SECRETOS PARA UNA BUENA COMUNICACIÓN 2.1 DESCRIPCIÓN:
COMUNICACIÓN FAMILIAR
1.5.2(b) Definition. The cocone(𝑓 | (𝑋, 𝑓) ∈ Φ) is astrict coproduct cocone inu�if it has the following properties:
• For every(𝑋, 𝑓) ∈ Φ,𝑓 : 𝑋 → 𝑌 is a quadrable morphism inu�.
• For every object(𝑇 , 𝑦)inu�∕𝑌, given pullback squares inu� of the form below
𝑆(𝑋,𝑓) 𝑋
𝑇 𝑌
𝑡(𝑋,𝑓) 𝑓
𝑦
for every(𝑋, 𝑓) ∈ Φ, the object𝑇 is a disjoint union of the following set:
{(𝑆(𝑋,𝑓), 𝑡(𝑋,𝑓)) | (𝑋, 𝑓) ∈ Φ}
Remark. In particular, whenΦ = ∅, this reduces to the notion ofstrict initial object. More explicitly,𝑌 is a strict initial object if and only if, for every morphism𝑦 : 𝑇 → 𝑌 inu�,𝑇 is an initial object inu�.
Pullbacks along strict coprod-uct cocones
Proposition. Assuming (𝑓 | (𝑋, 𝑓) ∈ Φ) is a strict coproduct cocone, the functor
u�∕𝑌 → ∏(𝑋,𝑓)∈Φu�∕𝑋 defined by pullback is fully faithful.
Proof. Straightforward. ⧫
1.5.3 ※For the remainder of this section,𝜅 is a regular cardinal.
1.5.4 ¶ Letu�be a𝜅-ary coherent category.
Recognition principle for disjoint unions in coherent categories
Lemma. Let𝑌 be an object inu�and letΦbe a𝜅-small set of subobjects of𝑌. The following are equivalent:
(i) 𝑌 is a disjoint union ofΦ.
(ii) (𝑌 ,id𝑌)is a coproduct ofΦin the category of subobjects of𝑌 and, for every commutative square inu� of the form below,
𝑇 𝑋1
𝑋0 𝑌
𝑓1 𝑓0
62
1.5. Strict coproducts if(𝑋0, 𝑓0)and(𝑋1, 𝑓1)are both inΦ, then either(𝑋0, 𝑓0) = (𝑋1, 𝑓1) or𝑇 is an initial object inu� (or both).
Proof. (i)⇒(ii). Immediate.
(ii)⇒(i). For the case whereΦhas two elements, see the proof of Propos-ition 1.4.3 in [Johnstone,2002, Part A]. The general case is similar. □ 1.5.5 Definition. A 𝜅-ary extensive category is a categoryu� with the
fol-lowing data:
• For every family(𝑋𝑖| 𝑖 ∈ 𝐼)of objects inu�where𝐼is a𝜅-small set,[1]
an object∐𝑖∈𝐼𝑋𝑖inu� and a strict coproduct cocone(𝑓𝑖| 𝑖 ∈ 𝐼)inu�
wheredom𝑓𝑖 = 𝑋𝑖andcodom𝑓𝑖= ∐𝑖∈𝐼𝑋𝑖.
Remark. In particular, a 𝜅-ary extensive category has a (chosen) strict initial object, say0.
1.5.6 Definition. A𝜅-ary pretopos is an exact category that is also a𝜅-ary extensive category.
1.5.6(a) Lemma. Every𝜅-ary pretopos is a𝜅-ary coherent category.
Proof. Straightforward. (Define the union of a set of subobjects to be the
exact image of their coproduct.) ⧫
1.5.6(b)
Criterion for existence of coproducts in coherent categories
Lemma. Letu� be a𝜅-ary coherent category and letℬbe a set of objects inu�. Assume the following hypotheses:
• u�is an exact category.
• For every object𝑋 inu�, there is an effective epimorphism𝑝 : ̃𝑋 ↠ 𝑋 inu� where𝑋̃ is inℬ.
• Strict coproducts of𝜅-small families of objects inℬexist inu�. Thenu�is a𝜅-ary pretopos.
[1] Since there is a proper class of𝜅-small sets, strictly speaking, one should restrict to e.g.
hereditarily𝜅-small sets here.
