• No results found

Notes on the Heuristic

Q 7 Organisations which do not have a diverse range of funding streams are at

8 DEVELOPMENT OF AN HEURISTIC

8.1 Notes on the Heuristic

Chapter III

Specificities

β€’ uοΏ½is closed under pullback.

β€’ For each object𝑋 inuοΏ½:

– 𝖩𝖿(𝑋)is the set of all finite and jointly surjective sinks on𝑋.

– π–©π–Ώπ—Š(𝑋) is the set of all finite sinks Ξ¦on 𝑋 such that the induced map∐(π‘ˆ,π‘₯)βˆˆΞ¦π‘ˆ β†’ 𝑋 is a universal topological quotient.

3.1.3 ΒΆ The following terminology is non-standard.

Definition. A continuous map𝑓 : 𝑋 β†’ π‘Œ issemiproper if it has the following property:

β€’ For every pullback square inTopof the form below,

𝑋′ 𝑋

π‘Œβ€² π‘Œ

𝑓′ 𝑓

ifπ‘Œβ€²is compact, then𝑋′is also compact.

Remark. Thus, from the relative point of view, a semiproper map of topological spaces is a continuous family of compact spaces.

Example. Every continuous map from a compact space to a Hausdorff space is semiproper. Indeed, given a pullback square in Top as in the definition, if𝑋 is compact andπ‘Œ is Hausdorff, then the comparison map 𝑋′ β†’ π‘Œβ€²Γ— 𝑋is a closed embedding, so𝑋′is compact whenπ‘Œβ€²is.

Properties of semiproper maps

Proposition.

(i) Every closed embedding of topological spaces is semiproper.

(ii) For every topological space𝑋, the codiagonalβˆ‡π‘‹ : 𝑋 β¨Ώ 𝑋 β†’ 𝑋 is semiproper.

(iii) The class of semiproper maps of topological spaces is a quadrable class of morphisms inTop.

(iv) The class of semiproper maps of topological spaces is closed under composition.

(v) The class of semiproper maps of topological spaces is closed under (possibly infinitary) coproduct inTop.

146

3.1. Compactness (vi) Given a surjective continuous map𝑓 : 𝑋 β†  π‘Œ and a continuous map𝑔 : π‘Œ β†’ 𝑍, if𝑔 ∘ 𝑓 : 𝑋 β†’ 𝑍 is semiproper, then𝑔 : π‘Œ β†’ 𝑍 is also semiproper.

Proof. Straightforward. ⧫

Corollary. Letβ„±π—Œπ—‰be the class of semiproper maps in uοΏ½. Then every morphism inuοΏ½ that is of β„±π—Œπ—‰-type𝖩𝖿-semilocally on the domain is semi-proper.

Proof. Applyproposition 3.1.3. β– 

3.1.4 ΒΆ We will see that the following is a specialisation of the notion of semi-proper map.

Definition. A continuous map𝑓 : 𝑋 β†’ π‘Œ isproper if it has the fol-lowing property:

β€’ For every pullback square inTopof the form below,

𝑋′ 𝑋

π‘Œβ€² π‘Œ

𝑓′ 𝑓

the map𝑓′ : 𝑋′ β†’ π‘Œβ€²is closed, i.e. the image of every closed sub-space of𝑋′is a closed subspace ofπ‘Œβ€².

Example. If𝑋is a compact topological space, then the unique map𝑋 β†’ 1is proper: this is the precisely the statement of the tube lemma.

Properties of proper maps

Proposition.

(i) An injective continuous map is proper if and only if it is a closed embedding.

(ii) For every topological space𝑋, the codiagonalβˆ‡π‘‹ : 𝑋 β¨Ώ 𝑋 β†’ 𝑋 is proper.

(iii) The class of proper maps of topological spaces is a quadrable class of morphisms inTop.

(iv) The class of proper maps of topological spaces is closed under com-position.

(v) The class of proper maps of topological spaces is closed under (pos-sibly infinitary) coproduct inTop.

(vi) Given a surjective continuous map𝑓 : 𝑋 β†  π‘Œ and a continuous map𝑔 : π‘Œ β†’ 𝑍, if𝑔 ∘ 𝑓 : 𝑋 β†’ 𝑍 is proper, then𝑔 : π‘Œ β†’ 𝑍 is also proper.

(vii) Given a pullback square inTopof the form below,

̃𝑋 𝑋

Μƒπ‘Œ π‘Œ

̃𝑓 𝑓

where Μƒπ‘Œ β†  π‘Œ is a universal topological quotient, if 𝑓 : ΜƒΜƒ 𝑋 β†’ Μƒπ‘Œ is proper, then𝑓 : 𝑋 β†’ π‘Œ is also proper.

Proof. Straightforward. ⧫

Corollary. Letℱ𝗉be the class of proper maps inuοΏ½. Then every morph-ism inuοΏ½ that isπ–©π–Ώπ—Š-semilocally ofℱ𝗉-type is proper.

