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3.4. Topological spaces (vi) (u�, 𝖤)satisfies the Shulman condition.^[1]

Proof. Straightforward. ⧫

Remark. In particular, every local homeomorphism inu� in the sense of definition 2.2.12is a member ofu�, so there is no danger of confusion in using the phrase ‘local homeomorphism inu�’.

3.4.3 ※For the remainder of this section:

• u� = Ex(u�, 𝖤).

• u�is the class of morphisms inu�corresponding to morphisms inPsh(u�) that are𝖩-locally ofu�-type.

• 𝖪is the𝜅-ary canonical coverage onu�.

• u�̂is the class ofu�-perfect morphisms inu�.

Furthermore, by abuse of notation, we will identifyu� with the image of the insertionu� →u�.

3.4.4 Proposition.

(i) (u�, ̂u�)is a𝜅-ary gros pretopos.

(ii) Moreover,(u�, ̂u�, 𝖪)satisfies the descent axiom.

(iii) A morphism inu� is a member ofu�if and only if it is a member of

̂u�.

Proof. This is a special case ofproposition 2.3.2. ■ 3.4.5 ¶ Consider the Yoneda representation h_• : Top → Psh(u�). Since u�

has pullbacks and the inclusion u� ↪ Toppreserves them,lemma a.2.6 implies that, for every topological space𝑋,h_𝑋is a𝖩-sheaf onu�. Thus, by proposition a.1.4, for every𝖩-sheaf𝐴onu�, there is a topological space

|𝐴| and a morphism𝜂_𝐴 : 𝐴 → h_|𝐴| inSh(u�, 𝖩) such that the following map is a bijection for every topological space𝑌:

Top(|𝐴|, 𝑌 ) →Hom_{Sh(u�,𝖩)}(𝐴,h_𝑌)

[1] Note that a morphism inu�is𝖩-covering if and only if it is𝖤-covering.

𝑓 ↦h_𝑓 ∘ 𝜂_𝐴

Indeed, we may take|𝐴| =lim−−→(𝑋,𝑎):El(𝐴)𝑋. This yields an adjunction:

Top Sh(u�, 𝖩)

h_•

⊥

|−|

It is clear (by construction) that the counit 𝜀_𝑋 : |h_{𝑋| → 𝑋} is a homeo-morphism for every object𝑋inu�. We would like to know if this happens for topological spaces that are not necessarily inu�.

3.4.5(a) Lemma. Let𝑋 be a topological space. The following are equivalent:

(i) The counit𝜀_𝑋 : |h_{𝑋| → 𝑋} is a homeomorphism.

(ii) For every topological space𝑌, the following is a bijection:

h_• :Top(𝑋, 𝑌 ) →Hom_{Sh(u�,𝖩)}(h_𝑋,h_𝑌)

Proof. Straightforward. ⧫

3.4.5(b) Lemma. The functor|−| :Sh(u�, 𝖩) →Toppreserves monomorphisms.

Proof. LetΓ :Sh(u�, 𝖩) → Setbe the evident functor defined on objects by𝐴 ↦ 𝐴(1). It is clear that1is a𝖩-local object inu�, so bylemma a.3.12, Γ : Sh(u�, 𝖩) → Set preserves colimits. On the other hand, proposi-tion a.1.4implies thatΓ :Sh(u�, 𝖩) →Setis isomorphic to the composite of|−| :Sh(u�, 𝖩) →Topand the forgetful functorTop→Set. Since the forgetful functorTop→Setis faithful, it follows that|−| : Sh(u�, 𝖩) →

Toppreserves monomorphisms. ■

3.4.5(c) Lemma. Let𝑓 : 𝑋 ↠ 𝑌 be a surjective local homeomorphism of topo-logical spaces and let (𝑅, 𝑑⁰, 𝑑₁) be the kernel pair of 𝑓 : 𝑋 ↠ 𝑌 in Top.

(i) The following is an exact fork inSh(u�, 𝖩): h_𝑅 ^𝑑0∘− h_𝑋 h_𝑌

𝑑₁∘−

𝑓∘−

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3.4. Topological spaces (ii) If both𝜀_𝑅 : |h_{𝑅| → 𝑅}^and𝜀_𝑋 : |h_{𝑋| → 𝑋} are homeomorphisms,

then𝜀_𝑌 : |h_{𝑌| → 𝑌} is also a homeomorphism.