Proof. Let(𝑋𝑖| 𝑖 ∈ 𝐼)be a family of objects inu�. For each𝑖 ∈ 𝐼, choose an effective epimorphism𝑝𝑖 : ̃𝑋𝑖 ↠ 𝑋𝑖inu�with𝑋̃𝑖inℬ, and then choose a pullback square inu�of the form below:
𝑅𝑖 𝑋̃𝑖
̃𝑋𝑖 𝑋𝑖
𝑑1 𝑑0
𝑝𝑖 𝑝𝑖
Let𝑋 = ∐̃ 𝑖∈𝐼𝑋̃𝑖. It is not hard to verify that the canonical morphism
(𝑗,𝑘)∈𝐼×𝐼∐
̃𝑋𝑗× ̃𝑋𝑘 → ̃𝑋 × ̃𝑋
in u� is an isomorphism. Thus, for each 𝑖 ∈ 𝐼, the composite 𝑅𝑖 →
̃𝑋𝑖× ̃𝑋𝑖 ↣ ̃𝑋 × ̃𝑋 is a monomorphism inu�. Let⟨𝑑1, 𝑑0⟩ : 𝑅 → ̃𝑋 × ̃𝑋 be the union. Then (𝑅, 𝑑0, 𝑑1) is an equivalence relation on𝑋̃, so there is an exact fork inu� of the form below:
𝑅 𝑑1 𝑋̃ 𝑋
𝑑0
𝑝
Moreover, for each𝑖 ∈ 𝐼, there is a unique morphism𝑚𝑖 : 𝑋𝑖 → 𝑋 inu�
such that the following diagram inu�commutes,
̃𝑋𝑖 𝑋𝑖
̃𝑋 𝑋
𝑝𝑖
𝑚𝑖 𝑝
(∗)
where𝑋̃𝑖 ↣ ̃𝑋 is the coproduct injection. Consider a commutative dia-gram inu� of the form below,
𝑅𝑗,𝑘 • 𝑋̃𝑘
• 𝑅 𝑋̃
̃𝑋𝑗 ̃𝑋 𝑋
𝑑1 𝑑0
𝑝 𝑝
where each square is a pullback square inu�. Then, either𝑗 = 𝑘or𝑅𝑗,𝑘is an initial object inu� (or both). In view oflemma 1.3.11, it follows that, 64
1.5. Strict coproducts for every pullback square inu� of form below,
𝑇 𝑋𝑘
𝑋𝑗 𝑋
𝑚𝑘 𝑚𝑗
either 𝑗 = 𝑘 or 𝑇 is an initial object in u� (or both). Hence, (∗) is a pullback square inu�, and by lemmas 1.3.8and1.3.9,𝑚𝑖 : 𝑋𝑖 → 𝑋 is a monomorphism. We then applylemma 1.5.4to deduce that𝑋is a disjoint union of{(𝑋𝑖, 𝑚𝑖) | 𝑖 ∈ 𝐼}, as required. ■ 1.5.7 ¶ Let u� be a category and let Fam𝜅(u�) be full subcategory ofFam(u�) spanned by those families𝑋such thatidx𝑋is a hereditarily𝜅-small set.
Proposition. Fam𝜅(u�)is a𝜅-ary extensive category.
Proof. Straightforward. ⧫
The free coprod-uct completion
Theorem. Letu�be a category with𝜅-ary coproducts, let𝐹 :u� →u�be a functor, and let𝛾 :u� →Fam𝜅(u�)be defined as inpropositiona.1.8.
(i) There exist a functor ̄𝐹 : Fam𝜅(u�) → u� that preserves𝜅-ary co-products and an isomorphism𝜂 : 𝐹 ⇒ ̄𝐹𝛾of functorsu� →u�.
(ii) Moreover, any such( ̄𝐹, 𝜂)is a left Kan extension of 𝐹 : u� → u�
along𝛾 :u� →Fam𝜅(u�). Proof. (i). Straightforward.
(ii). Let 𝐺 : Fam𝜅(u�) → u� be a functor and let 𝜑 : 𝐹 ⇒ 𝐺𝛾 be a natural transformation. For every object𝑋 inFam𝜅(u�), there is a unique morphism𝜃𝑋 : ̄𝐹(𝑋) → 𝐺(𝑋)making the diagram in u� shown below commute for every𝑖 ∈idx𝑋,
𝐹 (𝑋(𝑖)) ̄𝐹(𝑋)
𝐺(𝛾(𝑋)) 𝐺(𝑋)
𝜑𝑋(𝑖)
𝑣𝑖
𝜃𝑋
𝐺(𝑢𝑖)
where 𝑢𝑖 : 𝛾(𝑋(𝑖)) → 𝑋 and 𝑣𝑖 : 𝐹 (𝑋(𝑖)) → ̄𝐹(𝑋) are the respective coproduct insertions. It is clear that we obtain a natural transformation 𝜃 : ̄𝐹 ⇒ 𝐺, and it is unique one such that𝜃𝛾 ∙ 𝜂 = 𝜑. ■
1.5.8 ¶ Letu� be a category with a strict initial object 0and let 𝖤 be a unary coverage onu�.
Strict initial objects and locally 1-generable presheaves
Lemma. Let𝐴be a𝖤-locally 1-generable presheaf onu�. Then𝐴(0)has a unique element.
