Get scene camera motion

2.2 Ecumenae

Synopsis. We define various notions of categories equipped with struc-ture making it possible to interpret basic notions of topology such as open embeddings and local homeomorphisms.

Prerequisites. §§1.1,1.2,1.3,1.4,1.5,a.2.

2.2.1 Definition. Anecumene^[1] is a tuple(u�,u�, 𝖩)where:

• u�is a category.

• u�is a class of fibrations inu�.

• 𝖩is a coverage onu�.

• Every morphism inu� ofu�-type(u�, 𝖩)-semilocally on the domain is a member ofu�.

We will often abuse notation by referring to the category u� itself as an ecumene, omitting mention ofu�and𝖩.

Example. Letu�be a category with pullbacks, letu�be the class of quad-rable morphisms inu�, and let𝖩be any coverage onu�. Then(u�,u�, 𝖩)is an ecumene.

2.2.2 ¶ Letu�be a category and letℬbe a set of morphisms inu�.

Definition. A coverage𝖩onu�isℬ-adaptedif it has the following prop-erty:

• For every object 𝑋 in u� and every𝖩-covering sinkΦon 𝑋, there is ℬ-sinkΦ^′such thatΦ^′∈ 𝖩(𝑋)and↓(Φ^′) ⊆ ↓(Φ).

Example. Ifℬ=moru�, then every coverage onu� isℬ-adapted.

Remark. Let𝖩be a coverage onu� and let𝖩_ℬ(𝑋)be the set ofℬ-sinks on 𝑋 that are members of𝖩(𝑋). If𝖩isℬ-adapted, then𝖩_ℬis a coverage, and moreover the𝖩-covering sinks coincide with the𝖩_ℬ-covering sinks.

[1] — from Greek«οἰκουμένη», the inhabited world.

Properties of adapted coverages

Proposition. If𝖩is a ℬ-adapted coverage onu�, thenℬsatisfies the𝖩 -local collection axiom.

Proof. Immediate. ■

2.2.3 Definition. Thedescent axiomfor an ecumene(u�,u�, 𝖩)is the following:

• Every morphism inu�ofu�-type𝖩-semilocally on the base is a member ofu�.

The ecumene generated by a class of fibrations

Proposition. Letu� be a category, letℬbe a class of fibrations inu�, let 𝖩be a coverage onu�, and letu�be the class of morphisms inu� that are of ℬ-type(ℬ, 𝖩)-semilocally on the domain. Assuming every member ofu�is a quadrable morphism inu�:

(i) (u�,u�, 𝖩)is an ecumene.

In addition, assuming𝖩is aℬ-adapted coverage onu�: (ii) (u�,u�, 𝖩)satisfies the descent axiom.

(iii) 𝖩is au�-adapted coverage onu�.

Proof. (i). Byproposition 1.2.5andlemma 1.2.6,u�is a class of fibrations inu� andℬ ⊆ u�. It remains to be shown that every morphism inu� ofu� -type(u�, 𝖩)-semilocally on the domain is a member ofu�.

Let𝑓 : 𝑋 → 𝑌 be a morphism inu� and letΦbe a𝖩-covering sink on 𝑋such that, for every(𝑈, 𝑥) ∈ Φ, both𝑥 : 𝑈 → 𝑋and𝑓 ∘𝑥 : 𝑈 → 𝑌 are members ofu�. Since𝑥 : 𝑈 → 𝑋is a member ofu�, there is a𝖩-covering sinkΦ^′_(𝑈,𝑥) on𝑈 such that, for every(𝑇 , 𝑢) ∈ Φ^′_(𝑈,𝑥), both 𝑢 : 𝑇 → 𝑈 and𝑥 ∘ 𝑢 : 𝑇 → 𝑋are members ofℬ. Thus, byproposition a.2.14,

Φ^′ = ⋃_{(𝑈,𝑥)∈Φ}{(𝑇 , 𝑥 ∘ 𝑢) | (𝑇 , 𝑢) ∈ Φ^{′(𝑈,𝑥)}}

is a𝖩-covering sink on𝑋such that, for every(𝑇 , 𝑥^′) ∈ Φ, both𝑥^′: 𝑇 → 𝑋and𝑓 ∘𝑥^′ : 𝑇 → 𝑋are members ofℬ. Hence,𝑓 : 𝑋 → 𝑌 is ofℬ-type (ℬ, 𝖩)-semilocally on the domain. The claim follows.

