International Category Theory Conference CT 2006 White Point, Nova Scotia, June 25 - July 1, 2006 Galois theories of internal groupoids via congruence relations for Maltsev varieties Jo˜ ao J. Xarez jxarez@mat.ua.pt University of Aveiro 1
1 Coequalizer of the kernel pair C finitely-complete; ( F, ϕ ) pointed endofunctor on C , s.t. the kernel pair of ϕ A : A → F ( A ) has a coequalizer for every object A in C . π 2 ,A ✲ A × F ( A ) A A ✑ ✑ ✑ ✑ η A ✑ ✰ I ( A ) π 1 ,A ϕ A ◗◗◗◗◗ µ A ✸ ✑ ✑✑✑✑✑ η A ❄ ❄ s ϕ A ✲ F ( A ) A 2
Idempotency of ( I, η ) 2 Fix ( I, η ), Mono ( F, ϕ ) full subcategories of C . Lemma 2.1 ( I, η ) well-pointed endofunctor (i.e., Iη = ηI ); Fix ( I, η ) = Mono ( F, ϕ ) . Proposition 2.2 µ , Fη monics ⇒ ( I, η ) idempotent Remark 2.3 ( I, η ) idempotent ⇔ Iη = ηI and ηI iso ⇔ Fix ( I, η ) reflective in C 3
3 Stabilization and m.-l. factorization Proposition 3.1 All η A pullback stable regular epis and µ monic and Fη iso ✲ and F preserves C × I ( A ) A A η A ❄ ❄ g ✲ C I ( A ) ⇒ ( I, η ) idempotent with stable units; and ∀ B ∈ C ∃ p : E → B e.d.m. E ∈ Mono ( F, ϕ ) ⇒ ( E ′ , M ∗ ) factorization system (monotone-light). 4
4 First example: internal categories ( F, ϕ ) idempotent associated to the localization Cat ( S ) → LEqRel ( S ) ≃ S C �→ ∇ C 0 d 0 ✲ γ ✲ ✛ i C= C 1 × C 0 C 1 C 1 C 0 d 1 ✲ ϕ C = d C × d C d C 1 C 0 ❄ ❄ ❄ ✲ ✲ C 0 × C 0 ✛ ✲ C 0 ∇ C 0 = C 0 × C 0 × C 0 5
Lemma 4.1 S regular ⇒ for every C ∈ Cat ( S ) the kernel pair of ϕ C = ( d C , 1 C 0 ) has a coequalizer in Cat ( S ) . p C × p C q C × q C p C q C 1 C 0 d 0 ❄ ❄ ❄ ❄ ✲ ❄ γ i ✲ ✛ C 1 × C 0 C 1 C 1 C 0 d 1 ✲ 1 C 0 e C × e C e C d I ❄ ❄ ✲ ❄ 0 γ I ✲ ✛ e C i I ( C 1 ) × C 0 I ( C 1 ) I ( C 1 ) C 0 d I ✲ 1 6
Conclusion 4.2 S regular: Cat ( S ) → Preord ( S ) reflection with stable units; Grpd ( S ) → EqRel ( S ) reflection with stable units and monotone-light factorization, ( σ, d 1 ) : Eq ( d 0 ) → G , with σ = γ (1 G 1 × s ) , p 2 ✲ < 1 , 1 > p 1 × p 2 ✲ ✛ G 1 × G 0 G 1 × G 0 G 1 G 1 × G 0 G 1 G 1 p 1 ✲ σ × σ σ d 1 d 0 ❄ ❄ ✲ ❄ γ ✲ ✛ i . G 1 × G 0 G 1 G 1 G 0 d 1 ✲ ✒ ✐ s σ < 1 G 1 , id 0 > = 1 G 1 and d 1 i = 1 G 0 . 7
e.g. S = Set : Cat → Preord , ( E ′ , M ∗ ) = ( Full and Bijective on Objects , Faithful ). S Maltsev category: EqRel ( S ) = RRel ( S )( ⇒ Cat ( S ) = Grpd ( S )). S regular Maltsev category: Grpd ( S ) → EqRel ( S ) = RRel ( S ) reflection with stable units and monotone-light-factorization. A variety of universal algebras is Maltsev iff its theory has a Maltsev operator p : X × X × X → X , p ( x, y, y ) = x = p ( y, y, x ). e.g. Grp : p ( x, y, z ) = xy − 1 z ; Cat ( Grp ) = Grpd ( Grp ) ≃ CrossMod . 8
5 Geometric morphisms Corollary 5.1 C admits a (regular epi, mono)-factorization and ( F, ϕ ) idempotent ⇒ ( I, η ) idempotent. Corollary 5.2 C regular and ( F, ϕ ) idempotent ⇒ ( I, η ) idempotent; and F left exact ⇒ stable units; and ∀ B ∈ C ∃ p : E → B e.d.m. E ∈ Mono ( F, ϕ ) ⇒ m.-l. factorization. 9
Proposition 5.3 Let F : E → F be a geometric morphism between regular categories, F ∗ ⊣ F ∗ : E → F , which is an embedding. Then, the reflection I : F → Mono ( F ∗ ) , obtained from the localization F ∗ : F → E through the coequalizer of the kernel pair process, does have stable units. Moreover, there is a monotone-light factorization associated to the reflection I : F → Mono ( F ∗ ) provided the following four conditions also hold: 1. the category F is cocomplete; 2. the full subcategory Mono ( F ∗ ) is dense in F , i.e., every object of F is a colimit of objects of Mono ( F ∗ ) . 3. in F the coproduct of monomorphisms is a monomorphism; 4. regular epis are effective descent morphisms in F . 10
6 Second example: simplicial sets K : B → A fully faithful, S regular and complete S K : S A → S B ∆ op n ⊂ ∆ op , n ≥ 0, S = Set Smp → Smp n Smp → Mono ( F n ) ( F n , ϕ n ) �→ ( I n , η n ) Lemma 6.1 Every unit morphism of any representable functor ϕ n ∆( − , [ p ]) : ∆( − , [ p ]) → F n (∆( − , [ p ])) , p ≥ 0 , is a monomorphism in Smp = Set ∆ op . 11
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