Internal languages of finitely complete 
                
                  
                
                $$(\infty , 1)$$
                
                  
                    
                      (
                      ∞
                      ,
                      1
                      )
                    
                  
                
              -categories

Kapulkin, Chris; Szumiło, Karol

doi:10.1007/s00029-019-0480-0

Cited by 6 publications

(4 citation statements)

References 27 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Since this paper was first made publicly available in 2016, a part of Conjecture 3.7 was proven in [KS17]. Specifically, the ∞-category Lex ∞ is shown there to be equivalent to the ∞-category of comprehension categories with Id, 1, Σ-types, which reduces the conjecture to a comparison between contextual categories and comprehension categories with the appropriate structure.…”

Section: : Cxlcat Hottmentioning

confidence: 93%

The homotopy theory of type theories

Kapulkin

Lumsdaine

2018

Advances in Mathematics

Self Cite

View full text Add to dashboard Cite

We construct a left semi-model structure on the category of intensional type theories (precisely, on CxlCat Id,1,Σ(,Πext ) ). This presents an ∞-category of such type theories; we show moreover that there is an ∞-functor Cl∞ from there to the ∞-category of suitably structured quasi-categories.This allows a precise formulation of the conjectures that intensional type theory gives internal languages for higher categories, and provides a framework and toolbox for further progress on these conjectures.

show abstract

Section: : Cxlcat Hottmentioning

confidence: 93%

The homotopy theory of type theories

Kapulkin

Lumsdaine

2018

Advances in Mathematics

Self Cite

View full text Add to dashboard Cite

show abstract

“…The composition of the front square 2-cells κ : (12), and hence a natural equivalence in virtue of the associated Beck-Chevalley condition. We obtain two homotopy-commutative squares in Fun(I ∂∆ 1 , C) as follows.…”

Section: Higher Covering Diagramsmentioning

confidence: 99%

“…Now, the circumstance that formulas are modelled as subobjects is an immediate consequence of proof-irrelevance (and consequently extensionality) of first-order logic. In the ∞-categorical context of cartesian ∞-categories with possibly more logical structure however, the internal languages are proof-relevant and non-extensional type theories ( [12], [11], [18]), and hence require the modelling of formulas as type families which are generally far from being monic. On the flipside, this poses the question what notion of topology ought to be considered to reflect this.…”

Section: Introductionmentioning

confidence: 99%

Higher geometric sheaf theories

Stenzel¹

2022

Preprint

View full text Add to dashboard Cite

We introduce an ∞-category of geometric ∞-categories, whose objects are defined by dropping the effectivity condition on colimits in Anel and Joyal's definition of ∞-logos. We propose a theory of higher geometric sheaves on geometric ∞categories C, which will be shown to differ from the ordinary geometric sheaf theory on C by a cotopological fragment. We prove that this fragment is crucial: for instance every ∞-topos is the theory of higher geometric sheaves over itself, but the corresponding cotopological localization of its ordinary geometric sheaf theory is generally non-trivial. The notion of higher geometric sheaves over geometric ∞categories hence faithfully generalizes Lurie's definition of sheaves over ∞-toposes.We define this class of sheaf theories by way of an adaption of Anel and Leena Subramaniam's ∆-modulators. The according sheaves are characterized by a limit preservation property that is generally not captured by the classical sheaf condition as known from ordinary sheaf theory over topological spaces (or over geometric categories more generally). The latter arises as a special case instead. To motivate this class of limits, we first introduce and discuss simpler but analogous sheaf theories for extensive and regular ∞-categories.

show abstract

“…Recently, homotopy type theory (The Univalent Foundations Program 2013) is expected to provide an internal language for what should be called "elementary (∞, 1)-toposes." As a first step, Kapulkin and Szumiło (2019) showed that there is an equivalence between intensional Martin-Löf theories and finitely complete (∞, 1)-categories.…”

Section: Introductionmentioning

confidence: 99%

A general framework for the semantics of type theory

Uemura¹

2023

Math. Struct. Comp. Sci.

View full text Add to dashboard Cite

We propose an abstract notion of a type theory to unify the semantics of various type theories including Martin–Löf type theory, two-level type theory, and cubical type theory. We establish basic results in the semantics of type theory: every type theory has a bi-initial model; every model of a type theory has its internal language; the category of theories over a type theory is bi-equivalent to a full sub-2-category of the 2-category of models of the type theory.

show abstract

Internal languages of finitely complete $$(\infty , 1)$$ ( ∞ , 1 ) -categories

Cited by 6 publications

References 27 publications

The homotopy theory of type theories

The homotopy theory of type theories

Higher geometric sheaf theories

A general framework for the semantics of type theory

Contact Info

Product

Resources

About