Proof. Let(𝑋, 𝑎)be an𝖤-local generator of𝐴and let⊥𝑋 be the unique morphism0 → 𝐴inu�. Of course,𝑎 ⋅ ⊥𝑋 ∈ 𝐴(0); it remains to be shown that it is the unique element of 𝐴(0). Suppose𝑎′ ∈ 𝐴(0). Since every morphism inu�with codomain0is an isomorphism, bylemma 1.4.7, there is a morphism 𝑓′ : 0 → 𝑋 inu� such that𝑎′ = 𝑎 ⋅ 𝑓′. But𝑓′ = ⊥𝑋, so
we indeed have𝑎′= 𝑎 ⋅ ⊥𝑋. ■
The initial locally 1-generable presheaf
Proposition. The representable presheaf h0 is an initial object in the metacategory of𝖤-locally 1-generable presheaves onu�.
Proof. It is clear thath0is𝖤-locally 1-generable, and the Yoneda lemma
reduces the claim tolemma 1.5.8. ■
1.5.9 ※For the remainder of this section,u� is a𝜅-ary extensive category.
1.5.10 Definition. A complemented monomorphismin u� is a monomorph-ism𝑓 : 𝑋 → 𝑌 in u� for which there is an object (𝑋′, 𝑓′)inu�∕𝑌 such that𝑌 is a disjoint union of{(𝑋, 𝑓), (𝑋′, 𝑓′)}.
Properties of complemented monomorphisms
Proposition.
(i) Every isomorphism inu� is a complemented monomorphism inu�. (ii) The class of complemented monomorphisms in u� is closed under
composition.
(iii) The class of complemented monomorphisms in u� is a quadrable class of morphisms inu�.
(iv) The class of complemented monomorphisms in u� is closed under 𝜅-ary coproduct inu�.
Proof. Straightforward. ⧫
66
1.5. Strict coproducts 1.5.11 Definition. A unary coverage 𝖤 onu� is𝜅-summableif it has the
fol-lowing property:
• For every family (𝑓𝑖| 𝑖 ∈ 𝐼)of𝖤-covering morphisms inu�, if 𝐼 is a 𝜅-small set, then the coproduct
∐𝑖∈𝐼𝑓𝑖 : ∐𝑖∈𝐼dom𝑓𝑖 → ∐𝑖∈𝐼 codom𝑓𝑖 is also an𝖤-covering morphism inu�.
Remark. The above is a condition on the whole class of 𝖤-covering morphisms inu�, not just the members of𝖤.
1.5.12 ¶ Let𝖤be a𝜅-summable unary coverage onu�. 1.5.12(a)
Joins of locally 1-generable closed subpresheaves
Lemma. Let𝐵 be a 𝖤-locally 1-generable presheaf on u�. The set of𝖤 -locally 1-generable 𝖤-closed subpresheaves of 𝐵 (partially ordered by inclusion) has𝜅-ary joins.
Proof. Let(𝑌 , 𝑏)be an𝖤-local generator of𝐵and letΦbe a𝜅-small set of elements of𝐵. We must show that the set of 𝖤-closed subpresheaves of𝐵 generated by the members of Φadmit a join in the set of𝖤-locally 1-generable 𝖤-closed subpresheaves of 𝐵. By corollary 1.4.7, we may assume that, for each(𝑋, 𝑎) ∈ Φ, there is some morphism𝑓 : 𝑋 → 𝑌 in u� such that 𝑏 ⋅ 𝑓 = 𝑎. Let𝑋 = ∐̄ (𝑋,𝑎)∈Φ𝑋, let 𝑓 : ̄̄ 𝑋 → 𝑌 be the induced morphism inu�, let ̄𝑎 = 𝑏⋅ ̄𝑓, and let𝐴̄be the𝖤-closed subpresheaf of 𝐵 𝖤-locally generated by( ̄𝑋, ̄𝑎). Clearly, for every (𝑋, 𝑎) ∈ Φ, we have 𝑎 ∈ ̄𝐴(𝑋). It remains to be shown that 𝐴̄is the smallest𝖤-locally 1-generable𝖤-closed subpresheaf with this property.
Let𝐴′be an𝖤-closed subpresheaf of𝐵 𝖤-locally generated by(𝑋′, 𝑎′) and suppose, for every(𝑋, 𝑎) ∈ Φ, we have𝑎 ∈ 𝐴′(𝑋). As before, we may assume that 𝑎′ = 𝑏 ⋅ 𝑓′ for some morphism 𝑓′ : 𝑋′ → 𝑌 inu�. Bylemma 1.4.7, for each(𝑋, 𝑎) ∈ Φ, we have an𝖤-covering morphism 𝑝′(𝑋,𝑎)inu�and a morphism𝑓(𝑋,𝑎)′ inu�such that𝑎′⋅𝑓(𝑋,𝑎)′ = 𝑎⋅𝑝′(𝑋,𝑎). Let
̃𝑋 = ∐(𝑋,𝑎)∈Φdom𝑓′(𝑋,𝑎)and let 𝑓′: ̃𝑋 → 𝑌 be the induced morphism inu�. Then𝑎′⋅ 𝑓′ = ̄𝑎⋅ ∐(𝑋,𝑎)𝑓(𝑋,𝑎), so ̄𝑎 ∈ 𝐴′(𝑋). Thus,𝐴 ⊆ 𝐴̄ ′, so𝐴̄
is the desired join. ■
1.5.12(b)
Pullbacks of joins of locally 1-generable closed subpresheaves
Lemma. Assuming(u�, 𝖤)satisfies the Shulman condition,𝜅-ary joins of 𝖤-locally 1-generable𝖤-closed subpresheaves of𝖤-locally 1-presentable presheaves onu� are preserved by pullback.