90

2.2. Ecumenae (ii). Let𝑓 : 𝑋 → 𝑌 be a morphism inu� and letΨbe a𝖩-covering sink on𝑌. Suppose, for each(𝑉 , 𝑦) ∈ Ψ, there is a pullback square inu� of the form below,

𝑓^∗𝑉 𝑋

𝑉 𝑌

𝑦^∗𝑓

𝑓^∗𝑦 𝑓 𝑦

where𝑦^∗𝑓 : 𝑓^∗𝑉 → 𝑉 is a member of u�. Sinceu� is a quadrable class of morphisms and𝖩 is aℬ-adapted coverage on u�, we may assume that each 𝑦 : 𝑉 → 𝑌 is a member of ℬ. Then 𝑦 ∘ 𝑦^∗𝑓 : 𝑓^∗𝑉 → 𝑌 is a member of u� and 𝑓^∗𝑦 : 𝑓^∗𝑉 → 𝑋 is a member of ℬ. Moreover, {(𝑓^∗𝑉 , 𝑓^∗𝑦) | (𝑉 , 𝑦) ∈ Ψ}is a𝖩-covering sink on𝑋, so𝑓 : 𝑋 → 𝑌 is of u�-type(ℬ, 𝖩)-locally on the domain. Hence, by (i), 𝑓 : 𝑋 → 𝑌 is a member ofu�.

(iii). Immediate. ■

Remark. In particular, if (u�,u�, 𝖩) is an ecumene and 𝖩 is a u�-adapted coverage onu�, then the descent axiom is satisfied.

2.2.4 Definition. Aregulated ecumeneis an ecumene(u�,u�, 𝖩)with the fol-lowing additional data:

• For each morphism 𝑓 : 𝑋 → 𝑌 inu� that is a member ofu�, a mono-morphismim(𝑓) :Im(𝑓) ↣ 𝑌 inu�such that𝑓 =im(𝑓)∘𝜂_𝑓 for some 𝖩-covering morphism𝜂_𝑓 : 𝑋 ↠Im(𝑓)inu� and, for every commutat-ive diagram inu� of the form below,

𝑋^′ 𝑋

𝐼^′ Im(𝑓)

𝑌^′ 𝑌

𝑒^′ 𝜂_𝑓

𝑖^′ im(𝑓)

if both squares are pullback squares inu�, then 𝑒^′ : 𝑋^′ ↠ 𝐼^′ is an effective epimorphism inu�and𝑖^′ : 𝐼^′↣ 𝑌^′is a member ofu�.

Remark. In the above,𝑒 : 𝑋 ↠ 𝐼 is automatically a member ofu�, by lemma 1.1.3.

Proposition. Let(u�,u�, 𝖩)be a regulated ecumene.

(i) Every effective epimorphism in u� that is a member of u� is a 𝖩 -covering morphism inu�.

(ii) (u�,u�) is a regulated category, where u� is the subcategory of u� -perfect morphisms inu�.

Proof. (i). Let 𝑓 : 𝑋 → 𝑌 be an effective epimorphism in u� that is a member of u�. Then im(𝑓) : Im(𝑓) → 𝑌 must be an isomorphism in u�; but𝑓 : 𝑋 → 𝑌 factors as a𝖩-covering morphism inu� followed by im(𝑓) :Im(𝑓) → 𝑌, so𝑓 : 𝑋 → 𝑌 itself is also a 𝖩-covering morphism inu�.

