Definable and Non-definable Notions of Structure

@article{Swan2022DefinableAN,
  title={Definable and Non-definable Notions of Structure},
  author={Andrew W. Swan},
  journal={ArXiv},
  year={2022},
  volume={abs/2206.13643}
}
Definability is a key notion in the theory of Grothendieck fibrations that characterises when an external property of objects can be accessed from within the internal logic of the base of a fibration. In this paper we consider a generalisation of definability from properties of objects to structures on objects, introduced by Shulman under the name local representability. We first develop some general theory and show how to recover existing notions due to B´enabou and Johnstone as special cases. We… 

References

SHOWING 1-10 OF 51 REFERENCES

Lifting Problems in Grothendieck Fibrations

Many interesting classes of maps from homotopical algebra can be characterised as those maps with the right lifting property against certain sets of maps (such classes are sometimes referred to as

Algebraic Models of Dependent Type Theory

It has been observed [Awo16, Fio12] that the rules governing the essentially algebraic notion of a category with families [Dyb96] precisely match those of a representable natural transformation

Fibered Categories and the Foundations of Naive Category Theory

Any attempt to give “foundations”, for category theory or any domain in mathematics, could have two objectives: to provide a formal frame rich enough so that all the actual activity in the domain can be carried out within this frame, and consistent with a well-established and “safe” theory.

Axioms for Modelling Cubical Type Theory in a Topos

This work clarifies the definition and properties of the notion of uniform Kan filling that lies at the heart of the constructive interpretation of Voevodsky's univalence axiom and shows that there is a range of topos-theoretic models of homotopy type theory in this style.

Algebraic model structures

We define a new notion of an algebraic model structure, in which the cofibrations and fibrations are retracts of coalgebras for comonads and algebras for monads, and prove "algebraic" analogs of

W-Types with Reductions and the Small Object Argument

We define a simple kind of higher inductive type generalising dependent $W$-types, which we refer to as $W$-types with reductions. Just as dependent $W$-types can be characterised as initial algebras

Natural models of homotopy type theory

  • S. Awodey
  • Mathematics
    Mathematical Structures in Computer Science
  • 2016
It is shown that a category admits a natural model of type theory if it has a class of maps that behave like the abstract fibrations in axiomatic homotopy theory: They should be stable under pullback, closed under composition and relative products, and there should be weakly orthogonal factorizations into the class.

Nominal Presentation of Cubical Sets Models of Type Theory

This paper shows that a variant category of cubical sets with diagonals, equivalent to presheaves on Grothendieck's "smallest test category," has the pleasant property that path objects given by name abstraction are exponentials with respect to an interval object.

Identity Types in Algebraic Model Structures and Cubical Sets

We give a general technique for constructing a functorial choice of very good paths objects, which can be used to implement identity types in models of type theories in direct manner with little

Internal Universes in Models of Homotopy Type Theory

This work shows how to construct a universe that classifies the Cohen-Coquand-Huber-Mortberg (CCHM) notion of fibration from their cubical sets model, starting from the assumption that the interval is tiny - a property that the intervals in cubical set does indeed have.
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