Fredrik Bakke 2026 2 18 2026 3 9 https://samtoth.github.io/agda-synthetic-categories/Core.LeibnizAdjunction Core.LeibnizAdjunction /agda-synthetic-categories/Core.LeibnizAdjunction Leibniz Adjunction Module module Core.LeibnizAdjunction where open import Axioms.UF Fredrik Bakke2026218202639Imports open import Foundations.Prelude open import Core.Arrows open import Core.ArrowEquiv open import Core.Joins open import Core.PullbackPowers open import Core.PullbackPowersFamilies open import Core.PushoutSums open import Core.PushoutSumsFamilies Fredrik Bakke2026218202639https://samtoth.github.io/agda-synthetic-categories/stt-00FJ/stt-00FJ/agda-synthetic-categories/stt-00FJ/Iterated pullback powers and pushout sumsDefinition We define the dependent pullback power of the pushout sum of a doubly dependent type family, and the double dependent pullback power. module _ 𝓤1 𝓤2 𝓤3 𝓤4 𝓤5 𝓤6 : Level B'@A , B : Fam 𝓤1 𝓤2 Y'@X , Y : DepFam 𝓤3 𝓤4 B' V'@U , V : Dep²Fam 𝓤5 𝓤6 B' Y' where Π̂Σ̂ᶠ : Fam 𝓤1 ⊔ 𝓤2 ⊔ 𝓤3 ⊔ 𝓤4 ⊔ 𝓤5 ⊔ 𝓤6 𝓤1 ⊔ 𝓤2 ⊔ 𝓤3 ⊔ 𝓤4 ⊔ 𝓤6 Π̂Σ̂ᶠ = Π̂ᶠ Σ̂ᶠ B' Y' Σ̂ᵈᶠ B' Y' V' Π̂ᵈᶠ : DepFam 𝓤3 ⊔ 𝓤4 ⊔ 𝓤5 ⊔ 𝓤6 𝓤3 ⊔ 𝓤4 ⊔ 𝓤6 B' Π̂ᵈᶠ = λ a → Π̂ᶠ X a , Y a U a , V a .fst , λ a → Π̂ᶠ X a , Y a U a , V a .snd Π̂Π̂ᶠ : Fam 𝓤1 ⊔ 𝓤2 ⊔ 𝓤3 ⊔ 𝓤4 ⊔ 𝓤5 ⊔ 𝓤6 𝓤1 ⊔ 𝓤2 ⊔ 𝓤3 ⊔ 𝓤4 ⊔ 𝓤6 Π̂Π̂ᶠ = Π̂ᶠ B' Π̂ᵈᶠ module _ 𝓤1 𝓤2 𝓤3 𝓤4 𝓤5 𝓤6 : Level A : Type 𝓤1 B : Type 𝓤2 f : A → B X : B → Type 𝓤3 Y : B → Type 𝓤4 g : b : B → X b → Y b U : b : B → Y b → Type 𝓤5 V : b : B → Y b → Type 𝓤6 h : b : B y : Y b → U b y → V b y where Π̂Σ̂ : b , y : Σ B Y → U b y → Arrow-Π Σ̂ f g λ b , y → h b y Π̂Σ̂ = Π̂ Σ̂ f g λ b , y → h b y Π̂Π̂ : b : B y : Y b → U b y → Arrow-Π f λ b → Π̂ g b h b Π̂Π̂ = Π̂ f λ b → Π̂ g b h b Fredrik Bakke2026218202639https://samtoth.github.io/agda-synthetic-categories/stt-00FK/stt-00FK/agda-synthetic-categories/stt-00FK/The dependent Leibniz adjunction for familiesTheorem For all type families , dependent type families and doubly dependent type families , we have an equivalence of families where is the dependent pullback power and is the pushout sum of type families. We follow de Jong–Kraus–Ljungström [dJKL26] and prove the straightened case of the Leibniz adjunction first, although we generalise their result to the dependent case, and formalise it in general univalent foundations (with pushouts). For the proof, we first show that the base of the two families are equivalent, and then display a dependent equivalence over the base equivalence. Both are straight forward to construct. See respectively Leibnizᵈᶠ≃-base and Leibnizᵈᶠ≃-family below. We make use of two essential equivalences for the join: For the base we use the equivalence, for every and , Here means the type of null-homotopies of , I.e., elements together with homotopies . For the family we use the equivalence for every . The rest mostly speaks for itself. module _ 𝓤1 𝓤2 𝓤3 𝓤4 𝓤5 𝓤6 : Level B'@A , B : Fam 𝓤1 𝓤2 Y'@X , Y : DepFam 𝓤3 𝓤4 B' V'@U , V : Dep²Fam 𝓤5 𝓤6 B' Y' where Leibnizᵈᶠ≃-base : Π̂Σ̂ᶠ B' Y' V' .fst ≃ Π̂Π̂ᶠ B' Y' V' .fst Leibnizᵈᶠ≃-base = Π̂Σ̂ᶠ B' Y' V' .fst ≃⟨ Σ-≃ curry≃ λ _ → curry≃⟩ Σ u ∶ a : A x : X a → U a x a : A x : X a → B a * Y a x → V a x u a x ≃⟨ Σ-snd-≃ λ u → join-recᵈ≃null-homotopies ⟩ Σ u ∶ a : A x : X a → U a x Σ r ∶ a : A x : X a → Y a x → V a x u a x a : A b : B a x : X a → null-homotopy r a x ≃⟨ Σ-assoc e⁻¹ ∙e Σ-fst-≃-bwd Σ-Π-swap≃ λ a → x : X a → U a x λ a u → x : X a → Y a x → V a x u x ⟩ Π̂Π̂ᶠ B' Y' V' .fst ≃∎ Π̂Π̂ᶠ←Π̂Σ̂ᶠ-base : Π̂Σ̂ᶠ B' Y' V' .fst → Π̂Π̂ᶠ B' Y' V' .fst