A general limit lifting theorem for 2-dimensional monad theory (but - PowerPoint PPT Presentation

Limit lifting results Unifying morphisms Unifying limits Our result References A general limit lifting theorem for 2-dimensional monad theory (but don’t let the long title scare you!) Martin Szyld University of Buenos Aires - CONICET, Argentina CT 2017 @ UBC, Vancouver, Canada

� � � Limit lifting results Unifying morphisms Unifying limits Our result References Limit lifting along the forgetful functor T ⇒ T , T 2 m i K is a category, T is a monad on K ( K − → K , id ⇒ T ) T - Alg U creates lim F ≡ we can give lim F a F T -algebra structure such that it is lim F U (we lift the limit of F along U ) F A K

� � � Limit lifting results Unifying morphisms Unifying limits Our result References Limit lifting along the forgetful functor T ⇒ T , T 2 m i K is a category, T is a monad on K ( K − → K , id ⇒ T ) T - Alg U creates lim F ≡ we can give lim F a F T -algebra structure such that it is lim F U (we lift the limit of F along U ) F A K Previous results U 1 (from the V -enriched case) T - Alg − → K creates all limits.

� � � Limit lifting results Unifying morphisms Unifying limits Our result References Limit lifting along the forgetful functor T ⇒ T , T 2 m i K is a 2-category, T is a 2-monad on K ( K − → K , id ⇒ T ) T - Alg U creates lim F ≡ we can give lim F a F T -algebra structure such that it is lim F U (we lift the limit of F along U ) F A K Previous results U 1 (from the V -enriched case) T - Alg − → K creates all limits.

� � � Limit lifting results Unifying morphisms Unifying limits Our result References Limit lifting along the forgetful functor T ⇒ T , T 2 m i K is a 2-category, T is a 2-monad on K ( K − → K , id ⇒ T ) T - Alg s U creates lim F ≡ we can give lim F a F T -algebra structure such that it is lim F U (we lift the limit of F along U ) F A K The subindex ( s, p, ℓ ) indicates (strict, pseudo, lax) algebra morphisms Previous results U 1 (from the V -enriched case) T - Alg − → K creates all limits.

� � � Limit lifting results Unifying morphisms Unifying limits Our result References Limit lifting along the forgetful functor T ⇒ T , T 2 m i K is a 2-category, T is a 2-monad on K ( K − → K , id ⇒ T ) T - Alg s U creates lim F ≡ we can give lim F a F T -algebra structure such that it is lim F U (we lift the limit of F along U ) F A K The subindex ( s, p, ℓ ) indicates (strict, pseudo, lax) algebra morphisms Previous results U 1 (from the V -enriched case) T - Alg s − → K creates all limits.

� � � Limit lifting results Unifying morphisms Unifying limits Our result References Limit lifting along the forgetful functor T ⇒ T , T 2 m i K is a 2-category, T is a 2-monad on K ( K − → K , id ⇒ T ) T - Alg p U creates lim F ≡ we can give lim F a F T -algebra structure such that it is lim F U (we lift the limit of F along U ) F A K The subindex ( s, p, ℓ ) indicates (strict, pseudo, lax) algebra morphisms Previous results U 1 (from the V -enriched case) T - Alg s − → K creates all limits. U 2 T - Alg p − → K creates lax and pseudolimits [BKP,89].

� � � Limit lifting results Unifying morphisms Unifying limits Our result References Limit lifting along the forgetful functor T ⇒ T , T 2 m i K is a 2-category, T is a 2-monad on K ( K − → K , id ⇒ T ) T - Alg ℓ U creates lim F ≡ we can give lim F a F T -algebra structure such that it is lim F U (we lift the limit of F along U ) F A K The subindex ( s, p, ℓ ) indicates (strict, pseudo, lax) algebra morphisms Previous results U 1 (from the V -enriched case) T - Alg s − → K creates all limits. U 2 T - Alg p − → K creates lax and pseudolimits [BKP,89]. U 3 T - Alg ℓ − → K creates oplax limits [Lack,05].

� � � Limit lifting results Unifying morphisms Unifying limits Our result References Limit lifting along the forgetful functor T ⇒ T , T 2 m i K is a 2-category, T is a 2-monad on K ( K − → K , id ⇒ T ) T - Alg ? U creates lim F ≡ we can give lim F a F T -algebra structure such that it is lim F U (we lift the limit of F along U ) F A K The subindex ( s, p, ℓ ) indicates (strict, pseudo, lax) algebra morphisms Previous results U 1 (from the V -enriched case) T - Alg s − → K creates all limits. U 2 T - Alg p − → K creates lax and pseudolimits [BKP,89]. U 3 T - Alg ℓ − → K creates oplax limits [Lack,05]. Note: All these limits are weighted , and the projections of the limit are always strict morphisms.

