Sketch (Mathematics)

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

${\displaystyle M:D\to C}$

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.

• 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 .
• Toposes, Triples and Theories, Reprints in Theory and Applications of Categories, 12 (revised ed.), 2005, http://www.tac.mta.ca/tac/reprints/articles/12/tr12abs.html .
• 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, https://archive.org/details/handbookofcatego0000borc .
• "Esquisses et types des structures algébriques", Bul. Inst. Politehn. Iaşi, New Series 14 (18) (fasc. 1-2): 1–14, 1968 .
• Sketches of an elephant: a topos theory compendium. Vol. 2, Oxford Logic Guides, 44, Oxford: The Clarendon Press, Oxford University Press, 2002, ISBN 0-19-851598-7 .
• Accessible Categories: The Foundations of Categorical Model Theory, Contemporary Mathematics, 104, Providence, RI: American Mathematical Society, 1989, ISBN 0-8218-5111-X .

• Sketches: Outline with references (updated 2009).
• sketch in nLab

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