Galois theories of internal groupoids via congruence relations for - - PowerPoint PPT Presentation

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

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1 Coequalizer of the kernel pair

C finitely-complete; (F, ϕ) pointed endofunctor on C, s.t. the kernel pair of ϕA : A → F(A) has a coequalizer for every object A in C. I(A) ✑✑✑✑✑ ✑ ✸ ✑ ✑ ✑ ✑ ✑ ✰ ◗◗◗◗◗ s F(A) ϕA µA A ❄ ηA A ×F (A) A ✲ ❄ π2,A A ✲ ηA π1,A ϕA

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2 Idempotency of (I, η)

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

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3 Stabilization and m.-l. factorization

Proposition 3.1 All ηA pullback stable regular epis and µ monic and Fη iso and F preserves C C ×I(A) A I(A) A ηA g ✲ ✲ ❄ ❄ ⇒ (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).

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4 First example: internal categories

(F, ϕ) idempotent associated to the localization Cat(S) → LEqRel(S) ≃ S C → ∇C0 ∇C0 = C0 × C0 × C0 ✲ C0 × C0 ✲ ✛ ✲ C0 ϕC = dC × dC dC 1C0 ❄ ❄ ❄ C= C1 ×C0 C1 ✲ γ C1 ✲ ✛ ✲ d1 i d0 C0

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Lemma 4.1 S regular ⇒ for every C ∈ Cat(S) the kernel pair of ϕC = (dC, 1C0) has a coequalizer in Cat(S). ❄ pC × pC ❄ qC × qC ❄ pC ❄ qC ❄ 1C0 C1 ×C0 C1 ✲ γ C1 ✲ ✛ ✲ d1 i d0 C0 ❄ 1C0 ❄ eC × eC ❄ eC I(C1) ×C0 I(C1) ✲ γI I(C1) ✲ ✛ ✲ dI

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eCi dI C0

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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, d1) : Eq(d0) → G, with σ = γ(1G1 × s), G1 ×G0 G1 ✲ γ G1 ✐ ✒ s ✲ ✛ ✲ d1 i d0 G0 . σ × σ σ d1 ❄ ❄ ❄ G1 ×G0 G1 ×G0 G1 ✲ p1 × p2 G1 ×G0 G1 ✲ ✛ ✲ p1 < 1, 1 > p2 G1 σ < 1G1, id0 >= 1G1 and d1i = 1G0.

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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−1z; Cat(Grp) = Grpd(Grp) ≃ CrossMod.

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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.

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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
• f 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.

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6 Second example: simplicial sets

K : B → A fully faithful, S regular and complete SK : SA → SB ∆op

n ⊂ ∆op, n ≥ 0, S = Set

Smp → Smpn Smp → Mono(Fn) (Fn, ϕn) → (In, ηn) Lemma 6.1 Every unit morphism of any representable functor ϕn

∆(−,[p]) : ∆(−, [p]) → Fn(∆(−, [p])), p ≥ 0,

is a monomorphism in Smp = Set∆op.

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