Proof. (i). It is not hard to verify that h_𝑓 : h_𝑋 → h_𝑌 is a𝖩-locally sur-jective morphism inPsh(u�). Thus, bylemma a.3.10, we have the desired exact fork.

(ii). |−| : Sh(u�, 𝖩) → Toppreserves coequalisers, and𝑓 : 𝑋 ↠ 𝑌 is an effective epimorphism in Top, so it follows that 𝜀_𝑌 : |h_{𝑌| → 𝑌} is a homeomorphism if both 𝜀_𝑅 : |h_{𝑅| → 𝑅}and 𝜀_𝑋 : |h_{𝑋| → |}h_𝑌|are

homeomorphisms. ■

3.4.5(d) Lemma. Let(𝑋𝑖| 𝑖 ∈ 𝐼)be a family of topological spaces where 𝐼 is a 𝜅-small set and let𝑋 = ∐_𝑖∈𝐼𝑋_𝑖.

(i) h_𝑋is a coproduct of(h_{𝑋𝑖| 𝑖 ∈ 𝐼)}ⁱⁿSh(u�, 𝖩)(with the evident co-product injections).

(ii) If each𝜀_𝑋𝑖 : |h_{𝑋𝑖| → 𝑋𝑖}is a homeomorphism, then𝜀_𝑋 : |h_{𝑋| → 𝑋} is also a homeomorphism.

Proof. (i). Usinglemma 1.5.4, it is not hard to see that the Yoneda rep-resentationh_• :Top→Sh(u�, 𝖩)preserves𝜅-ary coproducts.

(ii). On the other hand,|−| : Sh(u�, 𝖩)also preserves (𝜅-ary) coproducts.

The claim follows. ■

3.4.5(e) Lemma. Let𝑋 be an object inu� and let𝑈 be an open subspace of𝑋. (i) h_𝑈 → h_𝑋 is a monomorphism in Psh(u�) that is 𝖩-semilocally of

u�-type.

(ii) 𝜀_𝑈 : |h_{𝑈| → 𝑈} is a homeomorphism inu�.

Proof. (i). By hypothesis, there is a𝜅-small setΦof open subspaces of 𝑉 such that:

• For each𝑉 ∈ Φ,𝑉 is homeomorphic to an object inu�.

• 𝑈 = ⋃_{𝑉 ∈Φ}𝑉.

It is clear that h_𝑈 → h_𝑋 is a monomorphism in Psh(u�), and it follows that h_𝑈 → h_𝑋 is 𝖩-semilocally of u�-type. Let ̄𝑉 = ∐_{𝑉 ∈Φ}𝑉 and let 𝑝 : ̄𝑉 ↠ 𝑈 be the evident projection. Clearly,𝑝 : ̄𝑉 ↠ 𝑈 is a surjective local homeomorphism. Let(𝑅, 𝑑⁰, 𝑑₁)be the kernel pair of𝑝 : ̄𝑉 ↠ 𝑉. Then𝑅 ≅ ∐_𝑉0∈Φ∐_𝑉1∈Φ𝑉₀∩𝑉₁, and each𝑉₀∩𝑉₁is homeomorphic to an object inu�, so bylemma 3.4.5(d), both𝜀_𝑅: |h_{𝑅| → 𝑅}and𝜀 _̄𝑉 : |h _{̄𝑉| → ̄𝑉} are homeomorphisms. Hence, bylemma 3.4.5(c),𝜀_𝑈 : |h_{𝑈| → 𝑈} is also

a homeomorphism. ■

3.4.6 ¶ In view of the discussion above, we make the following definition.

Definition. A topological space𝑋 isofu�-typeif there is a𝜅-small set Φof open subspaces of𝑋with the following properties:

• For every𝑈 ∈ Φ,𝑈 is homeomorphic to an object inu�.

• 𝑋 = ⋃_𝑈∈Φ𝑈.

We write ℳfor the metacategory of topological spaces of u�-type (and continuous maps).

Proposition.

(i) ℳis closed inTopunder𝜅-ary disjoint union.