Proof. Let ℎ : 𝐴 → 𝐵 be a morphism of𝖤-locally 1-presentable pre-sheaves on u�. By corollary 1.4.16, we may choose 𝖤-local generators (𝑋, 𝑎)and(𝑌 , 𝑏)of𝐴and𝐵 (respectively) and a morphism𝑓 : 𝑋 → 𝑌 inu�such that𝑏⋅𝑓 = ℎ(𝑎)and the induced morphismh𝑋 → 𝐴×𝐵h𝑌 is 𝖤-locally surjective. Note that, for every𝖤-locally 1-generable subpresheaf 𝐵′ ⊆ 𝐵,lemma 1.4.13andtheorem 1.4.16imply thatℎ−1𝐵′is a𝖤-locally 1-generable subpresheaf of𝐴. On the other hand, bycorollary 1.4.7, there is a morphism𝑦 : 𝑌′ → 𝑌 inu� such that 𝐵′ is𝖤-locally generated by (𝑌′, 𝑏 ⋅ 𝑦). We may then choose an𝖤-weak pullback square inu� of the form below,
𝑋′ 𝑌′
𝑋 𝑌
𝑥
𝑓′ 𝑦 𝑓
and by the weak pullback pasting lemma (lemma a.2.19), (𝑋′, 𝑎 ⋅ 𝑥)is an𝖤-local generator of𝐴′= ℎ−1𝐵′.
Now, let 𝐼 be a 𝜅-small set, and for each 𝑖 ∈ 𝐼, let𝑥𝑖 : 𝑋𝑖′ → 𝑋, 𝑦𝑖 : 𝑌𝑖′→ 𝑌, and𝑓𝑖′: 𝑋𝑖′→ 𝑌𝑖′be morphisms inu� such that:
• 𝑓 ∘ 𝑥𝑖 = 𝑦𝑖∘ 𝑓𝑖′.
• (𝑋𝑖′, 𝑎 ⋅ 𝑥𝑖) is an𝖤-local generator of𝐴′𝑖 = ℎ−1𝐵′𝑖, where 𝐵𝑖′ is the 𝖤-closed subpresheaf of𝐵 𝖤-locally generated by(𝑌𝑖′, 𝑏 ⋅ 𝑦𝑖).
We will show that the join of{𝐴′𝑖| 𝑖 ∈ 𝐼} is the pullback of the join of {𝐵𝑖′| 𝑖 ∈ 𝐼}. Let 𝑋′ = ∐𝑖∈𝐼𝑋𝑖′, let𝑌′ = ∐𝑖∈𝐼𝑌𝑖′, let𝑥 : 𝑋′ → 𝑋, 𝑦 : 𝑌′ → 𝑌, 𝑓′ : 𝑋′ → 𝑌′ be the induced morphisms in u�, let 𝐴′be the𝖤-closed subpresheaf of𝐴 𝖤-locally generated by(𝑋′, 𝑎 ⋅ 𝑥), and let 𝐵′ be the 𝖤-closed subpresheaf of 𝐵 𝖤-locally generated by (𝑌′, 𝑏 ⋅ 𝑦). Bylemma 1.5.12(a), 𝐴′is the join of {𝐴′𝑖| 𝑖 ∈ 𝐼}and𝐵′is the join of {𝐵𝑖′| 𝑖 ∈ 𝐼}, so𝐴′ ⊆ ℎ−1𝐵′. It remains to be shown thatℎ−1𝐵′⊆ 𝐴′. 68
1.5. Strict coproducts As in the first paragraph, choose an 𝖤-weak pullback square in u� of the form below.
̃𝑋′ 𝑌′
𝑋 𝑌
̃𝑥 ̃𝑓′ 𝑦 𝑓
Note that( ̃𝑋′, 𝑎 ⋅ ̃𝑥)is an 𝖤-local generator ofℎ−1𝐵′. To complete the proof, we must verify that𝑎 ⋅ ̃𝑥 ∈ 𝐴″( ̃𝑋′). Sinceu� is a𝜅-ary extensive category, we may assume that 𝑋̃′ = ∐𝑖∈𝐼𝑋̃𝑖′ and𝑓′̃ = ∐𝑖∈𝐼𝑓𝑖′̃ where each𝑓𝑖′̃ is a morphism𝑋̃𝑖′→ 𝑌𝑖′inu�. Let ̃𝑥𝑖 : ̃𝑋𝑖′→ 𝑋be the composite of the coproduct injection𝑋̃𝑖′→ ̃𝑋and ̃𝑥 : ̃𝑋 → 𝑋. Then,
ℎ(𝑎 ⋅ ̃𝑥𝑖) = 𝑏 ⋅ (𝑓 ∘ ̃𝑥𝑖) = 𝑏 ⋅ (𝑦𝑖∘ ̃𝑓𝑖′) so we have𝑎 ⋅ ̃𝑥𝑖 ∈ 𝐴′𝑖( ̃𝑋𝑖′) = ℎ−1𝐵𝑖′( ̃𝑋𝑖′).
