nLab
Artin gluing

Context

Topos Theory

Could not include topos theory - contents

Contents

Idea

Artin gluing is a fundamental construction in locale theory and topos theory. The original example is the way in which a topological space or locale XX may be glued together from an open subspace i:UXi \colon U \hookrightarrow X and its closed complement j:KXj \colon K \hookrightarrow X. The analogous construction for toposes gives the sheaf topos Sh(X)Sh(X) via a gluing together of Sh(U)Sh(U) and Sh(K)Sh(K), and applies more generally to give a sense of how to put two toposes together so that one becomes an open subtopos and the other a closed subtopos? of the gluing.

The topological case

Let us consider first the case of topological spaces. Let XX be a topological space, i:UXi \colon U \hookrightarrow X an open subspace, and j:KXj \colon K \hookrightarrow X the complementary closed subspace. Let O(X)O(X) denote the topology of XX. There is an injective map

i *,j *:O(X)O(U)×O(K)\langle i^\ast, j^\ast \rangle \colon O(X) \to O(U) \times O(K)
V(UV,KV)V \mapsto (U \cap V, K \cap V)

that is a map of frames. The general problem is to characterize the image of this map: in terms of structure pertaining to O(U)O(U) and O(K)O(K), which pairs (W,W)(W, W') of relatively open sets in UU and KK “glue together” to form an open set WWW \cup W' in XX?

Let int X:P(X)P(X)int_X: P(X) \to P(X) denote the interior operation, assigning to a subset of XX its interior; this is a left exact comonad on P(X)P(X). (Indeed, topologies on the set XX are in natural bijection with left exact comonads on P(X)P(X).) Our problem is to understand when the inclusion

WWint X(WW)W \cup W' \hookrightarrow int_X(W \cup W')

obtains. Since WO(U)W \in O(U) is already open when considered as a subset of XX, this condition boils down to the condition that

Wint X(WW).(1)W' \hookrightarrow int_X(W \cup W'). \qquad (1)
Proposition

A necessary and sufficient condition for (1) is that the inclusion Wint X(WK)W' \hookrightarrow int_X(W \cup K) obtains.

Proof

The necessity is clear since WKW' \subseteq K. The sufficiency is equivalent to having an inclusion

Wint X(WK)int X(WW).W' \cap int_X(W \cup K) \hookrightarrow int_X(W \cup W').

Since WW' is relatively open in the subspace KK, we may write W=KVW' = K \cap V for some VO(X)V \in O(X), and so we must check that there is an inclusion

(KV)int X(WK)int X(W(KV))(K \cap V) \cap int_X(W \cup K) \hookrightarrow int_X(W \cup (K \cap V))

or in other words, using distributivity and the fact that int Xint_X preserves intersections, an inclusion

KVint X(WK)int X(WK)int X(WV).K \cap V \cap int_X(W \cup K) \hookrightarrow int_X(W \cup K) \cap int_X(W \cup V).

But this is clear, since we have

KVint X(WK)int X(WK)K \cap V \cap int_X(W \cup K) \hookrightarrow int_X(W \cup K)

and

KVint X(WK)VWV=int X(WV)K \cap V \cap int_X(W \cup K) \hookrightarrow V \hookrightarrow W \cup V = int_X(W \cup V)

where to derive the last equation, we use the fact that WO(U)W \in O(U) and VV are open in XX.

Proposition

The operation

O(U)Wint X(WK)=int X(W¬U)O(X)O(U) \ni W \mapsto int_X(W \cup K) = int_X(W \cup \neg U) \in O(X)

is the right adjoint i *i_\ast to i *:O(X)O(U)i^\ast: O(X) \to O(U).

Proof

This is well-known. Indeed, for VO(X)V \in O(X) we have

Vint X(W¬U)in _O(X)_VW¬Uin _P(X)_\frac{V \subseteq int_X(W \cup \neg U) \qquad \text{in _O(X)_}}{V \subseteq W \cup \neg U \qquad \text{in _P(X)_}}

but the last condition is equivalent to having UVWU \cap V \subseteq W in P(X)P(X), or to i *(V)=UVWi^\ast(V) = U \cap V \subseteq W in O(X)O(X).

Summarizing, the gluing condition (1) above (for WO(K)W' \in O(K), WO(U)W \in O(U)) translates into saying that there is an inclusion

Wj *i *W.W' \hookrightarrow j^\ast i_\ast W.

where i *,j *i^\ast, j^\ast are restriction maps and i *i *i^\ast \dashv i_\ast. For future reference, observe that the operator j *i *:O(U)O(K)j^\ast i_\ast: O(U) \to O(K) is left exact.