(ii). Byproposition 1.1.11,u�is a class of separated fibrations inu�. Every effective epimorphism inu� is au�-calypsisa fortiori, so every member of u� is quadrablyu�-eucalyptic. Thus, every member of u�isu�-agathic, as

required. ■

2.2.5 ¶ Let𝜅be a regular cardinal.

Definition. A𝜅-ary extensive ecumeneis an ecumene(u�,u�, 𝖩)where:

• u�is a𝜅-ary extensive category.

• 𝖩is a𝜅-ary superextensive coverage onu�.

• Every complemented monomorphism inu� is a member ofu�.

Recognition principle for extensive ecumenae that satisfy the descent axiom

Lemma. Let(u�,u�, 𝖩)be an ecumene that satisfies the descent axiom. The following are equivalent:

(i) (u�,u�, 𝖩)is a𝜅-ary extensive ecumene.

(ii) u�is a𝜅-ary extensive category and𝖩is a𝜅-ary superextensive cov-erage onu�.

Proof. (i)⇒(ii). Immediate.

92

2.2. Ecumenae (ii)⇒(i). Let0be the initial object inu�. Then, for every object𝑋 inu�, the unique morphism0 → 𝑋is vacuously ofu�-type(u�, 𝖩)-locally on the domain, so it is a member ofu�. Similarly, given morphisms𝑓₀: 𝑋₀ → 𝑌₀ and𝑓₁ : 𝑋₁→ 𝑌₁inu�that are members ofu�,𝑓₀⨿𝑓₁ : 𝑋₀⨿𝑋₁→ 𝑌₀⨿𝑌₁ is ofu�-type𝖩-locally on the base, so it is also a member ofu�. Since every isomorphism inu� is a member ofu�, it follows that every complemented

monomorphism inu�is a member ofu�. ■

Properties of extensive ecumenae

Proposition. Let(u�,u�, 𝖩)be a𝜅-ary extensive ecumene.

(i) Given a family(𝑓^𝑖| 𝑖 ∈ 𝐼)where𝐼 is a𝜅-small set and each𝑓_𝑖is a morphism𝑋_𝑖 → 𝑌 inu�, if each𝑓_𝑖 : 𝑋_𝑖 → 𝑌 is a member ofu�, then the induced morphism𝑓 : ∐_𝑖∈𝐼𝑋_𝑖 → 𝑌 is also a member ofu�. (ii) u�is closed under𝜅-ary coproduct inu�.

(iii) Assuming𝖩isu�-adapted, a morphism𝑔 : 𝑌 → 𝑍inu�is𝖩-covering if and only if there is a morphism𝑓 : 𝑋 → 𝑌 inu� such that𝑔 ∘ 𝑓 : 𝑋 → 𝑍is𝖩-covering morphism inu� that is a member ofu�.

Proof. (i). By construction,𝑓 : ∐_𝑖∈𝐼𝑋_𝑖 → 𝑌 is ofu�-type(u�, 𝖩)-semi-locally on the domain, so it is a member ofu�.

(ii). Since coproduct injections are inu�andu�is closed under composition, the claim is a special case of (i).

(iii). Applylemma 1.5.16and (i). ■

2.2.6 ※For the remainder of this section,(u�,u�, 𝖩)is an ecumene.

2.2.7 ¶ It is convenient to introduce some terminology for properties of morph-isms related tou�.

2.2.7(a) Definition. A morphism inu� isgenialif it is a member ofu�. 2.2.7(b) Definition. A morphism inu� isétaleif it isu�-perfect.

2.2.7(c) Definition. A morphism inu�iseunoic^[2]if it is𝖩-semilocally ofu�-type.

[2] — from Greek«εὐνοϊκός», favourable.

Remark. Bylemma 1.2.15, eunoic morphisms are automatically𝖩-locally ofu�-type.

2.2.8 Definition. An equivalence relation (𝑅, 𝑑0, 𝑑₁) on an object𝑋 inu� is étaleif it has the following property:

• The projections𝑑₀, 𝑑₁ : 𝑅 → 𝑋are étale morphisms.

