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Internal Languages of Higher Categories
Karol Szumi lo
University of Leeds
2/18
Internal Languages of Higher Categories Karol Szumi lo University - - PowerPoint PPT Presentation
Internal Languages of Higher Categories Karol Szumi lo University - - PowerPoint PPT Presentation
Internal Languages of Higher Categories Karol Szumi lo University of Leeds HoTT 2019 1/18 Theorem (LambekScott) The category of -calculi is equivalent to the category of cartesian closed categories. 2/18 Theorem (LambekScott)
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Theorem (Lambek–Scott)
The category of λ-calculi is equivalent to the category of cartesian closed categories.
Theorem (Lambek–Scott)
The category of higher order intuitionistic type theories is (almost) equivalent to the category of elementary toposes with NNO.
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Theorem (Lambek–Scott)
The category of λ-calculi is equivalent to the category of cartesian closed categories.
Theorem (Lambek–Scott)
The category of higher order intuitionistic type theories is (almost) equivalent to the category of elementary toposes with NNO. What is a counterpart for HoTT?
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Joint work with Chris Kapulkin:
Theorem (2017)
The homotopy theory of comprehension categories with Id and Σ is equivalent to the homotopy theory of quasicategories with finite limits.
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Joint work with Chris Kapulkin:
Theorem (2017)
The homotopy theory of comprehension categories with Id and Σ is equivalent to the homotopy theory of quasicategories with finite limits.
Theorem (in progress)
The homotopy theory of comprehension categories with Id, Π and Σ satisfying functional extensionality is equivalent to the homotopy theory
- f locally cartesian closed quasicategories.
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A homotopy theory is modeled by a homotopical category, i.e., a category C with a subcategory of weak equivalences.
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A homotopy theory is modeled by a homotopical category, i.e., a category C with a subcategory of weak equivalences. C has an associated (∞,1)-category obtained by universally inverting the weak equivalences.
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A homotopy theory is modeled by a homotopical category, i.e., a category C with a subcategory of weak equivalences. C has an associated (∞,1)-category obtained by universally inverting the weak equivalences. A functor F∶C → D is homotopical if it preserves weak equivalences.
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A homotopy theory is modeled by a homotopical category, i.e., a category C with a subcategory of weak equivalences. C has an associated (∞,1)-category obtained by universally inverting the weak equivalences. A functor F∶C → D is homotopical if it preserves weak equivalences. It is a DK-equivalence if HoC → HoD is essentially surjective and LHC(X,Y ) → LHD(FX,FY ) are weak homotopy equivalences, i.e., it induces an equivalences of (∞,1)-categories.
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CompCatId,Π,Σ LCCQ
comprehension categories locally cartesian closed quasicategories
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CompCatId,Π,Σ LCCQ
comprehension categories locally cartesian closed quasicategories Π-tribes
Π-Trb
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CompCatId,Π,Σ LCCQ
comprehension categories locally cartesian closed quasicategories Π-tribes
Π-Trb Π-sTrb
semisimplicial Π-tribes
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CompCatId,Π,Σ LCCQ
comprehension categories locally cartesian closed quasicategories Π-tribes
Π-Trb Π-sTrb
semisimplicial Π-tribes
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A tribe is a category C with a subcategory of fibrations. T1 There is a terminal object and all objects are fibrant. T2 There are pullbacks along fibrations and fibrations are stable under pullback. T3 Every morphism factors as an anodyne morphism followed by a fibration. T4 Anodyne morphisms are stable under pullbacks along fibrations.