We can turn all this around. Suppose UU and KK are topological spaces, and suppose f:O(U)O(K)f: O(U) \to O(K) is left exact. Then we can manufacture a space XX which contains UU as an open subspace and KK as its closed complement, and (letting ii, jj being the inclusions as above) such that f=j *i *f = j^\ast i_\ast. The open sets of XX may be identified with pairs (W,W)O(U)×O(K)(W, W') \in O(U) \times O(K) such that Wf(W)W' \subseteq f(W); here we are thinking of (W,W)(W, W') as a stand-in for WWW \cup W'. In particular, open sets WW of UU give open sets (W,)(W, \emptyset) of XX, while open sets WW' of KK also give open sets UWU \cup W' of XX.

The localic case

The development given above generalizes readily to the context of locales. Thus, let XX be a locale, with corresponding frame denoted by O(X)O(X). Each element UO(X)U \in O(X) gives rise to two distinct frames:

  • The frame whose elements are algebras (fixed points) of the left exact idempotent monad U:O(X)O(X)U \vee - \colon O(X) \to O(X). The corresponding locale is the closed sublocale ¬U\neg U (more exactly, the frame surjection O(X)Alg(U)O(X) \to Alg(U \vee -) is identified with a sublocale ¬UX\neg U \to X).

  • The frame whose elements are algebras of the left exact idempotent monad U:O(X)O(X)U \Rightarrow - \colon O(X) \to O(X). (NB: for topological spaces, this is UV=int X(V¬U)U \Rightarrow V = int_X(V \cup \neg U). This is isomorphic as a frame (but not as a subset of O(X)O(X)) to the principal ideal of O(X)O(X) generated by UU, which is more obviously the topology of UU.) The sublocale corresponding to the frame surjection O(X)Alg(U)O(X) \to Alg(U \Rightarrow -) is the open sublocale corresponding to UU.

Put K=¬UK = \neg U, and let i *:O(X)O(U)i^\ast: O(X) \to O(U), j *:O(X)O(K)j^\ast: O(X) \to O(K) be the frame maps corresponding to the open and closed sublocales attached to UU, with right adjoints i *i_\ast, j *j_\ast. Again we have a left exact functor

f=j *i *:O(U)O(K).f = j^\ast i_\ast \colon O(U) \to O(K).

Observe that this gives rise to a left exact comonad

O(U)×O(K)O(U)×O(K):(W,W)(W,Wf(W))(2)O(U) \times O(K) \to O(U) \times O(K): (W, W') \mapsto (W, W' \wedge f(W)) \qquad (2)

whose coalgebras are pairs (W,W)(W, W') such that Wf(W)W' \leq f(W). The coalgebra category forms a frame.

Theorem

The frame map i *,j *:O(X)O(U)×O(K)\langle i^\ast, j^\ast \rangle \colon O(X) \to O(U) \times O(K) is identified with the comonadic functor attached to the comonad (2). In particular, O(X)O(X) can be recovered from O(U)O(U), O(K)O(K), and the comonad (2).

Since O(U+K)O(U)×O(K)O(U + K) \cong O(U) \times O(K), we can think of the frame map i *,j *\langle i^\ast, j^\ast \rangle as giving a localic surjection U+KXU + K \to X.

Again, we can turn all this around and say that given any two locales UU, KK and a left exact functor

f:O(U)O(K),f \colon O(U) \to O(K),

we can manufacture a locale XX whose frame O(X)O(X) is the category of coalgebras for the comonad

1 O(U)×(f×1 O(K)):O(U)×O(K)O(U)×O(K)(3)1_{O(U)} \times \wedge \circ (f \times 1_{O(K)}) \colon O(U) \times O(K) \to O(U) \times O(K) \qquad (3)

so that UU is naturally identified with an open sublocale of XX, KK with the corresponding closed sublocale, and with a localic surjection U+KXU + K \to X. This is the (Artin) gluing construction for ff.

The toposic case

Now suppose given toposes EE, EE' and a left exact functor Φ:EE\Phi \colon E \to E'. There is an induced left exact comonad

E×Eδ×1E×E×E1×Φ×1E×E×E1×prodE×E(3)E \times E' \stackrel{\delta \times 1}{\to} E \times E \times E' \stackrel{1 \times \Phi \times 1}{\to} E \times E' \times E' \stackrel{1 \times prod}{\to} E \times E' \qquad (3)

whose category of coalgebras is again (by a basic theorem of topos theory; see for instance here) a topos, called the Artin gluing construction for Φ\Phi, denoted Gl(Φ)\mathbf{Gl}(\Phi).