(ii) Given an object𝑋inℳ, if𝑈 is an open subspace of𝑋, then𝑈 is also an object inℳ.

(iii) For every object𝑌 inℳ, there exist an object 𝑋 in u� and a sur-jective local homeomorphism 𝑓 : 𝑋 ↠ 𝑌 such that 𝑋 ×_𝑌 𝑋 is homeomorphic to a𝜅-ary disjoint union of open subspaces of𝑋 and h_𝑓 :h_𝑋 →h_𝑌 is a morphism inPsh(u�)that is𝖩-semilocally ofu�-type.

(iv) For every object 𝑌 in ℳ, the counit 𝜀_𝑌 : |h_{𝑌| → 𝑌} is a homeo-morphism.

(v) The Yoneda representation ℳ → Sh(u�, 𝖩) is fully faithful, pre-serves𝜅-ary coproducts, and sends surjective local homeomorphisms inℳto effective epimorphisms inSh(u�, 𝖩).

Proof. (i) and (ii). Straightforward.

170

3.4. Topological spaces (iii). Let 𝑌 be an object inℳ. By definition, there is a𝜅-small setΨof open subspaces of𝑌 with the following properties:

• For every𝑉 ∈ Ψ,𝑉 is homeomorphic to an object inu�.

• 𝑌 = ⋃_{𝑉 ∈Ψ}𝑉.

Since u� is closed under 𝜅-ary disjoint union, ∐_{𝑉 ∈Ψ}𝑉 is also homeo-morphic to an object in u�, say 𝑋. There is an evident surjective local homeomorphism 𝑓 : 𝑋 ↠ 𝑌, and it is clear that𝑋 ×_𝑌 𝑋 is homeo-morphic to a𝜅-ary disjoint union of open subspaces of𝑋. Moreover, by proposition 1.2.13andlemma 3.4.5(e),h_𝑓 :h_𝑋 →h_𝑌 is𝖩-semilocally of u�-type, as claimed.

(iv). Apply lemmas3.4.5(c)and3.4.5(d)to (ii) and (iii).

(v). Bylemma 3.4.5(a)and (iv), the Yoneda representationℳ→Sh(u�, 𝖩) is fully faithful. We already know that the Yoneda representationTop→ Sh(u�, 𝖩) preserves 𝜅-ary coproducts and sends surjective local homeo-morphisms inTopto effective epimorphisms inSh(u�, 𝖩), so we are done.

■ 3.4.7 ¶ Bytheorem 2.1.14, the Yoneda representationu� → Sh(u�, 𝖩)is fully faithful and preserves limits of finite diagrams, 𝜅-ary coproducts, and exact quotients. Moreover, by lemma 2.1.16, a 𝖩-sheaf on u� is in the essential image of the Yoneda representation if and only if it is𝖩-locally 𝜅-presentable.

Lemma. If𝑥 : 𝑈 → 𝑋 is an open embedding inu� and𝑋is an object in u�, then|h_{𝑥| : |}h_{𝑈| → |}h_𝑋|is an open embedding of topological spaces.

Proof. Since h_𝑥 : h_𝑈 → h_𝑋 is a monomorphism in Psh(u�) that is 𝖩 -semilocally ofu�-type, there is a𝜅-small setΦof objects inu�_∕𝑈 with the following properties:

• For every (𝑉 , 𝑢) ∈ Φ, 𝑉 is an object inu� and𝑥 ∘ 𝑢 : 𝑉 → 𝑋 is an open embedding of topological spaces.

• The induced morphism𝑝 : ∐_{(𝑉 ,𝑢)∈Φ}𝑉 → 𝑈 inu� is an effective epi-morphism.

Thus,|h_{𝑝| : |}h_{∐(𝑉 ,𝑢)∈Φ𝑉| ↠ |𝑈|}is an effective epimorphism inTopand the composite |h_{𝑥| ∘ |}h_{𝑝| : |}h_{∐(𝑉 ,𝑢)∈Φ𝑉| → |𝑋|} is a local homeomorph-ism of topological spaces. On the other hand, bylemma 3.4.5(b), |h_{𝑥| :}

|h_{𝑈| → |}h_𝑋|is an injective continuous map. Thus,|h_{𝑥| : |}h_{𝑈| → |}h_𝑋|is indeed an open embedding of topological spaces. ■ Theorem. Letu�̄ be the class of local homeomorphisms inu�.