2.2.8(a)

Recognition principle for kernel pairs of étale morphisms

Lemma. Let(𝑅, 𝑑⁰, 𝑑₁)be a kernel pair of a morphism𝑓 : 𝑋 → 𝑌 inu�. The following are equivalent:

(i) 𝑓 : 𝑋 → 𝑌 is a genial morphism inu� and(𝑅, 𝑑0, 𝑑₁)is an étale equivalence relation on𝑋inu�.

(ii) 𝑓 : 𝑋 → 𝑌 is an étale morphism inu�.

Proof. (i)⇒(ii). Bylemma 1.1.10, the relative diagonal Δ_𝑓 : 𝑋 → 𝑅 is étale if either 𝑑₀ : 𝑅 → 𝑋 or 𝑑₁ : 𝑅 → 𝑋 is étale, in which case 𝑓 : 𝑋 → 𝑌 itself is étale.

(ii)⇒(i). The class of étale morphisms inu�is a quadrable class of morph-isms in u�, by proposition 1.1.11, so if 𝑓 : 𝑋 → 𝑌 is étale then both

𝑑₀, 𝑑₁ : 𝑅 → 𝑋are also étale. ■

2.2.8(b)

Recognition prin-ciple for kernel pairs of covering étale morphisms

Lemma. Let(𝑅, 𝑑0, 𝑑₁) be a kernel pair of a𝖩-covering morphism 𝑓 : 𝑋 ↠ 𝑌 inu�. Assuming(u�,u�, 𝖩)satisfies the descent axiom, the following are equivalent:

(i) (𝑅, 𝑑0, 𝑑₁)is an étale equivalence relation on𝑋 inu�. (ii) 𝑓 : 𝑋 ↠ 𝑌 is an étale morphism inu�.

Proof. By definition, the following is a pullback square inu�:

𝑅 𝑋

𝑋 𝑌

𝑑₁ 𝑑₀

𝑓 𝑓

Since 𝑓 : 𝑋 ↠ 𝑌 is a 𝖩-covering morphism in u�, the descent axiom implies that𝑓 : 𝑋 ↠ 𝑌 is a member ofu�if and only if either𝑑₀: 𝑅 → 𝑋 or𝑑₁ : 𝑅 → 𝑋 (or both) is a member of u�. Thus, the claims reduce to

lemma 2.2.8(a). ■

94

2.2. Ecumenae 2.2.9 ¶ The following generalises Proposition 1.6 in [JM].

Definition. The ecumene (u�,u�, 𝖩)is étale if every member of u� is an étale morphism inu�.

The induced étale ecumene

Proposition. Letu�be the class of étale morphisms inu�. (i) (u�,u�, 𝖩)is an étale ecumene.

(ii) If (u�,u�, 𝖩)satisfies the descent axiom, then (u�,u�, 𝖩)also satisfies the descent axiom.

(iii) (u�,u�, 𝖩)is𝜅-ary extensive if and only if(u�,u�, 𝖩)is𝜅-ary extensive.

Proof. (i). Byproposition 1.1.11,u�is a class of separated fibrations inu�. It remains to be shown that every morphism ofu�-type(u�, 𝖩)-semilocally on the domain is a member ofu�.

Let𝑓 : 𝑋 → 𝑌 be a morphism inu� and letΦbe a𝖩-coveringu�-sink on 𝑋 such that, for every (𝑈, 𝑥) ∈ Φ, 𝑓 ∘ 𝑥 : 𝑈 → 𝑌 is a member of u�. Then 𝑓 : 𝑋 → 𝑌 is of u�-type(u�, 𝖩)-semilocally on the domain, so by lemma 1.2.6, 𝑓 : 𝑋 → 𝑌 is a member ofu�. Since𝑥 : 𝑈 → 𝑋 is a member of u�, the induced morphism 𝑈 ×_𝑌 𝑈 → 𝑋 ×_𝑌 𝑋 is also a member ofu�. On the other hand,𝑓 ∘ 𝑥 : 𝑈 → 𝑌 is a member ofu�, so the relative diagonalΔ_𝑓∘𝑥 : 𝑈 → 𝑈 ×_𝑌 𝑈 is a member of u�. Thus, the relative diagonal Δ_𝑓 : 𝑋 → 𝑋 ×_𝑌 𝑋 is of u�-type(u�, 𝖩)-semilocally on the domain, therefore it is a member ofu�. Hence,𝑓 : 𝑋 → 𝑌 is indeed a member ofu�.

