In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, a categorical ring is, roughly, a
category
Category, plural categories, may refer to:
General uses
*Classification, the general act of allocating things to classes/categories Philosophy
* Category of being
* ''Categories'' (Aristotle)
* Category (Kant)
* Categories (Peirce)
* Category ( ...
equipped with addition and multiplication. In other words, a categorical ring is obtained by replacing the
underlying set
In mathematics, an algebraic structure or algebraic system 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 ...
of a
ring
(The) Ring(s) 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
Arts, entertainment, and media Film and TV
* ''The Ring'' (franchise), a ...
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
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, a morphism is a concept of category theory that generalizes structure-preserving maps such as homomorphism between algebraic structures, functions from a set to another set, and continuous functions between topological spaces. Al ...
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 automorphism ...
".
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) category theory, higher-dimensional algebra is the study of Categorification, categorified structures. It has applications in nonabelian algebraic topology, and generalizes abstract algebr ...
*
Lie n-algebra
In mathematics, a Lie ''n''-algebra is a generalization of a Lie algebra, a vector space with a bracket, to higher order operations. For example, in the case of a Lie 2-algebra, the Jacobi identity is replaced by an isomorphism
In mathematics, ...
Further reading
* John Baez
2-Rigs in Topology and Representation Theory
References
*
External links
*http://ncatlab.org/nlab/show/2-rig
Higher category theory
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