nLab theory of abelian groups



Type theory

Topos Theory

natural deduction metalanguage, practical foundations

  1. type formation rule
  2. term introduction rule
  3. term elimination rule
  4. computation rule

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

syntax object language

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

logiccategory theorytype theory
trueterminal object/(-2)-truncated objecth-level 0-type/unit type
falseinitial objectempty type
proposition(-1)-truncated objecth-proposition, mere proposition
proofgeneralized elementprogram
cut rulecomposition of classifying morphisms / pullback of display mapssubstitution
cut elimination for implicationcounit for hom-tensor adjunctionbeta reduction
introduction rule for implicationunit for hom-tensor adjunctioneta conversion
logical conjunctionproductproduct type
disjunctioncoproduct ((-1)-truncation of)sum type (bracket type of)
implicationinternal homfunction type
negationinternal hom into initial objectfunction type into empty type
universal quantificationdependent productdependent product type
existential quantificationdependent sum ((-1)-truncation of)dependent sum type (bracket type of)
equivalencepath space objectidentity type/path type
equivalence classquotientquotient type
inductioncolimitinductive type, W-type, M-type
higher inductionhigher colimithigher inductive type
-0-truncated higher colimitquotient inductive type
coinductionlimitcoinductive type
presettype without identity types
completely presented setdiscrete object/0-truncated objecth-level 2-type/set/h-set
setinternal 0-groupoidBishop set/setoid
universeobject classifiertype of types
modalityclosure operator, (idempotent) monadmodal type theory, monad (in computer science)
linear logic(symmetric, closed) monoidal categorylinear type theory/quantum computation
proof netstring diagramquantum circuit
(absence of) contraction rule(absence of) diagonalno-cloning theorem
synthetic mathematicsdomain specific embedded programming language

homotopy levels


topos theory



Internal Logic

Topos morphisms

Extra stuff, structure, properties

Cohomology and homotopy

In higher category theory




The theory of abelian groups TT is the logical theory whose models in a cartesian category are the abelian group objects.


The theory of abelian groups TT is the theory over the signature with one sort XX, one constant 00, two function symbols ++ (of sort X×XXX\times X\to X) and - (of sort XXX\to X) and equality with axioms:

  • xx+0=x\top\vdash_x x+0=x \quad
  • x,y,zx+(y+z)=(x+y)+z\top\vdash_{x,y,z} x+(y+z)=(x+y)+z\quad,
  • xx+(x)=0\top\vdash_x x+(-x)=0\quad.
  • x,yx+y=y+x\top\vdash_{x,y} x+y=y+x\quad.

The theory of abelian groups is algebraic. In the following we list some extensions that use increasingly stronger fragments of geometric logic.

Extensions of the theory of abelian groups

The theory of torsion-free abelian groups T T^\infty results from TT by addition of the following axioms:

  • For all n2n\geq 2: ((nx=0)) x(x=0))((nx=0))\vdash_x (x=0))\quad.

The resulting theory is a Horn theory.

The theory of divisible abelian groups T \T^\backslash results from TT by addition of the following regular axioms:

  • For all n2n\geq 2: x(y)(ny=x)\top\vdash_x (\exists y) (ny=x)\quad.

The theory of divisible torsion-free abelian groups T \T^{\backslash\infty} results from T \T^\backslash by adding the above axioms for torsion-freeness. The resulting theory is cartesian.

The theory of torsion abelian groups T ~T^~ is an infinitary geometric theory resulting from TT by addition of:

  • x n2(nx=0)\top\vdash_x \bigvee_{n\geq 2} (nx=0)\quad.


Last revised on October 29, 2014 at 12:58:28. See the history of this page for a list of all contributions to it.