Π̂Π̂ᶠ←Π̂Σ̂ᶠ-base = Leibnizᵈᶠ≃-base ._≃_.fwd Leibnizᵈᶠ≃-family : fq : Π̂Σ̂ᶠ B' Y' V' .fst → Π̂Σ̂ᶠ B' Y' V' .snd fq ≃ Π̂Π̂ᶠ B' Y' V' .snd Π̂Π̂ᶠ←Π̂Σ̂ᶠ-base fq Leibnizᵈᶠ≃-family f , q = Π̂Σ̂ᶠ B' Y' V' .snd f , q ≃⟨ curry≃ ⟩ a : A x : X a → null-homotopy q a , x ≃⟨ Π-fam-≃ λ a → join-null-homotopiesᵈ≃ λ x → q a , x ⟩ Π̂Π̂ᶠ B' Y' V' .snd Π̂Π̂ᶠ←Π̂Σ̂ᶠ-base f , q ≃∎ Leibnizᵈᶠ≃ : Π̂Σ̂ᶠ B' Y' V' ≃ᶠ Π̂Π̂ᶠ B' Y' V' Leibnizᵈᶠ≃ = Leibnizᵈᶠ≃-base , Leibnizᵈᶠ≃-family Fredrik Bakke2026218202639https://samtoth.github.io/agda-synthetic-categories/stt-00FL/stt-00FL/agda-synthetic-categories/stt-00FL/The nondependent Leibniz adjunction for familiesCorollary For all triples of type families , , and , we have an equivalence of families module _ 𝓤1 𝓤2 𝓤3 𝓤4 𝓤5 𝓤6 : Level B' : Fam 𝓤1 𝓤2 Y' : Fam 𝓤3 𝓤4 V' : Fam 𝓤5 𝓤6 where Leibnizᶠ≃ : ⟨ B' □ᶠ Y' ,ᶠ V' ⟩ ≃ᶠ ⟨ B' ,ᶠ ⟨ Y' ,ᶠ V' ⟩ ⟩ Leibnizᶠ≃ = Leibnizᵈᶠ≃ B' Fam→DepFam B' Y' Fam→Dep²Fam B' Y' V' Fredrik Bakke2026218202639https://samtoth.github.io/agda-synthetic-categories/stt-00FM/stt-00FM/agda-synthetic-categories/stt-00FM/The dependent Leibniz adjunction for arrowsCorollary For all arrows , dependent arrows and doubly dependent arrows , we have an equivalence of arrows module _ 𝓤1 𝓤2 𝓤3 𝓤4 𝓤5 𝓤6 : Level A : Type 𝓤1 B : Type 𝓤2 f : A → B X : B → Type 𝓤3 Y : B → Type 𝓤4 g : b : B → X b → Y b U : b : B → Y b → Type 𝓤5 V : b : B → Y b → Type 𝓤6 h : b : B y : Y b → U b y → V b y let B' = fibre-fam f let Y' = fibre-depfam g let V' = fibre-dep²fam h where Π̂Σ̂-bridge≃ : Arrow-equiv Π̂Σ̂ f g h arrow←Fam Π̂Σ̂ᶠ B' Y' V' Π̂Σ̂-bridge≃ = Π̂-Arrow-equiv←≃ᶠ Σ̂ f g λ b , y → h b y Π̂Σ̂ᶠ B' Y' V' Π̂ᶠ-ap-base-fam fibre-depfam λ b , y → h b y Σ̂-unstraighten≃ᶠ f g .snd Π̂Π̂-bridge≃ : Arrow-equiv Π̂Π̂ f g h arrow←Fam Π̂Π̂ᶠ B' Y' V' Π̂Π̂-bridge≃ = Π̂-Arrow-equiv←≃ᶠ f λ b → Π̂ g b h b Π̂Π̂ᶠ B' Y' V' Π̂ᶠ-≃ᶠ B' fibre-depfam λ b → Π̂ g b h b Π̂ᵈᶠ B' Y' V' λ b → Π̂-fibre-fam≃Π̂ᶠ-unstraight-fibre-fam g b h b Leibnizᵈ≃ : Arrow-equiv Π̂Σ̂ f g h Π̂Π̂ f g h Leibnizᵈ≃ = concat-Arrow-equiv concat-Arrow-equiv Π̂Σ̂-bridge≃ Arrow-equiv←≃ᶠ Π̂Σ̂ᶠ B' Y' V' Π̂Π̂ᶠ B' Y' V' Leibnizᵈᶠ≃ B' Y' V' inv-Arrow-equiv Π̂Π̂-bridge≃ Fredrik Bakke2026218202639https://samtoth.github.io/agda-synthetic-categories/stt-00FN/stt-00FN/agda-synthetic-categories/stt-00FN/The nondependent Leibniz adjunction for arrowsCorollary For all triples of arrows , , and , we have an equivalence of arrows module _ 𝓤1 𝓤2 𝓤3 𝓤4 𝓤5 𝓤6 : Level A : Type 𝓤1 B : Type 𝓤2 f : A → B C : Type 𝓤3 D : Type 𝓤4 g : C → D E : Type 𝓤5 B' : Type 𝓤6 h : E → B' where Leibniz≃ : Arrow-equiv ⟨ f □ g , h ⟩ ⟨ f , ⟨ g , h ⟩ ⟩ Leibniz≃ = Leibnizᵈ≃ f λ _ → g λ _ _ → h References Tom de Jong Nicolai Kraus Axel Ljungström https://samtoth.github.io/agda-synthetic-categories/dJKL26/ dJKL26 /agda-synthetic-categories/dJKL26/ The Leibniz Adjunction in Homotopy Type Theory, with an Application to Simplicial Type Theory Reference 10.48550/arXiv.2601.21843 Tom de Jong Nicolai Kraus Axel Ljungström Abstract Simplicial type theory extends homotopy type theory and equips types with a notion of directed morphisms. A Segal type is defined to be a type in which these directed morphisms can be composed. We show that all higher coherences can be stated and derived if simplicial type theory is taken to be homotopy type theory with a postulated interval type. In technical terms, this means that if a type has unique fillers for (2,1)-horns, it has unique fillers for all inner -horns. This generalizes a result of Riehl and Shulman for