Now, (by the pullback pasting lemma) there exist𝖤-covering morph-isms ̃𝑥′𝑖 : ̃𝑋𝑖″ → ̃𝑋𝑖′and𝑥′𝑖 : ̃𝑋𝑖″ → 𝑋𝑖′inu� such that𝑓𝑖′∘ 𝑥′𝑖 = ̃𝑓𝑖′∘ ̃𝑥′𝑖. Let𝑋̃″ = ∐𝑖∈𝐼𝑋̃𝑖″, let ̃𝑥′ = ∐𝑖∈𝐼 ̃𝑥′𝑖, and let𝑥′ = ∐𝑖∈𝐼𝑥′𝑖. Since𝖤is 𝜅-summable,𝑥′ : ̃𝑋″ → 𝑋′is an𝖤-covering morphism. Moreover, by construction,𝑥∘𝑥′ = ̃𝑥∘ ̃𝑥′, so𝑎⋅( ̃𝑥 ∘ ̃𝑥′) ∈ 𝐴′( ̃𝑋″). But𝐴′is𝖤-closed, so𝑎 ⋅ ̃𝑥 ∈ 𝐴′( ̃𝑋′), as required. ■ 1.5.13 ¶ The following is a partial generalisation of Proposition 2.1 in [Gran and Vitale,1998]. (Note that an extensive category satisfying the Shul-man condition with respect to the trivial coverage is weakly lextensive, by Proposition 1.2 in op. cit.)
The exact completion is a pretopos
Proposition. If𝖤is a𝜅-summable unary coverage onu� and(u�, 𝖤) sat-isfies the Shulman condition, then:
(i) Ex(u�, 𝖤)is a𝜅-ary pretopos.
(ii) The insertion functor 𝜄 : u� → Ex(u�, 𝖤) preserves limits of finite diagrams and coproducts of𝜅-small families of objects, and sends𝖤 -covering morphisms inu� to effective epimorphisms inEx(u�, 𝖤). Proof. Byproposition 1.4.23,Ex(u�, 𝖤)is a regular category, and by lem-mas 1.5.12(a) and 1.5.12(b), every 𝜅-small set of subobjects of every object has an exact union. Thus,Ex(u�, 𝖤)is a𝜅-ary coherent category.
We already know that𝜄 : u� → Ex(u�, 𝖤)preserves limits of finite dia-grams and sends𝖤-covering morphisms inu�to effective epimorphisms in Ex(u�, 𝖤). On the other hand, bylemma 1.5.4, it also preserves coproducts of𝜅-small families of objects. We then applylemma 1.5.6(b)to deduce thatEx(u�, 𝖤)has strict coproducts of all𝜅-small families of objects. ■ 1.5.14 ¶ For each object 𝑌 in u�, let 𝖪(𝑌 ) be the set of 𝜅-small subsets Φ ⊆
obu�∕𝑌 such that(𝑓 | (𝑋, 𝑓) ∈ Φ)is a (strict) coproduct cocone inu�. Definition. The𝜅-ary extensive coverageonu�is the coverage𝖪defined above.
Proposition. 𝖪as defined above is a composition-closed coverage onu�.
Proof. Straightforward. ⧫
1.5.15
Recognition prin-ciple for sheaves on extensive sites
Lemma. Let𝐴be a presheaf onu�. The following are equivalent:
(i) 𝐴is a𝖪-sheaf onu�.
(ii) 𝐴 : u�op →Setsends𝜅-ary coproducts inu� to𝜅-ary products in Set.
Proof. (i)⇒(ii). Let𝑌 be an object inu�, letΦbe a𝜅-small set of objects inu�∕𝑌, and suppose𝑌 is a disjoint union ofΦ. For each(𝑋, 𝑓) ∈ Φ, let 𝑎(𝑋,𝑓) be an element of𝐴(𝑋). Since𝐴 is𝖪-separated, there is at most one element𝑎of𝐴(𝑌 ) such that𝑎 ⋅ 𝑓 = 𝑎(𝑋,𝑓) for all(𝑋, 𝑓) ∈ Φ. Let u� be the sieve on𝑌 generated byΦ. Since 𝐴 : u�op → Set preserves terminal objects, for every commutative square inu� of the form below,
𝑇 𝑋0
𝑋1 𝑌
𝑥0 𝑥1
𝑓0 𝑓1
if(𝑋0, 𝑓0) and(𝑋1, 𝑓1)are inΦ, then𝑎(𝑋0,𝑓0)⋅ 𝑥0 = 𝑎(𝑋1,𝑓1)⋅ 𝑥1. But 𝐴 satisfies the sheaf condition with respect tou�, so there is indeed an element𝑎of𝐴(𝑌 )such that𝑎 ⋅ 𝑓 = 𝑎(𝑋,𝑓)for all(𝑋, 𝑓) ∈ Φ. Hence, the canonical map𝐴(𝑌 ) → ∏(𝑋,𝑓)∈Φ𝐴(𝑋)is a bijection.
