In
mathematics
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 ...
, 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 se ...
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 ...
are the elements of the
set
Set, The Set, SET or SETS may refer to:
Science, technology, and mathematics Mathematics
*Set (mathematics), a collection of elements
*Category of sets, the category whose objects and morphisms are sets and total functions, respectively
Electro ...
''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, morphisms a ...
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
In mathematics, an automorphism is an isomorphism from a mathematical object to itself. It is, in some sense, a symmetry of the object, and a way of mapping the object to itself while preserving all of its structure. The set of all automorphisms ...
" (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 natural ...
*
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 f ...
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