(i) If𝑋 is an object inℳ, then there is a(u�, ̄u�)-extent 𝐴inu� such thath_𝑋 ≅h_𝐴ⁱⁿSh(u�, 𝖩).

(ii) If 𝐴 is a (u�, ̄u�)-extent in u�, then |h_𝐴| is a topological space of u�-type.

(iii) The functor|h_{•| :} Xt(u�, ̄u�) → ℳis fully faithful and essentially surjective on objects.

Proof. (i). First, consider a subspace 𝑈 of an object 𝑋 inu�. Recalling the proof oflemma 3.4.5(e), we see that there is an open subobject𝐴of 𝑋 inu� such thath_𝑈 ≅ h_𝐴. Thus, byproposition 3.4.6, for every object 𝑋 in ℳ, there is an object 𝐴 in u� such that h_𝑋 ≅ h_𝐴. Moreover, by tracing the proof of that proposition, it is straightforward to verify that𝐴 is a(u�, ̄u�)-extent inu�.

(ii). In view of lemma 3.4.7, a similar argument shows that |h_𝐴| is a topological space ofu�-type if𝐴is a(u�, ̄u�)-extent inu�.

(iii). Hence, bylemma 3.4.5(a)andproposition 3.4.6,|h_{•| :}Xt(u�, ̄u�) → ℳis fully faithful and essentially surjective on objects. ■ 3.4.8 ¶ Byproposition 2.2.12, we haveu�̄ ⊆ ̂u�, soXt(u�, ̄u�) ⊆Xt(u�, ̂u�). We

will now see an explicit example where these inclusions are strict.

Example. Let u� be the category of topological spaces𝑋 such that the set of points of𝑋 is hereditarily𝜅-small. It is straightforward to check that the hypotheses ofproposition 2.3.14(b)are satisfied, so each(u�, ̄u�) -extent inu� is isomorphic to an object inu�.

On the other hand, consider the unit circle in the complex plane:

𝑆¹= {𝑧 ∈ ℂ | |𝑧| = 1}

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3.4. Topological spaces Let𝑅 = ℤ × 𝑆¹and let𝑑₀, 𝑑₁ : 𝑅 → 𝑆¹be defined as follows:

𝑑₀(𝑛, 𝑧) =exp(𝑖𝑛)𝑧 𝑑₁(𝑛, 𝑧) = 𝑧

Then𝑑₀, 𝑑₁ : 𝑅 → 𝑆¹are local homeomorphisms of topological spaces.

Moreover, (𝑅, 𝑑⁰, 𝑑₁) is an equivalence relation on 𝑆¹, but there does not exist a local homeomorphism 𝑓 : 𝑆¹ → 𝑌 such that 𝑓 ∘ 𝑑₀ = 𝑓 ∘ 𝑑₁: indeed, (𝑅, 𝑑0, 𝑑₁) is not a tractable equivalence relation. (Recall lemma 2.2.14(a).) Nonetheless, assuming𝑅and𝑆¹are objects in u�, an exact quotient of(𝑅, 𝑑0, 𝑑₁)exists inu�, andlemma 2.2.8(b)says that it is a(u�, ̂u�)-extent; but by the preceding discussion, it isnota(u�, ̄u�)-extent.

In this context, it is also worth noting that a(u�, ̄u�)-extent is the same thing as a u�̂-localic (u�, ̂u�)-extent. In other words, a(u�, ̂u�)-extent is a (u�, ̄u�)-extent precisely when it has enough open subobjects.

3.4.9 Example. Letu�be the category of Hausdorff spaces𝑋such that the set of points of𝑋is hereditarily𝜅-small. Then, bytheorem 3.4.7, the essential image of |−| : Xt(u�, ̄u�) → Top is spanned by the locally Hausdorff spaces𝑋 such that the set of points of𝑋 is𝜅-small. In particular, since there are locally Hausdorff spaces that are not Hausdorff spaces, assuming 𝜅 > ℵ₀, we have a(u�, ̄u�)-extent that is not isomorphic to any object in u�.