topos theory

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 $X$ may be glued together from an open subspace $i:U↪X$ and its closed complement $j:K↪X$. The analogous construction for toposes gives the sheaf topos $\mathrm{Sh}\left(X\right)$ via a gluing together of $\mathrm{Sh}\left(U\right)$ and $\mathrm{Sh}\left(K\right)$, 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 $X$ be a topological space, $i:U↪X$ an open subspace, and $j:K↪X$ the complementary closed subspace. Let $O\left(X\right)$ denote the topology of $X$. There is an injective map

$⟨{i}^{*},{j}^{*}⟩:O\left(X\right)\to O\left(U\right)×O\left(K\right)$\langle i^\ast, j^\ast \rangle \colon O(X) \to O(U) \times O(K)
$V↦\left(U\cap V,K\cap V\right)$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 $O\left(U\right)$ and $O\left(K\right)$, which pairs $\left(W,W\prime \right)$ of relatively open sets in $U$ and $K$ “glue together” to form an open set $W\cup W\prime$ in $X$?

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

$W\cup W\prime ↪{\mathrm{int}}_{X}\left(W\cup W\prime \right)$W \cup W' \hookrightarrow int_X(W \cup W')

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

$W\prime ↪{\mathrm{int}}_{X}\left(W\cup W\prime \right).\phantom{\rule{2em}{0ex}}\left(1\right)$W' \hookrightarrow int_X(W \cup W'). \qquad (1)
Proposition

A necessary and sufficient condition for (1) is that the inclusion $W\prime ↪{\mathrm{int}}_{X}\left(W\cup K\right)$ obtains.

Proof

The necessity is clear since $W\prime \subseteq K$. The sufficiency is equivalent to having an inclusion

$W\prime \cap {\mathrm{int}}_{X}\left(W\cup K\right)↪{\mathrm{int}}_{X}\left(W\cup W\prime \right).$W' \cap int_X(W \cup K) \hookrightarrow int_X(W \cup W').

Since $W\prime$ is relatively open in the subspace $K$, we may write $W\prime =K\cap V$ for some $V\in O\left(X\right)$, and so we must check that there is an inclusion

$\left(K\cap V\right)\cap {\mathrm{int}}_{X}\left(W\cup K\right)↪{\mathrm{int}}_{X}\left(W\cup \left(K\cap V\right)\right)$(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 ${\mathrm{int}}_{X}$ preserves intersections, an inclusion

$K\cap V\cap {\mathrm{int}}_{X}\left(W\cup K\right)↪{\mathrm{int}}_{X}\left(W\cup K\right)\cap {\mathrm{int}}_{X}\left(W\cup V\right).$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

$K\cap V\cap {\mathrm{int}}_{X}\left(W\cup K\right)↪{\mathrm{int}}_{X}\left(W\cup K\right)$K \cap V \cap int_X(W \cup K) \hookrightarrow int_X(W \cup K)

and

$K\cap V\cap {\mathrm{int}}_{X}\left(W\cup K\right)↪V↪W\cup V={\mathrm{int}}_{X}\left(W\cup V\right)$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 $W\in O\left(U\right)$ and $V$ are open in $X$.

Proposition

The operation

$O\left(U\right)\ni W↦{\mathrm{int}}_{X}\left(W\cup K\right)={\mathrm{int}}_{X}\left(W\cup ¬U\right)\in O\left(X\right)$O(U) \ni W \mapsto int_X(W \cup K) = int_X(W \cup \neg U) \in O(X)

is the right adjoint ${i}_{*}$ to ${i}^{*}:O\left(X\right)\to O\left(U\right)$.

Proof

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

$\frac{V\subseteq {\mathrm{int}}_{X}\left(W\cup ¬U\right)\phantom{\rule{2em}{0ex}}\text{in _O(X)_}}{V\subseteq W\cup ¬U\phantom{\rule{2em}{0ex}}\text{in _P(X)_}}$\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 $U\cap V\subseteq W$ in $P\left(X\right)$, or to ${i}^{*}\left(V\right)=U\cap V\subseteq W$ in $O\left(X\right)$.

