In category theory, a branch of mathematics, a dual object is an analogue of a dual vector space from linear algebra for objects in arbitrary monoidal categories. It is only a partial generalization, based upon the categorical properties of duality for finite-dimensional vector spaces. An object admitting a dual is called a dualizable object. In this formalism, infinite-dimensional vector spaces are not dualizable, since the dual vector space V∗ doesn't satisfy the axioms.[1] Often, an object is dualizable only when it satisfies some finiteness or compactness property.[2]
A category in which each object has a dual is called autonomous or rigid. The category of finite-dimensional vector spaces with the standard tensor product is rigid, while the category of all vector spaces is not.
^Ponto, Kate; Shulman, Michael (2014). "Traces in symmetric monoidal categories". Expositiones Mathematicae. 32 (3): 248–273. arXiv:1107.6032. Bibcode:2011arXiv1107.6032P. doi:10.1016/j.exmath.2013.12.003.
^Becker, James C.; Gottlieb, Daniel Henry (1999). "A history of duality in algebraic topology" (PDF). In James, I.M. (ed.). History of topology. North Holland. pp. 725–745. ISBN 978-0-444-82375-5.
theory, a branch of mathematics, a dualobject is an analogue of a dual vector space from linear algebra for objects in arbitrary monoidal categories....
square copula is a common example of the dual copula strategy used in reference to the "problem of nonexistent objects" as well as their relation to problems...
is the most characteristic feature of Plato's dualism; that noumena and the noumenal world are objects of the highest knowledge, truths, and values is...
and prefixes that express a range of properties, including subject and/or object agreement, tense and aspect, and evidentiality. Verbs are the central, obligatory...
the dualobject has the same nature as the source one (like in the Pontryagin duality itself), and the theories where the source object and its dual differ...
morphisms for each object). Morphisms may be some sort of function. For example, a monoid may be viewed as a category with a single object, whose morphisms...
an initial object of a category C is an object I in C such that for every object X in C, there exists precisely one morphism I → X. The dual notion is...
and Perst (McObject), available under dual open source and commercial licensing. 1966 MUMPS 1979 InterSystems M 1980 TORNADO – an object database for...
theory and in the theory of model categories. The dual notion is that of a projective object. An object Q {\displaystyle Q} in a category C {\displaystyle...
coalgebra is the dualobject of an algebra and conversely). If A is a finite-dimensional unital associative K-algebra, then its K-dual A∗ consisting of...
person of the object is marked by a suffix. If either subject or object is dual or plural, it is shown with a plural suffix following the object suffix. So...
mathematics a Lie coalgebra is the dual structure to a Lie algebra. In finite dimensions, these are dualobjects: the dual vector space to a Lie algebra naturally...
projective object generalizes the notion of a projective module. Projective objects in abelian categories are used in homological algebra. The dual notion...
Groot dualDual abelian variety Dual basis in a field extension Dual bundle Dual curve Dual (category theory) Dual graph Dual group DualobjectDual pair...
Dual identity can refer to: A secret identity, such as Clark Kent and Superman In mathematics, the coidentity of a dual group object or the counit of a...
subject–objectduality. The aim of the so-called break-through koan is to see the "nonduality of subject and object", in which "subject and object are no...
direct object's number. The singular object is expressed with the -ыл- suffix which changes depending on the mood and tense. The dualobject is expressed...
Dual consciousness (or Dual mind) is a hypothesis or concept in neuroscience. It is proposed that it is possible that a person may develop two separate...
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