In the mathematical theory of categories, a sketch is a category D, together with a set of cones intended to be limits and a set of cocones intended to be colimits. A model of the sketch in a category C is a functor
that takes each specified cone to a limit cone in C and each specified cocone to a colimit cocone in C. Morphisms of models are natural transformations. Sketches are a general way of specifying structures on the objects of a category, forming a category-theoretic analog to the logical concept of a theory and its models. They allow multisorted models and models in any category.
Sketches were invented in 1968 by Charles Ehresmann, using a different but equivalent definition. There are still other definitions in the research literature.
Referencesedit
Adámek, Jiří; Rosický, Jiří (1994), Locally Presentable and Accessible Categories, London Mathematical Society Lecture Note Series, vol. 189, Cambridge: Cambridge University Press, doi:10.1017/CBO9780511600579, ISBN0-521-42261-2, MR 1294136.
Barr, Michael; Wells, Charles (2005), Toposes, Triples and Theories, Reprints in Theory and Applications of Categories, vol. 12 (revised ed.), MR 2178101.
Borceux, Francis (1994), Handbook of Categorical Algebra. 2. Categories and Structures, Encyclopedia of Mathematics and its Applications, vol. 51, Cambridge: Cambridge University Press, ISBN0-521-44179-X, MR 1313497.
Ehresmann, Charles (1968), "Esquisses et types des structures algébriques", Bul. Inst. Politehn. Iaşi, New Series, 14 (18) (fasc. 1-2): 1–14, MR 0238918.
Johnstone, Peter T. (2002), Sketches of an elephant: a topos theory compendium. Vol. 2, Oxford Logic Guides, vol. 44, Oxford: The Clarendon Press, Oxford University Press, ISBN0-19-851598-7, MR 2063092.
Makkai, Michael; Paré, Robert (1989), Accessible Categories: The Foundations of Categorical Model Theory, Contemporary Mathematics, vol. 104, Providence, RI: American Mathematical Society, ISBN0-8218-5111-X, MR 1031717.
sketch, mathematics, mathematical, theory, categories, sketch, category, together, with, cones, intended, limits, cocones, intended, colimits, model, sketch, category, functor, displaystyle, rightarrow, that, takes, each, specified, cone, limit, cone, each, sp. In the mathematical theory of categories a sketch is a category D together with a set of cones intended to be limits and a set of cocones intended to be colimits A model of the sketch in a category C is a functor M D C displaystyle M D rightarrow C that takes each specified cone to a limit cone in C and each specified cocone to a colimit cocone in C Morphisms of models are natural transformations Sketches are a general way of specifying structures on the objects of a category forming a category theoretic analog to the logical concept of a theory and its models They allow multisorted models and models in any category Sketches were invented in 1968 by Charles Ehresmann using a different but equivalent definition There are still other definitions in the research literature References editAdamek Jiri Rosicky Jiri 1994 Locally Presentable and Accessible Categories London Mathematical Society Lecture Note Series vol 189 Cambridge Cambridge University Press doi 10 1017 CBO9780511600579 ISBN 0 521 42261 2 MR 1294136 Barr Michael Wells Charles 2005 Toposes Triples and Theories Reprints in Theory and Applications of Categories vol 12 revised ed MR 2178101 Borceux Francis 1994 Handbook of Categorical Algebra 2 Categories and Structures Encyclopedia of Mathematics and its Applications vol 51 Cambridge Cambridge University Press ISBN 0 521 44179 X MR 1313497 Ehresmann Charles 1968 Esquisses et types des structures algebriques Bul Inst Politehn Iasi New Series 14 18 fasc 1 2 1 14 MR 0238918 Johnstone Peter T 2002 Sketches of an elephant a topos theory compendium Vol 2 Oxford Logic Guides vol 44 Oxford The Clarendon Press Oxford University Press ISBN 0 19 851598 7 MR 2063092 Makkai Michael Pare Robert 1989 Accessible Categories The Foundations of Categorical Model Theory Contemporary Mathematics vol 104 Providence RI American Mathematical Society ISBN 0 8218 5111 X MR 1031717 External links editSketches Outline with references updated 2009 sketch at the nLab nbsp This category theory related article is a stub You can help Wikipedia by expanding it vte Retrieved from https en wikipedia org w index php title Sketch mathematics amp oldid 1170053588, wikipedia, wiki, book, books, library,