(ii). Let𝑓 : 𝑋 → 𝑌 be a morphism inu� and letΨbe a𝖩-covering sink on𝑌 such that, for every(𝑉 , 𝑦) ∈ Φ, we have a pullback square inu� of the form below,

𝑈 𝑋

𝑉 𝑌

𝑣 𝑥

𝑓 𝑦

where𝑣 : 𝑈 → 𝑉 is a member ofu�. The hypothesis implies𝑓 : 𝑋 → 𝑌 is a member of u�, and it remains to be shown that 𝑓 : 𝑋 → 𝑌 is u� -separated.

By the pullback pasting lemma, every face of the following diagram is a pullback square inu�:

𝑈 ×_𝑉 𝑈 𝑈

𝑈 𝑉

𝑋 ×_𝑌 𝑋 𝑋

𝑋 𝑌

𝑥×_𝑦𝑥

Hence, we have a pullback square inu� of the form below:

𝑈 𝑋

𝑈 ×_𝑉 𝑈 𝑋 ×_𝑌 𝑋

Δ_𝑣

𝑥

Δ_𝑓

Moreover, byproposition a.2.14,

{(𝑈 ×𝑉 𝑈, 𝑥 ×_𝑦𝑥) | (𝑉 , 𝑦) ∈ Ψ}

is a𝖩-covering sink on𝑋×_𝑌𝑋, so the relative diagonalΔ_𝑓 : 𝑋 → 𝑋×_𝑌𝑋 is ofu�-type𝖩-semilocally on the base. Thus,Δ_𝑓 : 𝑋 → 𝑋 ×_𝑌 𝑋 is a member ofu�, as required.

(iii). Immediate, because genial monomorphisms are the same as étale

monomorphisms. ■

2.2.10 ¶ Étale morphisms and eunoic morphisms are related as follows.

Recognition principle for étale morphisms

Lemma. Let 𝑓 : 𝑋 → 𝑌 be a quadrable morphism inu� such that the relative diagonal Δ_𝑓 : 𝑋 → 𝑋 ×_𝑌 𝑋 is also a quadrable morphism in u�. Assuming (u�,u�, 𝖩) satisfies the descent axiom, the following are equivalent:

(i) The morphism𝑓 : 𝑋 → 𝑌 is étale.

(ii) Both𝑓 : 𝑋 → 𝑌 and the relative diagonalΔ_𝑓 : 𝑋 → 𝑋 ×_𝑌 𝑋 are eunoic.

Proof. This is a special case ofproposition 1.2.20. ■

96

2.2. Ecumenae 2.2.11 ¶ It is convenient to introduce the following terminology.

2.2.11(a) Definition. Anopen embeddinginu� is a monomorphism inu�that is a member ofu�.

2.2.11(b) Definition. Thedescent axiom for open embeddingsin(u�,u�, 𝖩)is the following:

• Every monomorphisminu� ofu�-type 𝖩-semilocally on the base is an open embedding inu�.

Recognition principles for open embeddings

Lemma. Let 𝑓 : 𝑋 → 𝑌 be a quadrable morphism in u�. Assuming (u�,u�, 𝖩) satisfies the descent axiom for open embeddings, the following are equivalent:

(i) 𝑓 : 𝑋 → 𝑌 is an open embedding inu�. (ii) 𝑓 : 𝑋 → 𝑌 is an étale monomorphism inu�. (iii) 𝑓 : 𝑋 → 𝑌 is a eunoic monomorphism inu�.

