In
mathematics, a categorical ring is, roughly, a
category
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) ...
equipped with addition and multiplication. In other words, a categorical ring is obtained by replacing the
underlying set
In mathematics, an algebraic structure consists of a nonempty set ''A'' (called the underlying set, carrier set or domain), a collection of operations on ''A'' (typically binary operations such as addition and multiplication), and a finite set of ...
of a
ring
Ring may refer to:
* Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry
* To make a sound with a bell, and the sound made by a bell
:(hence) to initiate a telephone connection
Arts, entertainment and media Film and ...
by a category. For example, given a ring ''R'', let ''C'' be a category whose
objects
Object may refer to:
General meanings
* Object (philosophy), a thing, being, or concept
** Object (abstract), an object which does not exist at any particular time or place
** Physical object, an identifiable collection of matter
* Goal, an ai ...
are the elements of the
set ''R'' and whose
morphism
In mathematics, particularly in category theory, a morphism is a structure-preserving map from one mathematical structure to another one of the same type. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphis ...
s are only the identity morphisms. Then ''C'' is a categorical ring. But the point is that one can also consider the situation in which an element of ''R'' comes with a "nontrivial
automorphism" (cf. Lurie).
This line of generalization of a ring eventually leads to the notion of an
''E''''n''-ring.
See also
*
Categorification
In mathematics, categorification is the process of replacing set-theoretic theorems with category-theoretic analogues. Categorification, when done successfully, replaces sets with categories, functions with functors, and equations with natu ...
*
Higher-dimensional algebra
In mathematics, especially ( higher) category theory, higher-dimensional algebra is the study of categorified structures. It has applications in nonabelian algebraic topology, and generalizes abstract algebra.
Higher-dimensional categories
A ...
References
*Laplaza, M. Coherence for distributivity. Coherence in categories, 29-65. Lecture Notes in Mathematics 281, Springer-Verlag, 1972.
*Lurie, J. Derived Algebraic Geometry V: Structured Spaces
External links
*http://ncatlab.org/nlab/show/2-rig
{{algebra-stub
Higher category theory