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
\( M:D\rightarrow C \)
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
Adámek, Jiří; Rosický, Jiří (1994), Locally Presentable and Accessible Categories, London Mathematical Society Lecture Note Series, 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, 12 (revised ed.), MR 2178101.
Borceux, Francis (1994), Handbook of Categorical Algebra. 2. Categories and Structures, Encyclopedia of Mathematics and its Applications, 51, Cambridge: Cambridge University Press, ISBN 0-521-44179-X, MR 1313497.
Ehresmann, Charles (1968), "Esquisses et types des structures algébriques", Bul. Inst. Politehn. Iaşi (N.S.), 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, 44, Oxford: The Clarendon Press, Oxford University Press, ISBN 0-19-851598-7, MR 2063092.
Makkai, Michael; Paré, Robert (1989), Accessible Categories: The Foundations of Categorical Model Theory, Contemporary Mathematics, 104, Providence, RI: American Mathematical Society, ISBN 0-8218-5111-X, MR 1031717.
External links
Sketches: Outline with references (updated 2009).
sketch in nLab
Undergraduate Texts in Mathematics
Graduate Studies in Mathematics
Hellenica World - Scientific Library
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