• Corpus ID: 238582820

# Free Commutative Monoids in Homotopy Type Theory

@article{Choudhury2021FreeCM,
title={Free Commutative Monoids in Homotopy Type Theory},
author={Vikraman Choudhury and Marcelo P. Fiore},
journal={ArXiv},
year={2021},
volume={abs/2110.05412}
}
• Published 11 October 2021
• Mathematics
• ArXiv
We develop a constructive theory of finite multisets, defining them as free commutative monoids in Homotopy Type Theory. We formalise two algebraic presentations of this construction using 1-HITs, establishing the categorical universal property for each and thereby their equivalence. These presentations correspond to equational theories including a commutation axiom. In this setting, we prove important structural combinatorial properties of singleton multisets arising from concatenations and…
1 Citations
Symmetries in reversible programming: from symmetric rig groupoids to reversible programming languages
• Mathematics, Computer Science
Proc. ACM Program. Lang.
• 2022
This paper gives a denotational semantics for the Pi family of reversible programming languages for boolean circuits, using weak groupoids à la Homotopy Type Theory, and shows how to derive an equational theory for it, presented by 2-combinators witnessing equivalences of type isomorphisms.

## References

SHOWING 1-10 OF 54 REFERENCES
On Higher Inductive Types in Cubical Type Theory
• Mathematics
LICS
• 2018
A constructive semantics, expressed in a presheaf topos with suitable structure inspired by cubical sets, of some higher inductive types of spheres, torus, suspensions, truncations, and pushouts is described.
The Integers as a Higher Inductive Type
• Mathematics
LICS
• 2020
This paper considers higher inductive types using either a small universe or bi-invertible maps that represent integers without explicit set-truncation that are equivalent to the usual coproduct representation.
Semantics of higher inductive types
• Mathematics
Mathematical Proceedings of the Cambridge Philosophical Society
• 2019
Abstract Higher inductive types are a class of type-forming rules, introduced to provide basic (and not-so-basic) homotopy-theoretic constructions in a type-theoretic style. They have proven very
Constructing Infinitary Quotient-Inductive Types
• Mathematics
FoSSaCS
• 2020
It is shown that in dependent type theory with uniqueness of identity proofs, even the infinitary case of QW-types can be encoded using the combination of inductive-inductive definitions involving strictly positive occurrences of Hofmann-style quotient types, and Abel’s size types.
Higher Groups in Homotopy Type Theory
• Mathematics
LICS
• 2018
A development of the theory of higher groups, including infinity groups and connective spectra, in homotopy type theory is presented, which states that if an n-type can be delooped n + 2 times, then it is an infinite loop type.
Differential categories
• Mathematics
Mathematical Structures in Computer Science
• 2006
The notion of a categorical model of the differential calculus is introduced, and it is shown that it captures the not-necessarily-closed fragment of Ehrhard–Regnier differential $\lambda$-calculus.
Finite sets in homotopy type theory
• Mathematics, Computer Science
CPP
• 2018
It is shown how to use K(A) for an abstract interface for well-known finite set implementations such as tree- and list-like data structures and bounded quantification, which lifts a decidable property on A to one onK(A).
Higher inductive types in cubical computational type theory
• Mathematics
Proc. ACM Program. Lang.
• 2019
This work extends the cartesian cubical computational type theory introduced by Angiuli et al. with a schema for indexed cubical inductive types (CITs), an adaptation of higher induction types to the cubical setting, and proves a canonicity theorem with respect to these values.
Multisets in type theory
• H. Gylterud
• Mathematics, Computer Science
Mathematical Proceedings of the Cambridge Philosophical Society
• 2019
This work investigates the notion of iterative multisets, whereMultisets are iteratively built up from other multiset, in the context Martin–Löf Type Theory, inThe presence of Voevodsky’s Univalence Axiom.
Cubical Type Theory: A Constructive Interpretation of the Univalence Axiom
• Computer Science
TYPES
• 2015
A type theory in which it is possible to directly manipulate n-dimensional cubes based on an interpretation of dependenttype theory in a cubical set model that enables new ways to reason about identity types, for instance, function extensionality is directly provable in the system.