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 may be glued together from an open subspace and its closed complement . The analogous construction for toposes gives the sheaf topos via a gluing together of and , 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.
Let us consider first the case of topological spaces. Let be a topological space, an open subspace, and the complementary closed subspace. Let denote the topology of . There is an injective map
that is a map of frames. The general problem is to characterize the image of this map: in terms of structure pertaining to and , which pairs of relatively open sets in and “glue together” to form an open set in ?
Let denote the interior operation, assigning to a subset of its interior; this is a left exact comonad on . (Indeed, topologies on the set are in natural bijection with left exact comonads on .) Our problem is to understand when the inclusion
obtains. Since is already open when considered as a subset of , this condition boils down to the condition that
A necessary and sufficient condition for (1) is that the inclusion obtains.
The necessity is clear since . The sufficiency is equivalent to having an inclusion
Since is relatively open in the subspace , we may write for some , and so we must check that there is an inclusion
or in other words, using distributivity and the fact that preserves intersections, an inclusion
But this is clear, since we have
where to derive the last equation, we use the fact that and are open in .
is the right adjoint to .
This is well-known. Indeed, for we have
but the last condition is equivalent to having in , or to in .
Summarizing, the gluing condition (1) above (for , ) translates into saying that there is an inclusion
where are restriction maps and . For future reference, observe that the operator is left exact.
We can turn all this around. Suppose and are topological spaces, and suppose is left exact. Then we can manufacture a space which contains as an open subspace and as its closed complement, and (letting , being the inclusions as above) such that . The open sets of may be identified with pairs such that ; here we are thinking of as a stand-in for . In particular, open sets of give open sets of , while open sets of also give open sets of .
The frame whose elements are algebras (fixed points) of the left exact idempotent monad . The corresponding locale is the closed sublocale (more exactly, the frame surjection is identified with a sublocale ).
The frame whose elements are algebras of the left exact idempotent monad . (NB: for topological spaces, this is . This is isomorphic as a frame (but not as a subset of ) to the principal ideal of generated by , which is more obviously the topology of .) The sublocale corresponding to the frame surjection is the open sublocale corresponding to .
Put , and let , be the frame maps corresponding to the open and closed sublocales attached to , with right adjoints , . Again we have a left exact functor
Observe that this gives rise to a left exact comonad
whose coalgebras are pairs such that . The coalgebra category forms a frame.
The frame map is identified with the comonadic functor attached to the comonad (2). In particular, can be recovered from , , and the comonad (2).
Since , we can think of the frame map as giving a localic surjection .
Again, we can turn all this around and say that given any two locales , and a left exact functor
we can manufacture a locale whose frame is the category of coalgebras for the comonad
so that is naturally identified with an open sublocale of , with the corresponding closed sublocale, and with a localic surjection . This is the (Artin) gluing construction for .
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 , denoted .
Objects of are triples . A morphism from to consists of a pair of maps , which respects the maps (in the sense of an evident commutative square). In other words, the Artin gluing is just the comma category .
On the other hand, if is a topos and is a subterminal object, then it generates two subtoposes, an open subtopos whose reflector is , and a closed subtopos? whose reflector is the pushout . If is the topos of sheaves on a locale, then corresponds to an element of , hence an open sublocale with complement (say), and the open subtopos can be identified with and the closed one with .
Returning to the general case, let us denote the geometric embedding of the open subtopos by and that of the closed subtopos by . Then we have a composite functor
which is left exact.
Let be a subterminal object of a topos , as above. Then the left exact left adjoint
is canonically identified with the comonadic gluing construction . In particular, can be recovered from , , and the functor .
For a proof, see A4.5.6 in the Elephant.
Once again, the import of this theorem may be turned around. If is any left exact functor, then the projection
is easily identified with a logical functor where is the subterminal object . This realizes as an open subtopos of . On the other hand, for the same subterminal object , the corresponding classifying map
induces a Lawvere-Tierney topology given by
Then, the category of sheaves , or more exactly the left exact left adjoint to the category of sheaves, is naturally identified with the the projection
Artin gluing for toposes carries over in some slight extra generality, replacing left exact functors by pullback-preserving functors.
Examples of the gluing construction abound. Here are a few:
Let be an (elementary, not necessarily Grothendieck) topos, and let represent the terminal object – this of course is left exact. The gluing construction is called the scone (Sierpinski cone), or the Freyd cover, of .
If is a Grothendieck topos and is the (essentially unique) left exact left adjoint, then we have a gluing construction . This gluing may be regarded as the result of attaching a generic open point to .
Some of the material above refers to