2-ring

In mathematics, a categorical ring is, roughly, a category equipped with addition and multiplication. In other words, a categorical ring is obtained by replacing the underlying set of a ring by a category. For example, given a ring R, let C be a category whose objects are the elements of the set R and whose morphisms 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
• Higher-dimensional algebra

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

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Category theory

Key concepts

Key concepts Category Abelian Additive Concrete Pre-abelian Preadditive Bicategory Adjoint functors CCC Commutative diagram End Exponential Functor Kan extension Morphism Natural transformation Universal property Universal constructions Limits Terminal objects Products Equalizers Kernels Pullbacks Inverse limit Colimits Initial objects Coproducts Coequalizers Cokernels and quotients Pushout Direct limit Algebraic categories Sets Relations Magmas Groups Abelian groups Rings (Fields) Modules (Vector spaces) Constructions on categories Free category Functor category Kleisli category Opposite category Quotient category Product category Comma category Subcategory

Higher category theory

Key concepts Categorification Enriched category Higher-dimensional algebra Homotopy hypothesis Model category Simplex category String diagram Topos n-categories Weak n-categories Bicategory (pseudofunctor) Tricategory Tetracategory Kan complex ∞-groupoid ∞-topos Strict n-categories 2-category (2-functor) 3-category Categorified concepts 2-group 2-ring En-ring (Traced)(Symmetric) monoidal category n-group n-monoid

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