(ii)⇒(i). Straightforward. ■
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1.5. Strict coproducts
The Yoneda embedding into sheaves on an extensive site
Theorem.
(i) 𝖪is a subcanonical coverage onu�, i.e. every representable presheaf onu� is a𝖪-sheaf onu�.
(ii) The Yoneda embeddingh• :u� →Sh(u�, 𝖪)preserves𝜅-ary (strict) coproducts.
Proof. Applylemma 1.5.15. ■
1.5.16 Definition. A coverage 𝖩 on u� is 𝜅-ary superextensive if it has the following properties:
• For every object𝑋inu�, every element of𝖩(𝑋)is a𝜅-small sink on𝑋.
• For every object𝑋inu�,𝖪(𝑋) ⊆ 𝖩(𝑋), where𝖪is the𝜅-ary extensive coverage onu�.
Example. The 𝜅-ary extensive coverage on u� is 𝜅-ary superextensive, and by lemma 1.5.15and theorem 1.5.15, the𝜅-ary canonical coverage onu�is also𝜅-ary superextensive.
Recognition prin-ciple for covering sinks in superex-tensive coverages
Lemma. Let𝑋be an object inu�, letΦbe a𝜅-small set of objects inu�∕𝑋, let ̄𝑈 = ∐(𝑈,𝑥)∈Φ𝑈, and let ̄𝑥 : ̄𝑈 → 𝑋 be the induced morphism in u�. Assuming𝖩is a𝜅-ary superextensive coverage onu�, the following are equivalent:
(i) Φis a𝖩-covering sink on𝑋.
(ii) ̄𝑥 : ̄𝑈 → 𝑋is a𝖩-covering morphism inu�.
Proof. Straightforward. ⧫
The unary coverage asso-ciated with a superex-tensive coverage
Corollary. Let𝖩be a coverage on u� and𝖤 be the class of 𝖩-covering morphisms inu�. If𝖩is a𝜅-ary superextensive coverage onu�, then𝖤is a 𝜅-summable saturated unary coverage onu�.
Proof. First, we must verify that𝖤is a unary coverage. Let𝑓 : 𝑋 → 𝑌 be a morphism in u� and let 𝑞 : ̃𝑌 ↠ 𝑌 be a 𝖩-covering morphism in u�. Recallingparagraph a.2.13, we see that there is a𝜅-small𝖩-covering
sinkΦon𝑋such that↓(Φ) ⊆ 𝑓 ↓⟨𝑞⟩. Thus, bylemma 1.5.16, there is a commutative square inu� of the form below,
̃𝑋 ̃𝑌
𝑋 𝑌
𝑝 𝑞
𝑓
where𝑝 : ̃𝑋 ↠ 𝑋 is a𝖩-covering morphism inu�. It is straightforward to check that𝖤 is𝜅-summable. Finally, byproposition a.2.14, 𝖤is indeed
upward-closed and composition-closed. ■
1.5.17 ¶ Let𝖤be a𝜅-summable coverage onu�and, for each object𝑋 inu�, let 𝖩(𝑋)be the set of𝜅-small sinksΦon𝑋such that the induced morphism
∐(𝑈,𝑥)∈Φ𝑈 → 𝑋 inu� is a𝖤-covering morphism inu�.
Properties of the superex-tensive coverage generated by a summable unary coverage
Proposition.
(i) 𝖩is a𝜅-ary superextensive composition-closed coverage onu�. (ii) A morphism inu� is𝖤-covering if and only if it is𝖩-covering.
(iii) If𝖤is a subcanonical unary coverage onu�, then𝖩is a subcanonical coverage onu�.
(iv) Assuming 𝖪 is a 𝜅-ary superextensive coverage on u�, if every 𝖤 -covering morphism inu�is also𝖪-covering, then every𝖩-covering sink inu� is also𝖪-covering.
(v) Assuming𝐴 : u�op → Setsends 𝜅-ary coproducts in u� to 𝜅-ary products inSet,𝐴is an𝖤-sheaf onu� if and only if𝐴is a𝖩-sheaf on u�.
Proof. (i). It is clear that𝖩is a𝜅-ary superextensive coverage onu�, and 𝖩is composition-closed because𝖤is𝜅-summable.
(ii). By construction, every𝖤-covering morphism inu�is also𝖩-covering.
