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

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

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.