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

$W\prime ↪{j}^{*}{i}_{*}W.$W' \hookrightarrow j^\ast i_\ast W.

where ${i}^{*},{j}^{*}$ are restriction maps and ${i}^{*}⊣{i}_{*}$. For future reference, observe that the operator ${j}^{*}{i}_{*}:O\left(U\right)\to O\left(K\right)$ is left exact.

We can turn all this around. Suppose $U$ and $K$ are topological spaces, and suppose $f:O\left(U\right)\to O\left(K\right)$ is left exact. Then we can manufacture a space $X$ which contains $U$ as an open subspace and $K$ as its closed complement, and (letting $i$, $j$ being the inclusions as above) such that $f={j}^{*}{i}_{*}$. The open sets of $X$ may be identified with pairs $\left(W,W\prime \right)\in O\left(U\right)×O\left(K\right)$ such that $W\prime \subseteq f\left(W\right)$; here we are thinking of $\left(W,W\prime \right)$ as a stand-in for $W\cup W\prime$. In particular, open sets $W$ of $U$ give open sets $\left(W,\varnothing \right)$ of $X$, while open sets $W\prime$ of $K$ also give open sets $U\cup W\prime$ of $X$.

The localic case

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

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

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

Put $K=¬U$, and let ${i}^{*}:O\left(X\right)\to O\left(U\right)$, ${j}^{*}:O\left(X\right)\to O\left(K\right)$ be the frame maps corresponding to the open and closed sublocales attached to $U$, with right adjoints ${i}_{*}$, ${j}_{*}$. Again we have a left exact functor

$f={j}^{*}{i}_{*}:O\left(U\right)\to O\left(K\right).$f = j^\ast i_\ast \colon O(U) \to O(K).

Observe that this gives rise to a left exact comonad

$O\left(U\right)×O\left(K\right)\to O\left(U\right)×O\left(K\right):\left(W,W\prime \right)↦\left(W,W\prime \wedge f\left(W\right)\right)\phantom{\rule{2em}{0ex}}\left(2\right)$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 $\left(W,W\prime \right)$ such that $W\prime \le f\left(W\right)$. The coalgebra category forms a frame.

Theorem

The frame map $⟨{i}^{*},{j}^{*}⟩:O\left(X\right)\to O\left(U\right)×O\left(K\right)$ is identified with the comonadic functor attached to the comonad (2). In particular, $O\left(X\right)$ can be recovered from $O\left(U\right)$, $O\left(K\right)$, and the comonad (2).

Since $O\left(U+K\right)\cong O\left(U\right)×O\left(K\right)$, we can think of the frame map $⟨{i}^{*},{j}^{*}⟩$ as giving a localic surjection $U+K\to X$.

Again, we can turn all this around and say that given any two locales $U$, $K$ and a left exact functor

$f:O\left(U\right)\to O\left(K\right),$f \colon O(U) \to O(K),

we can manufacture a locale $X$ whose frame $O\left(X\right)$ is the category of coalgebras for the comonad

${1}_{O\left(U\right)}×\wedge \circ \left(f×{1}_{O\left(K\right)}\right):O\left(U\right)×O\left(K\right)\to O\left(U\right)×O\left(K\right)\phantom{\rule{2em}{0ex}}\left(3\right)$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 $U$ is naturally identified with an open sublocale of $X$, $K$ with the corresponding closed sublocale, and with a localic surjection $U+K\to X$. This is the (Artin) gluing construction for $f$.

