Higher Lenses

We show that total, very well-behaved lenses are not very well-behaved when treated proof-relevantly in the setting of homotopy type theory/univalent foundations. In their place we propose something more well-behaved: higher lenses. Such a lens contains an equivalence between the lens's source...

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Published in:Proceedings - Symposium on Logic in Computer Science Vol. 2021-June; pp. 1 - 13
Main Authors: Capriotti, Paolo, Danielsson, Nils Anders, Vezzosi, Andrea
Format: Conference Proceeding
Language:English
Published: IEEE 29.06.2021
Subjects:
Computer circuits
Computer science
Constant functions
Functions
Gettering
Homotopy types
Lenses
Optical fibers
Optics
Source types
ISSN:1043-6871
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Abstract We show that total, very well-behaved lenses are not very well-behaved when treated proof-relevantly in the setting of homotopy type theory/univalent foundations. In their place we propose something more well-behaved: higher lenses. Such a lens contains an equivalence between the lens's source type and the product of its view type and a remainder type, plus a function from the remainder type to the propositional truncation of the view type. It can equivalently be formulated as a getter function and a proof that its family of fibres is coherently constant, i.e. factors through propositional truncation.We explore the properties of higher lenses. For instance, we prove that higher lenses are equivalent to traditional ones for types that satisfy the principle of uniqueness of identity proofs. We also prove that higher lenses are n-truncated for n-truncated types, using a coinductive characterisation of coherently constant functions.
AbstractList We show that total, very well-behaved lenses are not very well-behaved when treated proof-relevantly in the setting of homotopy type theory/univalent foundations. In their place we propose something more well-behaved: higher lenses. Such a lens contains an equivalence between the lens's source type and the product of its view type and a remainder type, plus a function from the remainder type to the propositional truncation of the view type. It can equivalently be formulated as a getter function and a proof that its family of fibres is coherently constant, i.e. factors through propositional truncation.We explore the properties of higher lenses. For instance, we prove that higher lenses are equivalent to traditional ones for types that satisfy the principle of uniqueness of identity proofs. We also prove that higher lenses are n-truncated for n-truncated types, using a coinductive characterisation of coherently constant functions.
Author Capriotti, Paolo
Vezzosi, Andrea
Danielsson, Nils Anders
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  givenname: Nils Anders
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  fullname: Danielsson, Nils Anders
  organization: University of Gothenburg
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  givenname: Andrea
  surname: Vezzosi
  fullname: Vezzosi, Andrea
  organization: IT University Copenhagen
BackLink https://research.chalmers.se/publication/549468$$DView record from Swedish Publication Index (Chalmers tekniska högskola)
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Snippet We show that total, very well-behaved lenses are not very well-behaved when treated proof-relevantly in the setting of homotopy type theory/univalent...
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SubjectTerms Computer circuits
Computer science
Constant functions
Functions
Gettering
Homotopy types
Lenses
Optical fibers
Optics
Source types
Title Higher Lenses
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https://research.chalmers.se/publication/549468
Volume 2021-June
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