Sketch (mathematics)
Encyclopedia
A sketch is a category
D, together with a set of limit cone
s and a set of colimit cones. 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 transformation
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 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.
Category (mathematics)
In mathematics, a category is an algebraic structure that comprises "objects" that are linked by "arrows". A category has two basic properties: the ability to compose the arrows associatively and the existence of an identity arrow for each object. A simple example is the category of sets, whose...
D, together with a set of limit cone
Cone (category theory)
In category theory, a branch of mathematics, the cone of a functor is an abstract notion used to define the limit of that functor. Cones make other appearances in category theory as well.-Definition:...
s and a set of colimit cones. A model of the sketch in a category C is a functor
Functor
In category theory, a branch of mathematics, a functor is a special type of mapping between categories. Functors can be thought of as homomorphisms between categories, or morphisms when in the category of small categories....
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
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 of the categories involved. Hence, a natural transformation can be considered to be a "morphism of functors". Indeed this intuition...
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 and its models. They allow multisorted models and models in any category.
Sketches were invented in 1968 by Charles Ehresmann
Charles Ehresmann
Charles Ehresmann was a French mathematician who worked on differential topology and category theory. He is known for work on the topology of Lie groups, the jet concept , and his seminar on category theory.He attended the École Normale Supérieure in Paris before performing one year of military...
, using a different but equivalent definition. There are still other definitions in the research literature.