For the converse, suppose 𝑓 : 𝑋 → 𝑌 is a𝖩-covering morphism in u�. Recalling paragraph a.2.13, since 𝖩 is composition-closed, there is a 𝜅-small sinkΦon𝑋 such that the induced morphism∐(𝑈,𝑥)∈Φ𝑈 → 𝑋 → 𝑌 inu� is a𝖤-covering morphism inu�. But that implies𝑓 : 𝑋 → 𝑌 itself is a𝖤-covering morphism inu�, so we are done.
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1.5. Strict coproducts (iii) and (iv). Applylemma 1.5.16.
(v). Combinelemma 1.5.15and (iii). ■
Remark. Let 𝖪 be the𝜅-ary extensive coverage on u� and let𝖩′(𝑋) be defined as follows:
𝖩′(𝑋) = 𝖪(𝑋) ∪ {(𝑈, 𝑥) ∈obu�∕𝑋| 𝑥 ∈ 𝖤}
Then𝖩′is the smallest𝜅-ary superextensive coverage onu�containing𝖤.
In general,𝖩′ is strictly smaller than 𝖩, but the above proposition shows that they have the same covering sinks.
1.5.18 ¶ Let𝖩be a subcanonical𝜅-ary superextensive coverage onu� and let𝖪 be the𝜅-ary extensive coverage onu�.
Lemma. Letℎ : 𝐴 → 𝐵be a complemented monomorphism inSh(u�, 𝖩). If𝐵is a representable𝖩-sheaf, then𝐴is also a representable𝖩-sheaf.
Proof. By hypothesis, there is a (complemented) monomorphism ℎ′ : 𝐴′→ 𝐵inSh(u�, 𝖩)such that𝐵is the disjoint union of{(𝐴, ℎ), (𝐴′, ℎ′)}
inSh(u�, 𝖩). Thus, we have a unique morphism𝑝 : 𝐵 → 𝐵 ⨿ 𝐵such that the following diagram inSh(u�, 𝖩)commutes,
𝐴 𝐵 𝐴′
𝐵 𝐵 ⨿ 𝐵 𝐵
ℎ
ℎ
𝑝 ℎ′
ℎ′
where the bottom row is a coproduct diagram. By theorem 1.5.15, the Yoneda embeddingu� →Sh(u�, 𝖩)preserves𝜅-ary coproducts, so𝐵 ⨿ 𝐵 is a representable 𝖩-sheaf on u�. But coproduct injections are quadrable inu� and the Yoneda embeddingu� →Sh(u�, 𝖩)preserves pullbacks, so it follows that both𝐴and𝐴′are indeed representable𝖩-sheaves onu�. ■
Proposition. The inclusion Sh(u�, 𝖩) ↪ Sh(u�, 𝖪) preserves 𝜅-ary co-products.
Proof. Let(𝐴𝑖| 𝑖 ∈ 𝐼)be a family of 𝖩-sheavesu� where𝐼 is a 𝜅-small set and let𝐴be their coproduct inSh(u�, 𝖩). It suffices to show that the co-product cocone is jointly𝖪-locally surjective. Moreover, since coproduct cocones are preserved by pullback, we may assume that𝐴is a represent-able sheaf on u�. But thenlemma 1.5.18says that each𝐴𝑖 is also repre-sentable, so the coproduct cocone in question is indeed jointly𝖪-locally
surjective. ■
Remark. The above result is optimal in the following sense: if 𝜆 is a regular cardinal> 𝜅such thatu� is a non-trivial𝜆-ary extensive category and𝖫is the𝜆-ary extensive coverage onu�, then the inclusionSh(u�, 𝖫) ↪ Sh(u�, 𝖪)preserves𝜅-ary coproducts but not𝜆-ary coproducts.
1.5.19 ¶ In general, an exact category may not have coequalisers of all parallel pairs. However:
Coequalisers in pretoposes
Proposition. Letu�be a𝜅-ary pretopos and let𝖪be the𝜅-ary canonical coverage onu�. Assuming𝜅 > ℵ0:
(i) u� has coequalisers of all parallel pairs.
(ii) Ifu� is a𝜅-ary pretopos and𝐹 :u� →u� is a functor that preserves limits of finite diagrams,𝜅-ary coproducts, and exact quotients, then 𝐹 :u� →u� preserves colimits of𝜅-small diagrams.
(iii) In particular, the Yoneda embeddingu� → Sh(u�, 𝖪)preserves co-limits of𝜅-small diagrams.
Proof. (i) and (ii). See the proof of Lemma 1.4.19 in [Johnstone, 2002, Part A].
(iii). Since𝖪is a𝜅-ary superextensive coverage onu�, the Yoneda embed-dingu� →Sh(u�, 𝖪)preserves𝜅-ary coproducts, bytheorem 1.5.15. The Yoneda embedding also preserves exact quotients, bylemma a.3.10. □
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1.5. Strict coproducts 1.5.20 ¶ Let 𝜆 be a regular cardinal ≥ 𝜅. We define a category Fam𝜅𝜆(u�) as
follows:
• The objects are as inFam𝜆(u�).