Proof. Applyproposition 3.1.3. β– 

Remark. In the language of Β§2.2, what we have shown is that(uοΏ½,ℱ𝗉, π–©π–Ώπ—Š) is a finitary (i.e. β„΅0-ary) extensive regulated ecumene that satisfies the descent axiom and in which every eunoic morphism is genial.

3.1.5 ΒΆ Properness is closely related to compactness. For instance, suppose 𝑋 is a topological space such that the unique map𝑋 β†’ 1is proper. Let 𝑆 = {1 βˆ’ 𝑛+11 | 𝑛 ∈ β„•} βˆͺ {1} βŠ† ℝ and let (π‘₯𝑛| 𝑛 ∈ β„•) be a sequence of points of𝑋. Consider𝑇 = {(1 βˆ’ 𝑛+11 , π‘₯𝑛) | 𝑛 ∈ β„•} βŠ† 𝑆 Γ— 𝑋. The closure of𝑇 is ̄𝑇 = 𝑇 βˆͺ{1}×𝐴, where𝐴is the set of accumulation points of (π‘₯𝑛| 𝑛 ∈ β„•). Since ̄𝑇 is a closed subspace of 𝑆 Γ— 𝑋, its image is a closed subspace of𝑆. In particular,1is in the image of ̄𝑇, i.e.𝐴contains a point. Thus, every sequence in𝑋 contains a convergent subsequence, 148

3.1. Compactness i.e.𝑋 is sequentially compact. A similar argument using nets instead of sequences can be used to show that𝑋 is compact.

Much more generally, we have the following result.

Recognition principle for proper maps

Theorem. Let 𝑓 : 𝑋 β†’ π‘Œ be a continuous map. The following are equivalent:

(i) The map𝑓 : 𝑋 β†’ π‘Œ is proper.

(ii) For every topological space𝑇, the mapid𝑇 Γ— 𝑓 : 𝑇 Γ— 𝑋 β†’ 𝑇 Γ— π‘Œ is closed.

(iii) The map 𝑓 : 𝑋 β†’ π‘Œ is closed and, for every 𝑦 ∈ π‘Œ, π‘“βˆ’1{𝑦}is compact.

(iv) The map𝑓 : 𝑋 β†’ π‘Œ is closed and, for every subspaceπ‘Œβ€² βŠ† π‘Œ, if π‘Œβ€²is compact, thenπ‘“βˆ’1π‘Œβ€²is also compact.

(v) The map𝑓 : 𝑋 β†’ π‘Œ is closed and semiproper.

Proof. (i)β‡’(ii). Immediate.

(ii)β‡’(iii), (iii)β‡’(i). Seetag005Rin [Stacks].

(i)β‡’(v). Consider a pullback square inTopof the form below:

𝑋′ 𝑋

π‘Œβ€² π‘Œ

𝑓′ 𝑓

Suppose𝑓 : 𝑋 β†’ π‘Œ is proper andπ‘Œβ€² is compact. We must show that 𝑋′is compact. Then, byproposition 3.1.4,𝑓′: 𝑋′→ π‘Œβ€²is also proper.

Since the unique mapπ‘Œβ€² β†’ 1 is proper (by the tube lemma), it follows that 𝑋′ β†’ 1 is also proper. But we know (i) β‡’ (iii), so 𝑋′ is indeed compact.

(v)β‡’(iv), (iv)β‡’(iii). Immediate. β–‘

Example. If𝑋is a compact topological space andπ‘Œ is a Hausdorff space, then every continuous map𝑋 β†’ π‘Œ is proper: in view ofproposition 3.1.4 andtheorem 3.1.5, this is a special case oflemma 1.1.9.

3.1.6

When semiproper implies proper

Lemma. Let𝑓 : 𝑋 β†’ π‘Œ be a continuous map. Assumingπ‘Œ is a compactly generated Hausdorff space, the following are equivalent:

(i) The map𝑓 : 𝑋 β†’ π‘Œ is proper.

(ii) The map𝑓 : 𝑋 β†’ π‘Œ is semiproper.

(iii) For every subspace π‘Œβ€² βŠ† π‘Œ, if π‘Œβ€²is compact, then π‘“βˆ’1π‘Œβ€²is also compact.

Proof. (i)β‡’(ii). Seetheorem 3.1.5.

(ii)β‡’(iii). Immediate.

(iii)β‡’(i). In view of the theorem, it is enough to check that𝑓 : 𝑋 β†’ π‘Œ is a closed map.

Let 𝑋′be a closed subspace of 𝑋 and let π‘Œβ€² be its image inπ‘Œ. We wish to show that π‘Œβ€² is a closed subspace ofπ‘Œ. Since π‘Œ is compactly generated, it is enough to show thatπ‘Œβ€²βˆ© 𝑉 is a closed subspace of𝑉 for all compact subspaces𝑉 βŠ† π‘Œ.