Proof. (i) ⇒(ii). The relative diagonal Δ_𝑓 : 𝑋 → 𝑋 ×_𝑌 𝑋 is an iso-morphism inu�.

(ii)⇒(iii). Étale morphisms are eunoica fortiori. (iii)⇒(i). It suffices to verify the following:

• Every monomorphisminu� ofu�-type𝖩-semilocally on the domain is an open embedding inu�.

However, by lemma 1.2.19(a), every such monomorphism is automatic-ally ofu�-type(u�, 𝖩)-semilocally on the domain, hence is indeed a member

ofu�. ■

2.2.12 ¶ Letu�monobe the class of open embeddings inu�.

We will see that the following is a specialisation of the notion of étale morphism.

Definition. Alocal homeomorphisminu�is a morphism inu�ofu�mono -type(u�mono, 𝖩)-semilocally on the domain.

Remark. Bylemma 1.2.6, local homeomorphisms are automatically of u�mono-type(u�mono, 𝖩)-locally on the domain.

Proposition.

(i) Every open embedding inu� is a local homeomorphism inu�. (ii) The class of local homeomorphisms in u� is a quadrable class of

morphisms inu�.

(iii) The class of local homeomorphisms inu� is closed under composi-tion.

(iv) Given morphisms 𝑓 : 𝑋 → 𝑌 and 𝑔 : 𝑌 → 𝑍 in u�, if both 𝑔 : 𝑌 → 𝑍 and 𝑔 ∘ 𝑓 : 𝑋 → 𝑍 are local homeomorphisms, then 𝑓 : 𝑋 → 𝑌 is also a local homeomorphism.

(v) Every local homeomorphism inu� is an étale morphism inu�. Proof. (i). Immediate.

(ii). Clearly, every local homeomorphism in u� is of u�-type(u�, 𝖩) -semi-locally on the domain, hence is a member ofu�. In particular, local homeo-morphisms inu� are quadrable. The claim follows, byproposition 1.2.4.

(iii). In view oflemma 1.2.6, we may applyproposition 1.2.5.

(iv). LetΨbe a𝖩-coveringu�mono-sink on𝑌 such that, for every(𝑉 , 𝑦) ∈ Ψ, 𝑔 ∘ 𝑦 : 𝑉 → 𝑍 is an open embedding inu�. By proposition a.2.14 and (ii), there is a 𝖩-covering u�_mono-sink Φ on 𝑋 such that, for every (𝑈, 𝑥) ∈ Φ, 𝑔 ∘ 𝑓 ∘ 𝑥 : 𝑈 → 𝑍 is an open embedding inu� and factors through𝑔 ∘ 𝑦 : 𝑉 → 𝑍 for some(𝑉 , 𝑦) ∈ Ψ. Recalling thatu�monois a class of fibrations inu�, bylemma 1.1.3,𝑓 ∘ 𝑥 : 𝑈 → 𝑌 is also an open embedding inu�. Hence,𝑓 : 𝑋 → 𝑌 is a local homeomorphism inu�.

(v). If 𝑓 : 𝑋 → 𝑌 is a local homeomorphism in u�, then the relative diagonal Δ_𝑓 : 𝑋 → 𝑋 ×_𝑌 𝑋 is also a local homeomorphism inu�, by (i), (ii), and (iv). Thus, by lemma 1.1.6, 𝑓 : 𝑋 → 𝑌 is indeed an étale

morphism inu�. ■

98

2.2. Ecumenae 2.2.13

The ecumene of local homeo-morphisms

Proposition. Letu�be the class of local homeomorphisms inu�. (i) (u�,u�, 𝖩)is an étale ecumene.

(ii) (u�,u�, 𝖩)satisfies the descent axiom for open embeddings if and only if(u�,u�, 𝖩)satisfies the descent axiom for open embeddings.