• The morphisms𝑋 → 𝑌 are the morphisms
𝑖∈∐idx𝑋𝑋(𝑖) → ∐𝑗∈
idx𝑌𝑌 (𝑗)
inSh(u�, 𝖪)where𝖪is the𝜅-ary extensive coverage onu�.
• Composition and identities are inherited fromSh(u�, 𝖪).
We also define a functor𝛾 :u� →Fam𝜅𝜆(u�)as follows:
• For each object𝑋 inu�, we haveidx𝛾(𝑋) = {∗}and𝛾(𝑋)(∗) = 𝑋.
• For each morphism𝑓 : 𝑋 → 𝑌 inu�, we have𝛾(𝑓) =h𝑓.
Proposition. Fam𝜅𝜆(u�)is a𝜆-ary extensive category. Furthermore, 𝛾 : u� →Fam𝜅𝜆(u�)is fully faithful and preserves𝜅-ary (strict) coproducts.
Proof. It is clear thatFam𝜅𝜆(u�)has𝜆-ary coproducts. On the other hand, coproducts in Psh(u�) are always strict, so proposition a.2.18 and the-orem a.3.9 imply that coproducts inSh(u�, 𝖪)are also strict. Moreover, proposition 1.5.14andlemma a.3.11imply that any morphism𝑋 → 𝑌 in Fam𝜅𝜆(u�)must factor through the inclusion𝑌′ → 𝑌 for some𝑌′where
|idx𝑌′| ≤ max{|idx𝑋|, 𝜅}, so coproduct injections (of 𝜅-ary coprod-ucts) inFam𝜅𝜆(u�)are indeed quadrable. The remainder of the claim is a
consequence oftheorem 1.5.15. ■
The relative coproduct completion
Theorem. Letu�be a𝜆-ary extensive category and let𝐹 : u� →u� be a functor that preserves𝜅-ary coproducts.
(i) There exist a functor ̄𝐹 : Fam𝜅𝜆(u�) → u� that preserves𝜆-ary co-products and an isomorphism𝜂 : 𝐹 ⇒ ̄𝐹𝛾of functorsu� →u�.
(ii) Moreover, any such( ̄𝐹, 𝜂)is a left Kan extension of 𝐹 : u� → u�
along𝛾 :u� →Fam𝜅𝜆(u�).
Proof. Let𝖫be the𝜆-ary extensive topology onu�. Note that the functor 𝐹∗ :Psh(u�) →Psh(u�)sends𝖫-sheaves onu�to𝖪-sheaves onu�: this is
a consequence of lemma 1.5.15. Let𝐹! : Sh(u�, 𝖪) → Sh(u�, 𝖫) be (the functor part of) a left Kan extension of h𝐹 : u� → Sh(u�, 𝖫) along the Yoneda embedding h• : u� → Sh(u�, 𝖪). The unit of this Kan extension is automatically an isomorphism, because the Yoneda embedding is fully faithful. Moreover, by the earlier observation,𝐹! :Sh(u�, 𝖪) →Sh(u�, 𝖫) is a left adjoint of the functor𝐹∗ : Sh(u�, 𝖫) →Sh(u�, 𝖪), so it preserves all coproducts. Since every object in Fam𝜅𝜆(u�) is a 𝜆-ary coproduct of objects in the image of𝛾 :u� →Fam𝜅𝜆(u�), this yields the desired left Kan
extension of𝐹 :u� →u�. ■
1.5.21
Limits of finite diagrams in the relative coprod-uct completion
Proposition. With notation as inparagraph1.5.20:
(i) Ifu�has finitary products, thenFam𝜅𝜆(u�)also has finitary products.
(ii) Ifu� has equalisers, thenFam𝜅𝜆(u�)also has equalisers.
(iii) Ifu� has pullbacks, thenFam𝜅𝜆(u�)also has pullbacks.
Proof. (i). This is a consequence of the fact that binary products distribute over (possibly infinitary) coproducts inSh(u�, 𝖪).
(ii). Consider an equaliser diagram inSh(u�, 𝖪):
𝐴′ 𝐴 ℎ0 𝐵
ℎ1
Suppose both𝐴and𝐵are coproducts (inSh(u�, 𝖪)) of𝜆-small families of representable𝖪-sheaves onu�. We must show that𝐴′has the same prop-erty. It suffices to show that𝐴′is representable when𝐴is representable:
the general case follows by taking coproducts. But if𝐴is representable, thenproposition 1.5.20andlemma a.3.11imply that there is a represent-able𝖪-subsheaf𝐵′ ⊆ 𝐵 such that both ℎ0, ℎ1 : 𝐴 → 𝐵 factor through the inclusion𝐵′↪ 𝐵, so𝐴′is indeed representable.
(iii). A similar argument works. ■
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