Objects of Gl(Φ)\mathbf{Gl}(\Phi) are triples (e,e,f:eΦ(e))(e, e', f \colon e' \to \Phi(e)). A morphism from (e 0,e 0 ,f 0)(e_0, e_0^', f_0) to (e 1,e 1 ,f 1)(e_1, e_1^', f_1) consists of a pair of maps g:e 0e 1g \colon e_0 \to e_1, g:e 0 e 1 g': e_0^' \to e_1^' which respects the maps f 0,f 1f_0, f_1 (in the sense of an evident commutative square). In other words, the Artin gluing is just the comma category EΦE' \downarrow \Phi.

On the other hand, if EE is a topos and UEU\in E is a subterminal object, then it generates two subtoposes, an open subtopos whose reflector is () U(-)^U, and a closed subtopos? whose reflector is the pushout AA+ A×UUA\mapsto A +_{A\times U} U. If E=Sh(X)E=Sh(X) is the topos of sheaves on a locale, then UU corresponds to an element of O(X)O(X), hence an open sublocale with complement KK (say), and the open subtopos can be identified with Sh(U)Sh(U) and the closed one with Sh(K)Sh(K).

Returning to the general case, let us denote the geometric embedding of the open subtopos by i:E UEi\colon E_U \hookrightarrow E and that of the closed subtopos by j:E ¬UEj\colon E_{\neg U}\hookrightarrow E. Then we have a composite functor

E Ui *Ej *E ¬U E_U \xrightarrow{i_*} E \xrightarrow{j^*} E_{\neg U}

which is left exact.

Theorem

Let UU be a subterminal object of a topos EE, as above. Then the left exact left adjoint

i *,j *:EE U×E ¬U\langle i^\ast, j^\ast \rangle \colon E \to E_U \times E_{\neg U}

is canonically identified with the comonadic gluing construction Gl(j *i *)E U×E ¬U\mathbf{Gl}(j^\ast i_\ast) \to E_U \times E_{\neg U}. In particular, EE can be recovered from E UE_U, E ¬UE_{\neg U}, and the functor j *i *j^* i_*.

For a proof, see A4.5.6 in the Elephant.

Once again, the import of this theorem may be turned around. If f:EFf \colon E \to F is any left exact functor, then the projection

Gl(f)E×FprojE\mathbf{Gl}(f) \to E \times F \stackrel{proj}{\to} E

is easily identified with a logical functor Gl(f)Gl(f)/X\mathbf{Gl}(f) \to \mathbf{Gl}(f)/X where XX is the subterminal object (1,0,0f(1))(1, 0, 0 \to f(1)). This realizes EE as an open subtopos of Gl(f)\mathbf{Gl}(f). On the other hand, for the same subterminal object X1X \hookrightarrow 1, the corresponding classifying map

[X]:1Ω[X] \colon 1 \to \Omega

induces a Lawvere-Tierney topology jj given by

Ω1×Ω[X]×1Ω×ΩΩ.\Omega \cong 1 \times \Omega \stackrel{[X] \times 1}{\to} \Omega \times \Omega \stackrel{\wedge}{\to} \Omega.

Then, the category of sheaves Sh(j)Sh(j), or more exactly the left exact left adjoint Gl(f)Sh(j)\mathbf{Gl}(f) \to Sh(j) to the category of sheaves, is naturally identified with the the projection

Gl(f)E×FprojF,\mathbf{Gl}(f) \to E \times F \stackrel{proj}{\to} F,

thus realizing FF as equivalent to the closed subtopos? (Elephant, A.4.5, pp. 205-206) attached to the subterminal object XX.

Remarks:

  • Artin gluing for toposes carries over in some slight extra generality, replacing left exact functors ff by pullback-preserving functors.

  • Artin gluing applies also to other doctrines: regular categories, pretoposes, quasitoposes, etc. See Carboni-Johnstone.

Examples

Examples of the gluing construction abound. Here are a few:

  • Let EE be an (elementary, not necessarily Grothendieck) topos, and let hom(1,):ESet\hom(1, -): E \to Set represent the terminal object 11 – this of course is left exact. The gluing construction Gl(hom(1,))\mathbf{Gl}(\hom(1, -)) is called the scone (Sierpinski cone), or the Freyd cover, of EE.

  • If EE is a Grothendieck topos and Δ:SetE\Delta \colon Set \to E is the (essentially unique) left exact left adjoint, then we have a gluing construction EΔE \downarrow \Delta. This gluing may be regarded as the result of attaching a generic open point to EE.

References

  • Gavin Wraith, Artin Glueing , JPAA 4 (1974) pp.345-358.

  • Aurelio Carboni, Peter Johnstone, Connected limits, familial representability and Artin glueing , Mathematical Structures in Computer Science 5 (1995), 441–459.

Revised on November 14, 2014 18:57:58 by Thomas Holder (89.204.138.113)