Cosheaves and connectedness in formal topology

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1. X and π 0 have relations ≤ γ to make X locally connected. 2. p : CS(X; π 0 ) → X is a homeomorphism. 3. π 0 is strongly terminal in MX. peer-00821313, version 1 - 9 May 2013 Proof. The statement of the Theorem is in fact largely a rephrasing of [2, Proposition 6.2.5]. The statement there, after some notation changes, is as follows. For any B, the identity Id B is a discrete M-fibration (i.e. a complete spread for some cosheaf) iff B → 1 admits an M-adjoint (i.e. is an essential geometric morphism – see their Proposition 4.3.16; but in this case, where the codomain is 1, that is equivalent to B being locally connected). In this case the corresponding cosheaf 1 → M(B) is δ 1 ; r b . To see how this fits our phrasing, first suppose (2). Then we might as well replace p by Id X . Then the first part of their statement is our (1)⇔(2) (except for the part about the relations ≤ γ ). The second part of their statement then tells us that π 0 = δ 1 ; r b . But their r b : M1 → MX is by definition the right adjoint to M! : MX → M1. Also, δ 1 : 1 → M1 is a singleton set (a cosheaf over 1 is just a set) and δ 1 is right adjoint to ! : M1 → 1 because a singleton set is a terminal set. It follows that π 0 is right adjoint to ! : MX → 1, i.e. π 0 is strongly terminal. For the converse, if π 0 is a strongly terminal cosheaf, then the inverse image functor π0 ∗ : SMX → Set is left adjoint to ! ∗ : Set → SMX. But [2, Remark 4.3.3] δ ∗ : SMX → SX also has a left adjoint, so it follows that ! ∗ : Set → SX has a left adjoint and we recover the topos-theoretic definition of local connectedness. We now give a direct proof. (1)⇔(2): This is Proposition 34. For (2)⇔(3) we can use purely categorical methods analogous to those of Theorem 39, and using Lemma 42. Again, the difficult direction is (2)⇒(3). We calculate bicomma squares as – 〈x, β : δ(x) ⇒ π 0 , G〉 CS MX (X; (!; π 0 )) −→ MX ↓ p ′ ↓! 〈x, β : δ(x) ⇒ π 0 〉 CS(X; π 0 ) ! → 1 ↓ q ∗ π 0 ↓ π 0 〈x, F, α : δ(x) ⇒ F 〉 CS MX (X; Id) q −→ MX ↓ r ⇒ ↓ Id X −→ δ MX q ′ The diagram itself is analogous to that in Theorem 39, but down the left hand side we have described the points of the corresponding complete spreads. If p = (q ∗ π 0 ); r has an inverse, then, just as in Theorem 39, we find a map θ = 〈r; p −1 , q〉 : CS MX (X; Id) → CS MX (X; (!; π 0 )) over X × MX. It maps 〈x, F, α〉 to 〈x, β, F 〉 where 〈x, β〉 = p −1 (x). By Lemma 42 this gives us the 2-cell η : Id MX ⇒ !; π 0 . It remains to show that η π0 : π 0 ⇒ π 0 ; ! MX ; π 0 = π 0 is the identity. (Note that since 1 is localic, the isomorphism π 0 ; ! MX ∼ = Id1 must be equality.) To 30
calculate it we pull back θ along π 0 : 1 → MX. Consider, on top of the column of squares in the diagram, a further pullback square got by pulling back along π 0 : 1 → MX. Since π 0 ; ! MX = Id 1 , the vertex of the pullback square must be homeomorphic to CS(X; π 0 ), so we can take the square to be of the form CS(X; π 0 ) ! → 1 ↓ p ′′ ↓ π 0 CS MX (X; (!; π 0 )) q −→ ′ MX with p ′′ ; p ′ = Id CS(X;π0). Since (q ∗ π 0 ); r is a homeomorphism, there is a unique map CS(X; π 0 ) → CS(X; π 0 ) over X ×1, and so a unique homorphism π 0 → π 0 which must be the identity. peer-00821313, version 1 - 9 May 2013 8. Conclusions As far as I am aware, connectedness and local connectedness have not been formalized in full generality for formal topologies (not necessarily overt). On the other hand, in topos theory they have been extensively studied in great generality, being applied not only to locales (as the ordinary topos-theoretic notion of topological space) but also to toposes (as generalized spaces) and to geometric morphisms (as spaces relative to other spaces). The topos theory also links these notions to that of cosheaf, and studies the space of cosheaves (the symmetric topos) and cosheaves as maps (the complete spreads) analogous to local homeomorphisms. Much of the content of the present paper has been just to transfer a small part of this topos theory to formal topology. Where it deals with inductively generated formal topologies, it can be understood as translating the topostheoretic use of sites. Nonetheless, we have shown that there is much predicative content in a body of topos theory that is generally conducted impredicatively. Our main original contributions are as follows. First, we have imported the technology of cosheaves into formal topology. In the inductively generated case the definition is predicative (no quantification over subsets) and we have also imported the associated technology of complete spreads. Second, we have given general definitions of connectedness and local connectedness for formal topology. That of connectedness extends known definitions for the overt case, but is not predicative. For local connectedness we have used cosheaves to give a definition that (i) works with arbitrary bases, rather than requiring a special base of connected opens, and (ii) in the inductively generated case is predicative. Third, we have presented a new approach to a particular result from topos theory, that X is locally connected iff MX is totally connected (i.e. has a strongly terminal point). Our new proof is aimed at being predicative, and suggests a notion of “formal topos”, such as MX, as “generalized formal topology”. Thus we have sketched a glimpse of “formal topos theory”. 31
Page 1 and 2: Author manuscript, published in "An
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Page 5 and 6: in some detail in [6], but we have
Page 7 and 8: amounts to describing the inverse i
Page 9 and 10: The requirement in condition (3) th
Page 15 and 16: proving a ⊳ U we may assume that
Page 17 and 18: Now a ⊳ ⋃ γ∈π 0(a) (≤γ a
Page 21 and 22: Definition 30. We define a map p :
Page 23 and 24: Proof. (1)⇒(2): This follows by c
Page 25 and 26: corresponding sublocale is obtained
Page 27 and 28: Proof. (1)⇔(2): This is Lemma 37.
Page 29: In our current situation we are con
Page 33: [13] P. Johnstone, Open locales and

Cosheaves and connectedness in formal topology

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