Homotopy Type Theory Higher algebra > history (Rev #30)

Here we collect articles about doing higher algebra or homotopical algebra in HoTT.

General

• H-space
• A3-space
• grouplike A3-space
• commutative A3-space
• infinity-group
• loop space
• prespectrum
• spectrum
• connective spectrum?

Simplicial models

• totally ordered type
• simplex category
• simplicial type
• combinatorial spectrum?

Decidable objects

• booleans
• decidable universal quantifier
• decidable existential quantifier
• decidable directed graph
• decidable setoid
• decidable set
• decidable preordered type
• decidable strict order
• decidable dense strict order?

Truncations

• contractible type
• proposition
• set
• groupoid

Groupoid-truncated types

• groupoid enriched in monoids
• pointed connected groupoid

Set-truncated types

Magmas

• monoid
• commutative monoid
• group
• abelian group/Z-module
• divisible group
• torsion-free divisible group/Q-module
• action
• magma
• power-associative magma
• commutant
• abelian group homomorphism
• bilinear function

Rings, modules, and algebras

• Z-module/abelian group
• Q-module/torsion-free divisible group
• abelian group homomorphism
• bilinear function
• Z-algebra
• Q-algebra
• square function
• quadratic function
• characteristic
• unital Z-algebra
• ring
• commutative ring
• algebra (ring theory)
• commutative algebra (ring theory)
• polynomial ring
• univariate polynomial ring
• power function
• polynomial function
• function algebra
• module
• bimodule
• quadratic form
• diagonal
• diagonal form?
• Clifford algebra
• geometric algebra
• algebra (module theory)
• Hausdorff ring

Sequences

• sequence

• left shift operator

• right shift operator

• Z-module

• sequential polynomial

• series operator

• inverse series operator

• sequential derivative

• Q-module

• sequential antiderivative

• sequential differential operator?

• sequential linear differential equation?

• sequential exponential?

• sequential sine?

• sequential cosine?

• bidirectional sequence

• bidirectional shift operators?

Fields and cancellation algebras

• field (ring theory)
• residue field?
• Heyting field
• discrete field
• reciprocal function
• rational function
• category of partial functions
• difference quotient
• Hausdorff field?
• sequentially Hausdorff field?
• calculus field?
• ordered field
• Archimedean ordered field
• interval-complete Archimedean ordered field?
• Dedekind complete Archimedean ordered field
• Cauchy complete Archimedean ordered field
• integral domain
• residue integral domain?
• Heyting integral domain
• discrete integral domain
• ordered integral domain
• Archimedean ordered integral domain
• interval-complete Archimedean ordered integral domain?
• Dedekind complete Archimedean ordered integral domain
• skewfield
• residue skewfield?
• Heyting skewfield
• discrete skewfield
• domain (ring theory)
• residue domain?
• Heyting domain
• discrete domain
• commutative reciprocal ring
• commutative residue reciprocal ring?
• commutative Heyting reciprocal ring
• commutative discrete reciprocal ring
• commutative division ring
• commutative residue division ring?
• commutative Heyting division ring
• commutative discrete division ring
• commutative cancellation ring
• commutative residue cancellation ring?
• commutative Heyting cancellation ring
• commutative discrete cancellation ring
• reciprocal ring
• residue reciprocal ring?
• Heyting reciprocal ring
• discrete reciprocal ring
• division ring
• residue division ring?
• Heyting division ring
• discrete division ring
• cancellation ring
• residue cancellation ring?
• Heyting cancellation ring
• discrete cancellation ring
• reciprocal Z-algebra
• residue reciprocal Z-algebra?
• Heyting reciprocal Z-algebra
• discrete reciprocal Z-algebra
• division Z-algebra
• residue division Z-algebra?
• Heyting division Z-algebra
• discrete division Z-algebra
• cancellation Z-algebra
• residue cancellation Z-algebra?
• Heyting cancellation Z-algebra
• discrete cancellation Z-algebra

References

• HoTT Book