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A tribe is a category C with a subcategory of fibrations. T1 There is a terminal object and all objects are fibrant. T2 There are pullbacks along fibrations and fibrations are stable under pullback. T3 Every morphism factors as an anodyne morphism followed by a fibration. T4 Anodyne morphisms are stable under pullbacks along fibrations. anodyne = LLP with respect to fibrations
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A tribe is a category C with a subcategory of fibrations. T1 There is a terminal object and all objects are fibrant. T2 There are pullbacks along fibrations and fibrations are stable under pullback. T3 Every morphism factors as an anodyne morphism followed by a fibration. T4 Anodyne morphisms are stable under pullbacks along fibrations. anodyne = LLP with respect to fibrations a b
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A tribe is a category C with a subcategory of fibrations. T1 There is a terminal object and all objects are fibrant. T2 There are pullbacks along fibrations and fibrations are stable under pullback. T3 Every morphism factors as an anodyne morphism followed by a fibration. T4 Anodyne morphisms are stable under pullbacks along fibrations. anodyne = LLP with respect to fibrations a b a Pa a ×b a
∼
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A tribe is a category C with a subcategory of fibrations. T1 There is a terminal object and all objects are fibrant. T2 There are pullbacks along fibrations and fibrations are stable under pullback. T3 Every morphism factors as an anodyne morphism followed by a fibration. T4 Anodyne morphisms are stable under pullbacks along fibrations. anodyne = LLP with respect to fibrations a b a Pa a ×b a
∼
a Pa Pa c
∼
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A path object on a is given by a factorization a ↣
∼ Pa ↠ a × a.
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A path object on a is given by a factorization a ↣
∼ Pa ↠ a × a.
A homotopy between morphisms a → b is a morphism a → Pb.
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A path object on a is given by a factorization a ↣
∼ Pa ↠ a × a.
A homotopy between morphisms a → b is a morphism a → Pb. A homotopy equivalence is a morphism with an inverse up to homotopy.
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A path object on a is given by a factorization a ↣
∼ Pa ↠ a × a.
A homotopy between morphisms a → b is a morphism a → Pb. A homotopy equivalence is a morphism with an inverse up to homotopy. A homomorphism of tribes is a functor that preserves fibrations, anodyne morphisms, terminal object and pullbacks along fibrations.
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A path object on a is given by a factorization a ↣
∼ Pa ↠ a × a.
A homotopy between morphisms a → b is a morphism a → Pb. A homotopy equivalence is a morphism with an inverse up to homotopy. A homomorphism of tribes is a functor that preserves fibrations, anodyne morphisms, terminal object and pullbacks along fibrations. A Π-tribe is a tribe T such that for all fibrations p∶a ↠ b, the pullback functor p∗∶T ↡b → T ↡a has a right adjoint Πp that is a homomorphism.
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A path object on a is given by a factorization a ↣
∼ Pa ↠ a × a.
A homotopy between morphisms a → b is a morphism a → Pb. A homotopy equivalence is a morphism with an inverse up to homotopy. A homomorphism of tribes is a functor that preserves fibrations, anodyne morphisms, terminal object and pullbacks along fibrations. A Π-tribe is a tribe T such that for all fibrations p∶a ↠ b, the pullback functor p∗∶T ↡b → T ↡a has a right adjoint Πp that is a homomorphism. A Π-homomorphism is a homomorphism that preserves Π.
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A path object on a is given by a factorization a ↣
∼ Pa ↠ a × a.
A homotopy between morphisms a → b is a morphism a → Pb. A homotopy equivalence is a morphism with an inverse up to homotopy. A homomorphism of tribes is a functor that preserves fibrations, anodyne morphisms, terminal object and pullbacks along fibrations. A Π-tribe is a tribe T such that for all fibrations p∶a ↠ b, the pullback functor p∗∶T ↡b → T ↡a has a right adjoint Πp that is a homomorphism. A Π-homomorphism is a homomorphism that preserves Π. It is a weak equivalence if it induces an equivalence on homotopy categories.
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A semisimplicial set is a “simplicial set without degeneracies”.
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A semisimplicial set is a “simplicial set without degeneracies”. Semisimplicial sets carry a symmetric monoidal geometric product.
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A semisimplicial set is a “simplicial set without degeneracies”. Semisimplicial sets carry a symmetric monoidal geometric product. A semisimplicial Π-tribe is a Π-tribe that is enriched in semisimplicial sets such that ▸ cotensors by finite semisimplicial sets exist and the “pullback cotensor property” is satisfied; ▸ cotensors preserve anodyne morphisms; ▸ adjunctions p∗ ⊢ Πp are semisimplicial.
