nLab ring groupoid

Contents

Context

Algebra

Type theory

natural deduction metalanguage, practical foundations

judgement
hypothetical judgement, sequent
- antecedents $\vdash$ consequent, succedents

type theory (dependent, intensional, observational type theory, homotopy type theory)

calculus of constructions

syntax object language

theory, axiom
proposition/type (propositions as types)
definition/proof/program (proofs as programs)
theorem

computational trinitarianism =
propositions as types +programs as proofs +relation type theory/category theory

logic	category theory	type theory
true	terminal object/(-2)-truncated object	h-level 0-type/unit type
false	initial object	empty type
proposition	(-1)-truncated object	h-proposition, mere proposition
proof	generalized element	program
cut rule	composition of classifying morphisms / pullback of display maps	substitution
cut elimination for implication	counit for hom-tensor adjunction	beta reduction
introduction rule for implication	unit for hom-tensor adjunction	eta conversion
logical conjunction	product	product type
disjunction	coproduct ((-1)-truncation of)	sum type (bracket type of)
implication	internal hom	function type
negation	internal hom into initial object	function type into empty type
universal quantification	dependent product	dependent product type
existential quantification	dependent sum ((-1)-truncation of)	dependent sum type (bracket type of)
equivalence	path space object	identity type/path type
equivalence class	quotient	quotient type
induction	colimit	inductive type, W-type, M-type
higher induction	higher colimit	higher inductive type
-	0-truncated higher colimit	quotient inductive type
coinduction	limit	coinductive type
preset		type without identity types
completely presented set	discrete object/0-truncated object	h-level 2-type/set/h-set
set	internal 0-groupoid	Bishop set/setoid
universe	object classifier	type of types
modality	closure operator, (idempotent) monad	modal type theory, monad (in computer science)
linear logic	(symmetric, closed) monoidal category	linear type theory/quantum computation
proof net	string diagram	quantum circuit
(absence of) contraction rule	(absence of) diagonal	no-cloning theorem
	synthetic mathematics	domain specific embedded programming language

homotopy levels

semantics

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Categorification

Idea

A ring groupoid is a 1-truncated $A_\infty$ -ring.

Equivalently: A groupoid $G$ that is both a symmetric 2-group $(G, 0, \oplus)$ and a monoidal groupoid $(G, 1, \otimes)$ such that $\otimes$ distributes over $\oplus$ satisfying certain higher coherence laws (given in Kelly74).

Properties

The discrete groupoid of integers $\mathbb{Z}$ is the initial ring groupoid.
The contractible groupoid or the trivial ring groupoid $0$ is the terminal ring groupoid.

References

G. M. Kelly, Coherence theorems for lax algebras and distributive laws, Lecture Notes in Mathematics 420, Springer Verlag, Berlin(1974) 281-375 $[$ doi:10.1007/BFb0063106 $]$

Last revised on May 20, 2022 at 20:29:15. See the history of this page for a list of all contributions to it.

nLab ring groupoid

Context

Algebra

Algebraic theories

Algebras and modules

Higher algebras

Model category presentations

Geometry on formal duals of algebras

Theorems

Type theory

Categorification

Background

Contents

Examples

Category theory

Concepts

Universal constructions

Theorems

Extensions

Applications

Contents

Idea

Properties

See also

References