(iii) (u�,u�, 𝖩)is𝜅-ary extensive if and only if(u�,u�, 𝖩)is𝜅-ary extensive.

Proof. (i). Byproposition 2.2.12,u�is a class of separated fibrations inu�. It remains to be shown that every morphism ofu�-type(u�, 𝖩)-semilocally on the domain is a member ofu�.

Let 𝑓 : 𝑋 → 𝑌 be a morphism in u� and let Φ be a 𝖩-covering u�-sink on 𝑋 such that, for every (𝑈, 𝑥) ∈ Φ, 𝑓 ∘ 𝑥 : 𝑈 → 𝑌 is a local homeomorphism in u�. So, for each (𝑈, 𝑥) ∈ Φ, there is a 𝖩-covering u�_monosinkΘ_(𝑈,𝑥)on𝑈such that, for every(𝑇 , 𝑢) ∈ Θ_(𝑈,𝑥),𝑥∘𝑢 : 𝑇 → 𝑋 is an open embedding inu�. Consider the sinkΦ^′defined as follows:

Φ^′= ⋃_{(𝑈,𝑥)∈Φ}{(𝑇 , 𝑥 ∘ 𝑢) | (𝑇 , 𝑢) ∈ Θ(𝑈,𝑥)}

By proposition a.2.14, Φ^′ is a𝖩-coveringu�mono-sink on 𝑋. Moreover, each𝑓 ∘𝑥∘𝑢 : 𝑇 → 𝑌 is a local homeomorphism inu�, so𝑓 : 𝑋 → 𝑌 is of u�-type(u�_mono, 𝖩)-semilocally on the domain. Thus, byproposition 1.2.4, 𝑓 : 𝑋 → 𝑌 itself is indeed a local homeomorphism inu�.

(ii) and (iii). Immediate. ■

2.2.14 ¶ We will now characterise local homeomorphisms as genial morphisms whose kernel pair have a certain property.

Definition. An equivalence relation (𝑅, 𝑑0, 𝑑₁) on an object𝑋 inu� is tractableif it has the following properties:

• The projections𝑑₀, 𝑑₁ : 𝑅 → 𝑋are members ofu�.

• There is a𝖩-coveringu�mono-sinkΦon𝑋 such that, for every(𝑈, 𝑥) ∈ Φand every object(𝑇 , 𝑟)inu�_∕𝑅, if both𝑑₀∘ 𝑟, 𝑑₁∘ 𝑟 : 𝑇 → 𝑋 factor through𝑥 : 𝑈 → 𝑋, then𝑑₀∘ 𝑟 = 𝑑₁∘ 𝑟.

Remark. Byproposition 2.2.12, if(𝑅, 𝑑0, 𝑑₁)is a tractable equivalence relation on𝑋, then the relative diagonalΔ : 𝑋 → 𝑅is an open embedding inu�. However, the converse is not true in general, even when we assume that the projections𝑑₀, 𝑑₁ : 𝑅 → 𝑋are local homeomorphisms inu�. 2.2.14(a)

A sufficient criterion for tractability

Lemma. Let(𝑅, 𝑑0, 𝑑₁)be an equivalence relation on an object𝑋 inu�. Assuming the projections𝑑₀, 𝑑₁ : 𝑅 → 𝑋 are genial morphisms inu�, if there is a local homeomorphism𝑓 : 𝑋 → 𝑌 inu�such that𝑓 ∘𝑑₀= 𝑓 ∘𝑑₁, then(𝑅, 𝑑0, 𝑑₁)is tractable.