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A frame in a Π-tribe T is a Reedy fibrant homotopically constant semisimplicial object in T .
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A frame in a Π-tribe T is a Reedy fibrant homotopically constant semisimplicial object in T . The category of frames Fr T is a semisimplicial Π-tribe.
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A frame in a Π-tribe T is a Reedy fibrant homotopically constant semisimplicial object in T . The category of frames Fr T is a semisimplicial Π-tribe. Π-sTrb Π-Trb
U Fr
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A frame in a Π-tribe T is a Reedy fibrant homotopically constant semisimplicial object in T . The category of frames Fr T is a semisimplicial Π-tribe. Π-sTrb Π-Trb
U Fr
U Fr T T
∼
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A frame in a Π-tribe T is a Reedy fibrant homotopically constant semisimplicial object in T . The category of frames Fr T is a semisimplicial Π-tribe. Π-sTrb Π-Trb
U Fr
U Fr T T
∼ ̂ Fr
Fr UT ′ ̂ FrT ′ T ′
∼ ∼
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A quasicategory is a simplicial set C with the RLP with respect to the inner horn inclusions. Quasicategories model (∞,1)-categories.
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A quasicategory is a simplicial set C with the RLP with respect to the inner horn inclusions. Quasicategories model (∞,1)-categories. C has finite limits if for all finite simplicial sets K and x∶K → C, the functor CK(const ,x) is representable.
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A quasicategory is a simplicial set C with the RLP with respect to the inner horn inclusions. Quasicategories model (∞,1)-categories. C has finite limits if for all finite simplicial sets K and x∶K → C, the functor CK(const ,x) is representable. C is cartesian closed if for all x,y ∈ C, the functor C( ×x,y) is representable.
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A quasicategory is a simplicial set C with the RLP with respect to the inner horn inclusions. Quasicategories model (∞,1)-categories. C has finite limits if for all finite simplicial sets K and x∶K → C, the functor CK(const ,x) is representable. C is cartesian closed if for all x,y ∈ C, the functor C( ×x,y) is representable. C is locally cartesian closed if for all x ∈ C, C ↓ x is cartesian closed.
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A quasicategory is a simplicial set C with the RLP with respect to the inner horn inclusions. Quasicategories model (∞,1)-categories. C has finite limits if for all finite simplicial sets K and x∶K → C, the functor CK(const ,x) is representable. C is cartesian closed if for all x,y ∈ C, the functor C( ×x,y) is representable. C is locally cartesian closed if for all x ∈ C, C ↓ x is cartesian closed. A morphism of LCCQs is a functor (simplicial map) that preserves finite limits and local exponentials.
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A quasicategory is a simplicial set C with the RLP with respect to the inner horn inclusions. Quasicategories model (∞,1)-categories. C has finite limits if for all finite simplicial sets K and x∶K → C, the functor CK(const ,x) is representable. C is cartesian closed if for all x,y ∈ C, the functor C( ×x,y) is representable. C is locally cartesian closed if for all x ∈ C, C ↓ x is cartesian closed. A morphism of LCCQs is a functor (simplicial map) that preserves finite limits and local exponentials. It is an categorical equivalence if it has an inverse up to natural equivalence.
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LCCQ Π-Trb Π-sTrb
U Fr
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LCCQ Π-Trb Π-sTrb
U Fr
Outline of the proof: ▸ Show that Π-sTrb and LCCQ are fibration categories.
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LCCQ Π-Trb Π-sTrb
U Nf Fr
Outline of the proof: ▸ Show that Π-sTrb and LCCQ are fibration categories. ▸ Implement the rightward functors with a convenient construction: the quasicategory of frames.
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LCCQ Π-Trb Π-sTrb
U Nf Fr
Outline of the proof: ▸ Show that Π-sTrb and LCCQ are fibration categories. ▸ Implement the rightward functors with a convenient construction: the quasicategory of frames. ▸ Show that Nf is an exact functor with Approximation Properties.
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LCCQ Π-Trb Π-sTrb
U Nf Fr
Outline of the proof: ▸ Show that Π-sTrb and LCCQ are fibration categories. ▸ Implement the rightward functors with a convenient construction: the quasicategory of frames. ▸ Show that Nf is an exact functor with Approximation Properties. ▸ For that use tribes of representable injectively fibrant presheaves.