the case , . Our main technical tool is the Leibniz adjunction: the pushout-product is left adjoint to the pullback-hom in the wild category of types. While this adjunction is well known for ordinary categories, it is much more involved for higher categories, and the fact that it can be proved for the wild category of types (a higher category without stated higher coherences) is non-trivial. We make profitable use of the equivalence between the wild category of maps and that of families. We have formalized the results in Cubical Agda. Context Backlinks Fredrik Bakke 2026 2 24 https://samtoth.github.io/agda-synthetic-categories/stt-00FG/ stt-00FG /agda-synthetic-categories/stt-00FG/ Leibniz adjunction PR56 Fredrik Bakke 2026 2 24 Summary We formalise the dependent Leibniz adjunction following the proof approach of de Jong–Kraus–Ljungström [dJKL26]. This adjunction says that, given any arrow (i.e., function between types) , dependent arrow , and doubly dependent arrow , we have an equivalence of arrows Here is the dependent pullback power and is the pushout sum. These are defined via the two squares and The dependent Leibniz adjunction generalises the Leibniz adjunction, which we also formalise, in addition to straightened variants of both (dependent, nondependent). In fact, we derive the unstraightened results from the straightened ones. Our result generalises that of de Jong–Kraus–Ljungström [dJKL26] from the nondependent to the dependent case, and we formalise it in the general framework of univalent foundations (with pushouts). The Leibniz adjunction is a fundamental tool in the theory of lifting conditions, and hence it is very useful in the development of synthetic category theory, indeed, homotopy theory more broadly. The dependent variant is less known, and at the time of writing I don't know of a source that discusses it. It is, however, still a useful generalisation that allows us to consider more refined, dependent lifting conditions. Fredrik Bakke 2026 2 24 New modules Module Module Module Module Module Module https://samtoth.github.io/agda-synthetic-categories/Core-index/ Core-index /agda-synthetic-categories/Core-index/ Core Namespace Equivalences of arrows Retracts of maps The category of maps of types Bipointed wedge sums Functoriality of bipointed wedge sums Properties of the canonical pushouts Canonical Sequential Colimits Cartesian displayed spans Cartesian squares Straightening for cartesian span maps Cocartesian squares (co)diagonal maps Coequalisers Cofibres of maps Orthogonality of Coslice maps Coslices of types Commuting cubes Descent for pushouts Descent for surjective maps Pullbacks using Identity systems Fibred joins and the modified image construction The flattening lemma for coequalisers Flattening pushouts Flattening for Sequential Colimits Functoriality of pullbacks Pushouts are functorial over the span diagram Global classes of maps Images of maps The homotopy interval type Joins Leibniz Adjunction Lifting squares Orthogonal maps Closure properties of orthogonal maps Pi types as sections (Pre)Postwhiskering The pullback power The pullback power of families Pushouts of equivalent types Pushout sums Pushout sum of families Relative suspensions Functoriality of relative suspensions Characterising the equality of sequence maps Set Quotients Slices of types Small types Maps between span diagrams Spheres Surjective maps Suspensions Universal fibrations Wedge sums Functoriality of wedge sums Related Contributions