Proof. LetΦbe a𝖩-coveringu�mono-sink on𝑋such that, for every(𝑈, 𝑥) ∈ Φ,𝑓 ∘ 𝑥 : 𝑈 → 𝑌 is an open embedding in u�. For every object(𝑇 , 𝑟)in u�_∕𝑅, we have𝑓 ∘ 𝑑₀∘ 𝑟 = 𝑓 ∘ 𝑑₁∘ 𝑟, so if both𝑑₀∘ 𝑟, 𝑑₁∘ 𝑟 : 𝑇 → 𝑋 factor through𝑥 : 𝑈 → 𝑋, then𝑑₀∘ 𝑟 = 𝑑₁∘ 𝑟. Hence(𝑅, 𝑑⁰, 𝑑₁) is indeed a

tractable equivalence relation on𝑋 inu�. ■

2.2.14(b)

Recognition prin-ciple for kernel pairs of local homeomorphisms

Lemma. Let(𝑅, 𝑑0, 𝑑₁)be a kernel pair of a morphism𝑓 : 𝑋 → 𝑌 inu�. The following are equivalent:

(i) 𝑓 : 𝑋 → 𝑌 is a genial morphism inu�and(𝑅, 𝑑0, 𝑑₁)is a tractable equivalence relation on𝑋inu�.

(ii) 𝑓 : 𝑋 → 𝑌 is a local homeomorphism inu�.

Proof. (i) ⇒(ii). LetΦbe a𝖩-coveringu�mono-sink on𝑋 such that, for every (𝑈, 𝑥) ∈ Φ and every object (𝑇 , 𝑟) in u�_∕𝑅, if both 𝑑₀ ∘ 𝑟, 𝑑₁ ∘ 𝑟 : 𝑇 → 𝑋 factor through𝑥 : 𝑈 → 𝑋, then𝑑₀∘ 𝑟 = 𝑑₁ ∘ 𝑟. Then𝑓 ∘ 𝑥 : 𝑈 → 𝑌 is a monomorphism in u�: indeed, given 𝑢₀, 𝑢₁ : 𝑇 → 𝑈 inu�, if𝑓 ∘ 𝑥 ∘ 𝑢₀ = 𝑓 ∘ 𝑥 ∘ 𝑢₁, then we may apply the hypothesis to deduce that𝑢₀ = 𝑢₁. Since𝑓 : 𝑋 → 𝑌 is a member ofu�andu� is closed under composition, it follows that𝑓 ∘ 𝑥 : 𝑈 → 𝑌 is an open embedding inu�. Hence𝑓 : 𝑋 → 𝑌 is indeed a local homeomorphism inu�.

(ii)⇒(i). This is a special case oflemma 2.2.14(a). ■

100

2.2. Ecumenae 2.2.14(c)

Recognition principle for kernel pairs of covering local homeomorphisms

Lemma. Let(𝑅, 𝑑0, 𝑑₁) be a kernel pair of a𝖩-covering morphism 𝑓 : 𝑋 ↠ 𝑌 in u�. Assuming (u�,u�, 𝖩) satisfies the descent axiom for open embeddings, the following are equivalent:

(i) (𝑅, 𝑑0, 𝑑₁)is a tractable equivalence relation on𝑋 inu�. (ii) 𝑓 : 𝑋 ↠ 𝑌 is a local homeomorphism inu�.

Proof. (i)⇒(ii). By definition, the following is a pullback square inu�:

𝑅 𝑋

𝑋 𝑌

𝑑₁ 𝑑₀

𝑓 𝑓

Suppose𝑥 : 𝑈 → 𝑋 is an open embedding inu�such that𝑓 ∘ 𝑥 : 𝑈 → 𝑌 is a monomorphism in u�. Then the projection𝑋 ×_𝑌 𝑈 → 𝑋 is also an open embedding in u�, and since 𝑓 : 𝑋 ↠ 𝑌 is a 𝖩-covering morphism in u�, 𝑓 ∘ 𝑥 : 𝑈 → 𝑌 is an open embedding in u�. Thus, following the argument of lemma 2.2.14(b), we see that𝑓 : 𝑋 → 𝑌 itself is a local homeomorphism inu�.

(ii)⇒(i). This is a special case oflemma 2.2.14(a). ■

In document 3D head motion, point-of-regard and encoded gaze fixations in real scenes: next-generation portable video-based monocular eye tracking (Page 134-143)