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
:
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
- {{citation
| last1 = Adámek | first1 = Jiří
| last2 = Rosický | first2 = Jiří
| doi = 10.1017/CBO9780511600579
| isbn = 0-521-42261-2
| location = Cambridge
| mr = 1294136
| publisher = Cambridge University Press
| series = London Mathematical Society Lecture Note Series
| title = Locally Presentable and Accessible Categories
| volume = 189
| year = 1994}}.
- {{citation
| last1 = Barr | first1 = Michael | author1-link = Michael Barr (mathematician)
| last2 = Wells | first2 = Charles | author2-link = Charles Wells (mathematician)
| edition = revised
| series = Reprints in Theory and Applications of Categories
| mr = 2178101
| title = Toposes, Triples and Theories
| url = http://www.tac.mta.ca/tac/reprints/articles/12/tr12abs.html
| volume = 12
| year = 2005}}.
- {{citation
| last = Borceux
| first = Francis
| isbn = 0-521-44179-X
| location = Cambridge
| mr = 1313497
| publisher = Cambridge University Press
| series = Encyclopedia of Mathematics and its Applications
| title = Handbook of Categorical Algebra. 2. Categories and Structures
| volume = 51
| year = 1994
| url-access = registration
| url = https://archive.org/details/handbookofcatego0000borc
}}.
- {{citation
| last = Ehresmann | first = Charles | author-link = Charles Ehresmann
| issue = fasc. 1-2
| journal = Bul. Inst. Politehn. Iaşi |series=New Series
| mr = 0238918
| pages = 1–14
| title = Esquisses et types des structures algébriques
| volume = 14 (18)
| year = 1968}}.
- {{citation
| last = Johnstone | first = Peter T. | author-link = Peter Johnstone (mathematician)
| isbn = 0-19-851598-7
| location = Oxford
| mr = 2063092
| publisher = The Clarendon Press, Oxford University Press
| series = Oxford Logic Guides
| title = Sketches of an elephant: a topos theory compendium. Vol. 2
| volume = 44
| year = 2002}}.
- {{citation
| last1 = Makkai | first1 = Michael | author1-link = Michael Makkai
| last2 = Paré | first2 = Robert
| isbn = 0-8218-5111-X
| location = Providence, RI
| mr = 1031717
| publisher = American Mathematical Society
| series = Contemporary Mathematics
| title = Accessible Categories: The Foundations of Categorical Model Theory
| volume = 104
| year = 1989}}.
External links
- [http://www.cwru.edu/artsci/math/wells/pub/pdf/Sketch.pdf Sketches: Outline with references] (updated 2009).
- {{nlab|id=sketch}}
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