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A fibration category is a category with subcategories of weak equivalences and fibrations. F1 There is a terminal object and all objects are fibrant. F2 Pullbacks along fibrations exist and (acyclic) fibrations are stable under pullback. F3 Every morphism factors as a weak equivalence followed by a fibration. F4 Weak equivalences satisfy the 2-out-of-6 property.
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A fibration category is a category with subcategories of weak equivalences and fibrations. F1 There is a terminal object and all objects are fibrant. F2 Pullbacks along fibrations exist and (acyclic) fibrations are stable under pullback. F3 Every morphism factors as a weak equivalence followed by a fibration. F4 Weak equivalences satisfy the 2-out-of-6 property. An exact functor between fibration categories is a functor that preserves weak equivalences, fibrations, terminal object and pullbacks along fibrations.
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A fibration category is a category with subcategories of weak equivalences and fibrations. F1 There is a terminal object and all objects are fibrant. F2 Pullbacks along fibrations exist and (acyclic) fibrations are stable under pullback. F3 Every morphism factors as a weak equivalence followed by a fibration. F4 Weak equivalences satisfy the 2-out-of-6 property. An exact functor between fibration categories is a functor that preserves weak equivalences, fibrations, terminal object and pullbacks along fibrations. Every tribe is a fibration category and every homomorphism is exact.
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A fibration category is a category with subcategories of weak equivalences and fibrations. F1 There is a terminal object and all objects are fibrant. F2 Pullbacks along fibrations exist and (acyclic) fibrations are stable under pullback. F3 Every morphism factors as a weak equivalence followed by a fibration. F4 Weak equivalences satisfy the 2-out-of-6 property. An exact functor between fibration categories is a functor that preserves weak equivalences, fibrations, terminal object and pullbacks along fibrations. Every tribe is a fibration category and every homomorphism is exact. LCCQ is a fibration category (fibrations are inner isofibrations).
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A fibration between tribes F∶S ↠ T : S T
F
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A fibration between tribes F∶S ↠ T : S T
F
a b Fa y
≅ ≅
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A fibration between tribes F∶S ↠ T : S T
F
a b Fa y
≅ ≅
a b c Fa y Fc
∼ ∼
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A fibration between tribes F∶S ↠ T : S T
F
a b Fa y
≅ ≅
a b c Fa y Fc
∼ ∼
a b c d Fa Fb Fc Fd
∼ ∼
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A fibration between tribes F∶S ↠ T : S T
F
a b Fa y
≅ ≅
a b c Fa y Fc
∼ ∼
a b c d Fa Fb Fc Fd
∼ ∼
a b c Fa Fb Fc
∼ ∼
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A fibration between tribes F∶S ↠ T : S T
F
a b Fa y
≅ ≅
a b c Fa y Fc
∼ ∼
a b c d Fa Fb Fc Fd
∼ ∼
a b c Fa Fb Fc
∼ ∼
The category Π-sTrb is a fibration category.
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A fibration between tribes F∶S ↠ T : S T
F
a b Fa y
≅ ≅
a b c Fa y Fc
∼ ∼
a b c d Fa Fb Fc Fd
∼ ∼
a b c Fa Fb Fc
∼ ∼
The category Π-sTrb is a fibration category. A path object on T ∈ Π-sTrb: T T Z
R
T × T
∼
where Z = {● ←
∼ ● → ∼ ●}.
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If C is a quasicategory, let IC denote the opposite of its category of elements with face operators only.
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If C is a quasicategory, let IC denote the opposite of its category of elements with face operators only. The “initial projection” functor NIC → C creates weak equivalences in IC.
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If C is a quasicategory, let IC denote the opposite of its category of elements with face operators only. The “initial projection” functor NIC → C creates weak equivalences in IC. IC is a homotopical inverse category and NIC → C is a localization.