The toposic case

Now suppose given toposes $E$, $E\prime$ and a left exact functor $\Phi :E\to E\prime$. There is an induced left exact comonad

$E×E\prime \stackrel{\delta ×1}{\to }E×E×E\prime \stackrel{1×\Phi ×1}{\to }E×E\prime ×E\prime \stackrel{1×\mathrm{prod}}{\to }E×E\prime \phantom{\rule{2em}{0ex}}\left(3\right)$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 $\Phi$, denoted $\mathrm{Gl}\left(\Phi \right)$.

Objects of $\mathrm{Gl}\left(\Phi \right)$ are triples $\left(e,e\prime ,f:e\prime \to \Phi \left(e\right)\right)$. A morphism from $\left({e}_{0},{e}_{0}^{\prime },{f}_{0}\right)$ to $\left({e}_{1},{e}_{1}^{\prime },{f}_{1}\right)$ consists of a pair of maps $g:{e}_{0}\to {e}_{1}$, $g\prime :{e}_{0}^{\prime }\to {e}_{1}^{\prime }$ which respects the maps ${f}_{0},{f}_{1}$ (in the sense of an evident commutative square). In other words, the Artin gluing is just the comma category $E\prime ↓\Phi$.

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

Returning to the general case, let us denote the geometric embedding of the open subtopos by $i:{E}_{U}↪E$ and that of the closed subtopos by $j:{E}_{¬U}↪E$. Then we have a composite functor

${E}_{U}\stackrel{{i}_{*}}{\to }E\stackrel{{j}^{*}}{\to }{E}_{¬U}$E_U \xrightarrow{i_*} E \xrightarrow{j^*} E_{\neg U}

which is left exact.

Theorem

Let $U$ be a subterminal object of a topos $E$, as above. Then the left exact left adjoint

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

is canonically identified with the comonadic gluing construction $\mathrm{Gl}\left({j}^{*}{i}_{*}\right)\to {E}_{U}×{E}_{¬U}$. In particular, $E$ can be recovered from ${E}_{U}$, ${E}_{¬U}$, and the functor ${j}^{*}{i}_{*}$.

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

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

$\mathrm{Gl}\left(f\right)\to E×F\stackrel{\mathrm{proj}}{\to }E$\mathbf{Gl}(f) \to E \times F \stackrel{proj}{\to} E

is easily identified with a logical functor $\mathrm{Gl}\left(f\right)\to \mathrm{Gl}\left(f\right)/X$ where $X$ is the subterminal object $\left(1,0,0\to f\left(1\right)\right)$. This realizes $E$ as an open subtopos of $\mathrm{Gl}\left(f\right)$. On the other hand, for the same subterminal object $X↪1$, the corresponding classifying map

$\left[X\right]:1\to \Omega$[X] \colon 1 \to \Omega

induces a Lawvere-Tierney topology $j$ given by

$\Omega \cong 1×\Omega \stackrel{\left[X\right]×1}{\to }\Omega ×\Omega \stackrel{\wedge }{\to }\Omega .$\Omega \cong 1 \times \Omega \stackrel{[X] \times 1}{\to} \Omega \times \Omega \stackrel{\wedge}{\to} \Omega.

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

$\mathrm{Gl}\left(f\right)\to E×F\stackrel{\mathrm{proj}}{\to }F,$\mathbf{Gl}(f) \to E \times F \stackrel{proj}{\to} F,

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

Remarks:

• Artin gluing for toposes carries over in some slight extra generality, replacing left exact functors $f$ 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 $E$ be an (elementary, not necessarily Grothendieck) topos, and let $\mathrm{hom}\left(1,-\right):E\to \mathrm{Set}$ represent the terminal object $1$ – this of course is left exact. The gluing construction $\mathrm{Gl}\left(\mathrm{hom}\left(1,-\right)\right)$ is called the scone (Sierpinski cone), or the Freyd cover, of $E$.

• If $E$ is a Grothendieck topos and $\Delta :\mathrm{Set}\to E$ is the (essentially unique) left exact left adjoint, then we have a gluing construction $E↓\Delta$. This gluing may be regarded as the result of attaching a generic open point to $E$.

References

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