the two operators, there must be some connection between them to replace the de Morgan relationship found in the linear logic context. This turns out to be the notion of tensorial strength. We analyze coherence for this situation, using a modification of the usual nets due to Danos, which is a form suitable for linear logic with exponentials but without negation.
Differential categories
- R. F. BLUTE, J. R. B. COCKETT, R. A. G. SEELY
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- Mathematical Structures in Computer Science / Volume 16 / Issue 6 / December 2006
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- 01 November 2006, pp. 1049-1083
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Following work of Ehrhard and Regnier, we introduce the notion of a differential category: an additive symmetric monoidal category with a comonad (a ‘coalgebra modality’) and a differential combinator satisfying a number of coherence conditions. In such a category one should imagine the morphisms in the base category as being linear maps and the morphisms in the coKleisli category as being smooth (infinitely differentiable). Although such categories do not necessarily arise from models of linear logic, one should think of this as replacing the usual dichotomy of linear vs. stable maps established for coherence spaces.
After establishing the basic axioms, we give a number of examples. The most important example arises from a general construction, a comonad $S_\infty$ on the category of vector spaces. This comonad and associated differential operators fully capture the usual notion of derivatives of smooth maps. Finally, we derive additional properties of differential categories in certain special cases, especially when the comonad is a storage modality, as in linear logic. In particular, we introduce the notion of a categorical model of the differential calculus, and show that it captures the not-necessarily-closed fragment of Ehrhard–Regnier differential $\lambda$-calculus.
The logic of linear functors
- R. BLUTE, J. R. B. COCKETT, R. A. G. SEELY
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- Mathematical Structures in Computer Science / Volume 12 / Issue 4 / August 2002
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- 16 September 2002, pp. 513-539
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This paper describes a family of logics whose categorical semantics is based on functors with structure rather than on categories with structure. This allows the consideration of logics that contain possibly distinct logical subsystems whose interactions are mediated by functorial mappings. For example, within one unified framework, we shall be able to handle logics as diverse as modal logic, ordinary linear logic, and the ‘noncommutative logic’ of Abrusci and Ruet, a variant of linear logic that has both commutative and noncommutative connectives.
Although this paper will not consider in depth the categorical basis of this approach to logic, preferring instead to emphasise the syntactic novelties that it generates in the logic, we shall focus on the particular case when the logics are based on a linear functor, in order to give a definite presentation of these ideas. However, it will be clear that this approach to logic has considerable generality.
Bart Jacobs. Categorical logic and type theory. Studies in logic and the foundations of mathematics, vol. 141. Elsevier, Amsterdam etc. 1999, xvii + 760 pp.
- R. A. G. Seely
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- Journal:
- Bulletin of Symbolic Logic / Volume 6 / Issue 2 / June 2000
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- 15 January 2014, pp. 225-229
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- June 2000
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Introduction
- M. BARR, P. J. SCOTT, R. A. G. SEELY
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- Mathematical Structures in Computer Science / Volume 10 / Issue 2 / April 2000
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- 01 April 2000, pp. 97-98
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On December 5, 1997, a small conference was held at McGill on the occasion of Jim Lambek's 75th birthday. Subsequently it was decided to publish two volumes of papers contributed in his honour to mark this occasion: this issue of Mathematical Structures in Computer Science is one of the volumes; the other is Volume 6 of the journal Theory and Applications of Categories. At the December 1997 conference, a brief biographical essay was presented by Michael Barr and appears in the TAC volume. However, we wish to make some further remarks here.
Jim completed his Ph.D. at McGill under Hans Zassenhaus in 1950, and has remained at McGill since then. But it is of interest to note that Jim wrote two theses: the second involved biquaternions in mathematical physics, and so foreshadows a significant feature of his career: Jim has consistently shown a remarkable range of interests, from physics to linguistics, from algebra to logic, from the history and philosophy of mathematics to the theory of computing science (although he never touches a computer, to this day!). Let us just review a small sample of his more than 100 published papers.
Introduction to linear bicategories
- J. R. B. COCKETT, J. KOSLOWSKI, R. A. G. SEELY
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- Mathematical Structures in Computer Science / Volume 10 / Issue 2 / April 2000
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- 01 April 2000, pp. 165-203
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Linear bicategories are a generalization of bicategories in which the one horizontal composition is replaced by two (linked) horizontal compositions. These compositions provide a semantic model for the tensor and par of linear logic: in particular, as composition is fundamentally non-commutative, they provide a suggestive source of models for non-commutative linear logic.
In a linear bicategory, the logical notion of complementation becomes a natural linear notion of adjunction. Just as ordinary adjoints are related to (Kan) extensions, these linear adjoints are related to the appropriate notion of linear extension.
There is also a stronger notion of complementation, which arises, for example, in cyclic linear logic. This sort of complementation is modelled by cyclic adjoints. This leads to the notion of a *ast;-linear bicategory and the coherence conditions that it must satisfy. Cyclic adjoints also give rise to linear monads: these are, essentially, the appropriate generalization (to the linear setting) of Frobenius algebras and the ambialgebras of Topological Quantum Field Theory.
A number of examples of linear bicategories arising from different sources are described, and a number of constructions that result in linear bicategories are indicated.
Categorical semantics for higher order polymorphic lambda calculus
- R. A. G. Seely
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- The Journal of Symbolic Logic / Volume 52 / Issue 4 / December 1987
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- 12 March 2014, pp. 969-989
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- December 1987
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A categorical structure suitable for interpreting polymorphic lambda calculus (PLC) is defined, providing an algebraic semantics for PLC which is sound and complete. In fact, there is an equivalence between the theories and the categories. Also presented is a definitional extension of PLC including “subtypes”, for example, equality subtypes, together with a construction providing models of the extended language, and a context for Girard's extension of the Dialectica interpretation.
Locally cartesian closed categories and type theory
- R. A. G. Seely
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- Journal:
- Mathematical Proceedings of the Cambridge Philosophical Society / Volume 95 / Issue 1 / January 1984
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- 24 October 2008, pp. 33-48
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- January 1984
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It is well known that for much of the mathematics of topos theory, it is in fact sufficient to use a category C whose slice categories C/A are cartesian closed. In such a category, the notion of a ‘generalized set’, for example an ‘A-indexed set’, is represented by a morphism B → A of C, i.e. by an object of C/A. The point about such a category C is that C is a C-indexed category, and more, is a hyper-doctrine, so that it has a full first order logic associated with it. This logic has some peculiar aspects. For instance, the types are the objects of C and the terms are the morphisms of C. For a given type A, the predicates with a free variable of type A are morphisms into A, and ‘proofs’ are morphisms over A. We see here a certain ‘ambiguity’ between the notions of type, predicate, and term, of object and proof: a term of type A is a morphism into A, which is a predicate over A; a morphism 1 → A can be viewed either as an object of type A or as a proof of the proposition A.