Let𝑉 be a compact subspace ofπ‘Œ. Thenπ‘“βˆ’1𝑉 is a compact subspace of𝑋. Sinceπ‘‹β€²βˆ© π‘“βˆ’1𝑉 is a closed subspace ofπ‘“βˆ’1𝑉, it is compact. The image ofπ‘‹β€²βˆ© π‘“βˆ’1𝑉 in𝑉 isπ‘Œβ€²βˆ© 𝑉, and since𝑉 is Hausdorff, it follows thatπ‘Œβ€²βˆ© 𝑉 is indeed a closed subspace of𝑉. β–  3.1.7 Definition. A continuous map 𝑓 : 𝑋 β†’ π‘Œ is perfect if it has the

following properties:

β€’ 𝑓 : 𝑋 β†’ π‘Œ is proper.

β€’ 𝑓 : 𝑋 β†’ π‘Œ is separated, i.e. the relative diagonalΔ𝑓 : 𝑋 β†’ 𝑋 Γ—π‘Œ 𝑋 is a closed embedding.

Example. For a topological space𝑋, the unique map𝑋 β†’ 1 is perfect if and only if𝑋 is a compact Hausdorff space.

Properties of perfect maps

Proposition.

(i) An injective continuous map is perfect if and only if it is a closed embedding.

150

3.1. Compactness (ii) For every topological space𝑋, the codiagonalβˆ‡π‘‹ : 𝑋 β¨Ώ 𝑋 β†’ 𝑋

is perfect.

(iii) The class of perfect maps of topological spaces is a quadrable class of morphisms inTop.

(iv) The class of perfect maps of topological spaces is closed under composition.

(v) The class of perfect maps of topological spaces is closed under (possibly infinitary) coproduct inTop.

(vi) Given continuous maps 𝑓 : 𝑋 β†’ π‘Œ and 𝑔 : π‘Œ β†’ 𝑍, if both 𝑔 : π‘Œ β†’ 𝑍 and𝑔 ∘ 𝑓 : 𝑋 β†’ 𝑍 are perfect, then𝑓 : 𝑋 β†’ π‘Œ is also perfect.

(vii) Given a surjective continuous map𝑓 : 𝑋 β†  π‘Œ and a continuous map𝑔 : π‘Œ β†’ 𝑍, if𝑓 : 𝑋 β†  π‘Œ is proper and𝑔 βˆ˜π‘“ : 𝑋 β†’ 𝑍is perfect, then𝑔 : π‘Œ β†’ 𝑍 is also perfect.

(viii) Given a pullback square inTopof the form below,

̃𝑋 𝑋

Μƒπ‘Œ π‘Œ

̃𝑓 𝑓

where Μƒπ‘Œ β†  π‘Œ is a universal topological quotient, if 𝑓 : ΜƒΜƒ 𝑋 β†’ Μƒπ‘Œ is perfect, then𝑓 : 𝑋 β†’ π‘Œ is also perfect.

Proof. (i)–(vi). Straightforward. (Recall propositions1.1.11and3.1.4.) (vii). Under the hypotheses,𝑔 : π‘Œ β†’ 𝑍is proper. It remains to be shown that𝑔 : π‘Œ β†’ 𝑍is separated.

Consider the following commutative square inTop:

𝑋 π‘Œ

𝑋 ×𝑍 𝑋 π‘Œ ×𝑍 π‘Œ

Ξ”π‘”βˆ˜π‘“

𝑓

Δ𝑔

𝑓×𝑍𝑓

Since𝑓 : 𝑋 β†  π‘Œ is proper, so too is𝑓 ×𝑍 𝑓 : 𝑋 ×𝑍 𝑋 β†’ π‘Œ ×𝑍 π‘Œ. On the other hand, since 𝑔 ∘ 𝑓 : 𝑋 β†’ 𝑍 is separated, the relative diagonal

Ξ”π‘”βˆ˜π‘“ : 𝑋 β†’ 𝑋 ×𝑍 𝑋 is a closed embedding. Hence,Δ𝑔 : π‘Œ β†’ π‘Œ ×𝑍 π‘Œ is indeed a closed embedding.

(viii). Under the hypotheses, the following is a pullback square inTop,

̃𝑋 𝑋

̃𝑋 Γ— Μƒπ‘Œ ̃𝑋 𝑋 Γ—π‘Œ 𝑋

Δ𝑓̃ Δ𝑓

and the claim follows. β– 

Corollary. Letu�𝗉be the class of perfect maps inuοΏ½. Then every morph-ism inuοΏ½ that is(u�𝗉, π–©π–Ώπ—Š)-semilocally ofu�𝗉-type is perfect.

Remark. In the language of Β§2.2, what we have shown is that(uοΏ½,u�𝗉, π–©π–Ώπ—Š) is an Γ©tale finitary (i.e.β„΅0-ary) extensive regulated ecumene that satisfies the descent axiom.

152

Download now "Sustainability in volu..."

Outline

Related documents