� � � Limit lifting results Unifying morphisms Unifying limits Our result References Limit lifting along the forgetful functor T ⇒ T , T 2 m i K is a 2-category, T is a 2-monad on K ( K − → K , id ⇒ T ) T - Alg ? U creates lim F ≡ we can give lim F a F T -algebra structure such that it is lim F U (we lift the limit of F along U ) F A K The subindex ( s, p, ℓ ) indicates (strict, pseudo, lax) algebra morphisms Previous results U 1 (from the V -enriched case) T - Alg s − → K creates all limits. U 2 T - Alg p − → K creates lax and pseudolimits [BKP,89]. U 3 T - Alg ℓ − → K creates oplax limits [Lack,05]. Note: All these limits are weighted , and the projections of the limit are always strict morphisms. We will present a theorem which unifies and generalizes these results.

� � Limit lifting results Unifying morphisms Unifying limits Our result References Ω-morphisms of T -algebras f A lax morphism A − → B between T -algebras has a structural 2-cell T f � TA TB 1 lax ( ℓ ) morphism: f any 2-cell. a ⇓ f b � B A f

� � Limit lifting results Unifying morphisms Unifying limits Our result References Ω-morphisms of T -algebras f A lax morphism A − → B between T -algebras has a structural 2-cell T f � TA TB 1 lax ( ℓ ) morphism: f any 2-cell. 2 pseudo ( p ) morphism: f invertible. a ⇓ f b � B A f

� � Limit lifting results Unifying morphisms Unifying limits Our result References Ω-morphisms of T -algebras f A lax morphism A − → B between T -algebras has a structural 2-cell T f � TA TB 1 lax ( ℓ ) morphism: f any 2-cell. 2 pseudo ( p ) morphism: f invertible. a ⇓ f b 3 strict ( s ) morphism: f an identity. � B A f

� � Limit lifting results Unifying morphisms Unifying limits Our result References Ω-morphisms of T -algebras f A lax morphism A − → B between T -algebras has a structural 2-cell T f � TA TB 1 lax ( ℓ ) morphism: f any 2-cell. 2 pseudo ( p ) morphism: f invertible. a ⇓ f b 3 strict ( s ) morphism: f an identity. � B A f Fix a family Ω of 2-cells of K . f is a Ω-morphism if f ∈ Ω.

� � Limit lifting results Unifying morphisms Unifying limits Our result References Ω-morphisms of T -algebras f A lax morphism A − → B between T -algebras has a structural 2-cell T f � TA TB 1 lax ( ℓ ) morphism: f any 2-cell. 2 pseudo ( p ) morphism: f invertible. a ⇓ f b 3 strict ( s ) morphism: f an identity. � B A f Fix a family Ω of 2-cells of K . f is a Ω-morphism if f ∈ Ω. Considering Ω ℓ = 2-cells( K ), Ω p = { invertible 2-cells } , Ω s = { identities } , we recover the three cases above.

Limit lifting results Unifying morphisms Unifying limits Our result References A general notion of weighted limit. The conical case (Gray,1974) We fix A , B 2-categories, Σ ⊆ Arrows( A ), Ω ⊆ 2-cells( B )

� � � � Limit lifting results Unifying morphisms Unifying limits Our result References A general notion of weighted limit. The conical case (Gray,1974) We fix A , B 2-categories, Σ ⊆ Arrows( A ), Ω ⊆ 2-cells( B ) F • σ - ω -natural transformation: A � B , θ is a lax natural θ ⇓ G θ A FA GA transformation such that θ f is in Ω when f is in Σ. F f ⇓ θ f Gf � GB FB θ B

� � � � � � � � Limit lifting results Unifying morphisms Unifying limits Our result References A general notion of weighted limit. The conical case (Gray,1974) We fix A , B 2-categories, Σ ⊆ Arrows( A ), Ω ⊆ 2-cells( B ) F • σ - ω -natural transformation: A � B , θ is a lax natural θ ⇓ G θ A FA GA transformation such that θ f is in Ω when f is in Σ. F f ⇓ θ f Gf � GB FB θ B △ E • σ - ω -cone (for F , with vertex E ∈ B ): is a σ - ω -natural A � B , θ ⇓ F FA θ A i.e. such that θ f is in Ω when f is in Σ. E ⇓ θ f F f θ B FB

Limit lifting results Unifying morphisms Unifying limits Our result References • σ - ω -limit: is the universal σ - ω -cone, in the sense that the following is an isomorphism π ∗ B ( E, L ) − → σ - ω -Cones( E, F )

� � � � � Limit lifting results Unifying morphisms Unifying limits Our result References • σ - ω -limit: is the universal σ - ω -cone, in the sense that the following is an isomorphism π ∗ B ( E, L ) − → σ - ω -Cones( E, F ) � θ On objects: ϕ � θ A FA π A E L ⇓ θ f ⇓ π f F f π B FB θ B

� � � � � Limit lifting results Unifying morphisms Unifying limits Our result References • σ - ω -limit: is the universal σ - ω -cone, in the sense that the following is an isomorphism π ∗ B ( E, L ) − → σ - ω -Cones( E, F ) � θ On objects: ϕ � θ A FA π A ⇓ θ f � E L ⇓ π f F f ∃ ! ϕ π B FB θ B