In the
mathematical
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
theory of
categories
Category, plural categories, may refer to:
Philosophy and general uses
*Categorization, categories in cognitive science, information science and generally
*Category of being
* ''Categories'' (Aristotle)
*Category (Kant)
* Categories (Peirce)
* ...
, a sketch is a category ''D'', together with a set of
cone
A cone is a three-dimensional geometric shape that tapers smoothly from a flat base (frequently, though not necessarily, circular) to a point called the apex or vertex.
A cone is formed by a set of line segments, half-lines, or lines con ...
s 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
In mathematics, specifically category theory, a functor is a mapping between categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, and m ...
:
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 transformation
In category theory, a branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure (i.e., the composition of morphisms) of the categories involved. Hence, a natur ...
s. 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
A theory is a rational type of abstract thinking about a phenomenon, or the results of such thinking. The process of contemplative and rational thinking is often associated with such processes as observational study or research. Theories may be ...
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
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External links
Sketches: Outline with references(updated 2009).
*
Category theory
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