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If C is a quasicategory, let IC denote the opposite of its category of elements with face operators only. The “initial projection” functor NIC → C creates weak equivalences in IC. IC is a homotopical inverse category and NIC → C is a localization. If T is a Π-tribe, define a simplicial set Nf T (quasicategory of frames): (Nf )m = {x∶I[m] → T ∣ x is homotopical and Reedy fibrant}.
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If C is a quasicategory, let IC denote the opposite of its category of elements with face operators only. The “initial projection” functor NIC → C creates weak equivalences in IC. IC is a homotopical inverse category and NIC → C is a localization. If T is a Π-tribe, define a simplicial set Nf T (quasicategory of frames): (Nf )m = {x∶I[m] → T ∣ x is homotopical and Reedy fibrant}. Nf T is a locally cartesian closed quasicategory and Nf ∶Π-sTrb → LCCQ is an exact functor.
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Theorem (Cisinski)
If F∶C → D is an exact functor between fibration categories, then the following are equivalent. ▸ F is a DK-equivalence. ▸ F is a weak equivalence. ▸ F satisfies the Approximation Properties:
App1 it reflects weak equivalences, App2 blah: x Fb b x ′ Fa a.
∼ ∼
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Theorem (Cisinski)
If F∶C → D is an exact functor between fibration categories, then the following are equivalent. ▸ F is a DK-equivalence. ▸ F is a weak equivalence. ▸ F satisfies the Approximation Properties:
App1 it reflects weak equivalences, App2 blah: x Fb b x ′ Fa a.
∼ ∼
We need to show that Nf satisfies App2.
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C Nf T T C′ Nf S S
F ∼ ∼
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C Nf T T C′ Nf S S
F ∼ ∼
A simplicial presheaf A over a simplicial category A is representable if there is A(,a) →
∼ A.
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C Nf T T C′ Nf S S
F ∼ ∼
A simplicial presheaf A over a simplicial category A is representable if there is A(,a) →
∼ A.
Consider the hammock localization LH IC. Then the category RC of representable injectively fibrant simplicial presheaves over LH IC is a Π-tribe (with injective fibrations). Moreover, Nf RC is equivalent to C.
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An object of S consists of ▸ presheaves X, ̃ X over LH IC with X injectively fibrant, ▸ an object a ∈ T , ▸ an injective fibration ̃ X ↠ X × LHT (F ,a) such that ▸ ̃ X → X is a weak equivalence, ▸ there is a representation LH IC(,x) →
∼ ̃
X that induces a weak equivalence Fx → a.
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An object of S consists of ▸ presheaves X, ̃ X over LH IC with X injectively fibrant, ▸ an object a ∈ T , ▸ an injective fibration ̃ X ↠ X × LHT (F ,a) such that ▸ ̃ X → X is a weak equivalence, ▸ there is a representation LH IC(,x) →
∼ ̃
X that induces a weak equivalence Fx → a. S is a Π-tribe.
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An object of S consists of ▸ presheaves X, ̃ X over LH IC with X injectively fibrant, ▸ an object a ∈ T , ▸ an injective fibration ̃ X ↠ X × LHT (F ,a) such that ▸ ̃ X → X is a weak equivalence, ▸ there is a representation LH IC(,x) →
∼ ̃
X that induces a weak equivalence Fx → a. S is a Π-tribe. An m-simplex of C′ consists of ▸ an m-simplex (X, ̃ X,a) of Nf S, ▸ an m-simplex x of C, ▸ a natural choice of representations LH IC(,xϕ) →
∼ ̃
Xϕ for ϕ∶[k] ↪ [m] inducing weak equivalences Fxϕ → aϕ.
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C Nf T T C′ Nf S S
F
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C Nf T T C′ Nf S S
F ∼
C′ → C is an acyclic fibration. In particular, C′ is a LCCQ.
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C Nf T T C′ Nf S S
F ∼ ∼
C′ → C is an acyclic fibration. In particular, C′ is a LCCQ. C′ → Nf S preserves finite limits and satisfies quasicategorical Approximation Properties.
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C Nf T T C′ Nf S S
F ∼ ∼
̂ FrT Fr T ̂ FrS Fr S
∼ ∼ ∼ ∼