Free Higher Groups in Homotopy Type Theory

Kraus, Nicolai; Altenkirch, Thorsten

doi:10.1145/3209108.3209183

Cited by 12 publications

(23 citation statements)

References 17 publications

(13 reference statements)

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“…Of course, the intermediate goal that this paper is working towards is the statement that 2LTT can eat HoTT. Even a much weaker question, namely whether the initial ∞-CwF has trivial fundamental group, would be very interesting; but already this seemingly much simpler problem appears to be highly non-trivial, and similar results for seemingly simpler situations [65]- [67] suggest that a solution will require new techniques.…”

Section: Open Problems and Future Directionsmentioning

confidence: 99%

Internal ∞-Categorical Models of Dependent Type Theory : Towards 2LTT Eating HoTT

Kraus

2021

2021 36th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS)

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Using dependent type theory to formalise the syntax of dependent type theory is a very active topic of study and goes under the name of "type theory eating itself" or "type theory in type theory." Most approaches are at least loosely based on Dybjer's categories with families (CwF's) and come with a type Con of contexts, a type family Ty indexed over it modelling types, and so on. This works well in versions of type theory where the principle of unique identity proofs (UIP) holds. In homotopy type theory (HoTT) however, it is a long-standing and frequently discussed open problem whether the type theory "eats itself" and can serve as its own interpreter. The fundamental underlying difficulty seems to be that categories are not suitable to capture a type theory in the absence of UIP.In this paper, we develop a notion of ∞-categories with families (∞-CwF's). The approach to higher categories used relies on the previously suggested semi-Segal types, with a new construction of identity substitutions that allow for both univalent and non-univalent variations. The type-theoretic universe as well as the internalised (set-level) syntax are models, although it remains a conjecture that the latter is initial. To circumvent the known unsolved problem of constructing semisimplicial types, the definition is presented in two-level type theory (2LTT).Apart from introducing ∞-CwF's, the paper explains the shortcomings of 1-categories in type theory without UIP as well as the difficulties of and approaches to internal higher-dimensional categories.

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Section: Open Problems and Future Directionsmentioning

confidence: 99%

Internal ∞-Categorical Models of Dependent Type Theory : Towards 2LTT Eating HoTT

Kraus

2021

2021 36th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS)

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“…This means that a normalisation function is only definable for sets a : A with a decidable equality. The general problem of whether F(A) is a set, if A is, is still open -in [KA18] it is shown to be the case, if we 1-truncate the HIT.…”

Section: Introductionmentioning

confidence: 99%

The Integers as a Higher Inductive Type

Altenkirch

Scoccola

2020

Proceedings of the 35th Annual ACM/IEEE Symposium on Logic in Computer Science

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We consider the problem of defining the integers in Homotopy Type Theory (HoTT). We can define the type of integers as signed natural numbers (i.e., using a coproduct), but its induction principle is very inconvenient to work with, since it leads to an explosion of cases. An alternative is to use set-quotients, but here we need to use set-truncation to avoid non-trivial higher equalities. This results in a recursion principle that only allows us to define function into sets (types satisfying UIP). In this paper we consider higher inductive types using either a small universe or bi-invertible maps. These types represent integers without explicit set-truncation that are equivalent to the usual coproduct representation. This is an interesting example since it shows how some coherence problems can be handled in HoTT. We discuss some open questions triggered by this work. The proofs have been formally verified using cubical Agda.

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“…This construction is completely reasonable without the assumption that A is a set, and it therefore defines the free higher group (cf. [21]); from now on, we write F A for it: F A :≡ Ω(hcolim(A ⇒ 1)).…”

Section: Introductionmentioning

confidence: 99%

“…It is well-known in the community how to circumvent the problem if A has decidable equality. However, the only piece of progress on the general question that we are aware of is the result in [21], where it is shown that all fundamental groups [28,Chp 6.11] are trivial. In other words: Instead of showing that everything above truncation level 0 is trivial, the result shows that a single level is trivial.…”

Section: Introductionmentioning

confidence: 99%

“…In other words: Instead of showing that everything above truncation level 0 is trivial, the result shows that a single level is trivial. The proof in [21] uses a rather intricate construction which is precisely tailored to the situation. The construction of functions (A/∼) → X is the guiding motivation for the results that we present in this paper.…”

Section: Introductionmentioning

confidence: 99%

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Coherence via Well-Foundedness

Kraus

Raumer

2020

Proceedings of the 35th Annual ACM/IEEE Symposium on Logic in Computer Science

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Suppose we are given a graph and want to show a property for all its cycles (closed chains). Induction on the length of cycles does not work since sub-chains of a cycle are not necessarily closed. This paper derives a principle reminiscent of induction for cycles for the case that the graph is given as the symmetric closure of a locally confluent and (co-)well-founded relation. We show that, assuming the property in question is sufficiently nice, it is enough to prove it for the empty cycle and for cycles given by local confluence.Our motivation and application is in the field of homotopy type theory, which allows us to work with the higher-dimensional structures that appear in homotopy theory and in higher category theory, making coherence a central issue. This is in particular true for quotienting -a natural operation which gives a new type for any binary relation on a type and, in order to be wellbehaved, cuts off higher structure (set-truncates). The latter makes it hard to characterise the type of maps from a quotient into a higher type, and several open problems stem from this difficulty.We prove our theorem on cycles in a type-theoretic setting and use it to show coherence conditions necessary to eliminate from set-quotients into 1types, deriving approximations to open problems on free groups and pushouts. We have formalised the main result in the proof assistant Lean.

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Free Higher Groups in Homotopy Type Theory

Abstract: Given a type A in homotopy type theory (HoTT), we can define the free ∞-group on A as the loop space of the suspension of A + 1. Equiv-

Cited by 12 publications

References 17 publications

Internal ∞-Categorical Models of Dependent Type Theory : Towards 2LTT Eating HoTT

Internal ∞-Categorical Models of Dependent Type Theory : Towards 2LTT Eating HoTT

The Integers as a Higher Inductive Type

Coherence via Well-Foundedness

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