The classifying topos for a given type of mathematical structure — for example the structures: “group”, “torsor”, “ring”, “category” etc. — is a (Grothendieck) topos such that geometric morphisms are the same as structures of this sort in the topos , i.e. groups internal to , torsors internal to , etc. In other words, a classifying topos is a representing object for the functor which sends a topos to the category of structures of the desired sort in .
In particular for a sheaf topos on a topological space and a (bare, i.e. discrete) group, a -torsor in is a -principal bundle over . There is a classifying topos denoted , such that the groupoid of -principal bundles over is equivalent to geometric morphims :
Hence one can think of classifying topoi as a grand generalization of the notion of classifying space in topology.
In a tautological way, every topos is the classifying topos for something, namely for the categories of geometric morphisms into it. The concept of geometric theory allows one to usefully interpret these categories as categories of certain structures in :
And structures of type in is what geometric morphisms classify.
So the classifying topos for the geometric theory is a Grothendieck topos equipped with a “universal model of ”. This means that for any Grothendieck topos together with a model of in , there exists a unique (up to isomorphism) geometric morphism such that maps the -model to the model . More precisely, for any Grothendieck topos , the category of -models in is equivalent to the category of geometric morphisms .
The fact that a classifying topos is like the ambient set theory but equipped with that universal model is essentially the notion of forcing in logic: the passage to the internal logic of the classifying topos forces the universal model to exist.
Its universality property means that any geometric functor
factors essentially uniquely as
for the universal model and the left adjoint part of a geometric morphism. More precisely, composition with defines an equivalence between the category of geometric morphisms and the category of geometric functors .
More specifically, for any cartesian theory, regular theory or coherent theory (which in ascending order are special cases of each other and all of geometric theories), the corresponding syntactic category comes equipped with the structure of a syntactic site (see there) and the classifying topos for is the sheaf topos .
The notion of classifying topos is part of a trend, begun by Lawvere, of viewing a mathematical theory in logic as a category with suitable exactness properties and which contains a “generic model”, and a model of the theory as a functor which preserves those properties. This is described in more detail at internal logic and type theory, but here are some simple examples to give the flavor. The original example is that of a ‘finite products theory’:
Finite products theory. Roughly speaking, a ‘finite products theory’, ‘Lawvere theory’, or ‘algebraic theory’ is a theory describing some mathematical structure that can be defined in an arbitrary category with finite products. An example would be the theory of groups. As explained in the entry for Lawvere theory, for each such theory there is a category with finite products – the syntactic category, which serves as a “classifying category” for , in that models of in the category of sets correspond to product-preserving functors . More generally, for any category with finite products, say , models of in correspond to product-preserving functors .
Finite limits theory. Next up the line is the notion of ‘finite limits theory’, sometimes called an essentially algebraic theory. This is roughly a theory describing some structure that can be defined in an arbitrary category with finite limits (also called a finitely complete category). An example of a finite limits theory would be the theory of categories. (The notion of ‘category’ requires finite limits, while the notion of ‘group’ does not, because categories but not groups involve a partially defined operation, namely composition of morphisms.) Every finite limits theory admits a classifying category : a finitely complete category such that models of in a category with finite limits correspond to functors that preserve finite limits. (Such functors are called left exact, or ‘lex’ for short.)
Geometric theory. Further up the line, a geometric theory is roughly a theory which can be formulated in that fragment of first-order logic that deals in finite limits and arbitrary (small) colimits, plus certain exactness properties the details of which need not concern us. The point is that a category with finite limits, small colimits, and appropriate exactness is just a Grothendieck topos, and a functor preserving finite limits and small colimits is just the inverse image part of a geometric morphism. Just as in the previous two cases, any ‘geometric theory’ has a classifying category (which is now a Grothendieck topos) which possesses a “generic object” for that theory, and T-models in any other Grothendieck topos E can be identified with geometric morphisms , or specifically with their inverse image parts.
Each type of theory may be considered a -theory, or doctrine. Furthermore, each type of theory can be promoted to a theory “further up the line”, by freely adding the missing structure to the classifying category. This can always be done purely formally, but in a few cases this promotion also has other, more explicit descriptions.
For instance, to go from a finite products theory to the corresponding finite limits theory, we can take the opposite of the category of finitely presentable models of in , thanks to Gabriel-Ulmer duality. Similarly, to go from a finite limits theory to the classifying topos of the corresponding geometric theory, we can take the category of presheaves on the classifying category of the finite limits theory.
The fact that classifying toposes are what they all comes down, if spelled out explicitly, to the fact that geometric morphism of toposes can be identified with certain morphism of sites , for these toposes, going the other way round, , and having certain properties. If here is the syntactic site of some theory and we choose to be the canonical site of (itself equipped with the canonical coverage) this makes manifest why the geometric morphisms in correspond to models of in .
is induced by a morphism of sites
This appears as (Johnstone, lemma C2.3.8).
It suffices to observe that the factorization, if it exists, is a morphism of sites.
This appears as (Johnstone, cor. C2.3.9).
We list and discuss explicit examples of classifying toposes.
The presheaf topos on the opposite category of FinSet is the classifying topos for the theory of objects, sometimes called the “object classifier” This is not to be confused with the notion of an object classifier in an (∞,1)-topos and maybe better called in full the classifying topos for the theory of objects.
We discuss the finite product theory of groups. This theory has a classifying category . is a category with finite products equipped with an object , the “walking group”, a morphism describing multiplication, a morphism describing inverses, and a morphism describing the identity element of , obeying the usual group axioms. For any category with finite products, say , a finite-product-preserving functor is the same as a group object in . For more details, see Lawvere theory.
We can promote to a category with finite limits, , by adjoining all finite limits. As mentioned above, one way to do this is to take the category of models of in Set, which is simply , and then take the full subcategory of finitely presentable groups. By Gabriel-Ulmer duality, the opposite of this is . For any category with finite products, say , a left exact functor is the same as a group object in .
For any Grothendieck topos, say , a left exact left adjoint functor is the same as a group object in .
The discussion above for groups can be repeated verbatim for rings, since they too are described by a finite products theory.
This appears as (Moerdijk 95, prop. 5.4).
are equivalently linear orders. Evidently, such a functor is in particular a simplicial set and we will show that being flat is equivalent to this simplicial set being the nerve of an inhabited linear order regarded as a category (a (0,1)-category).
First assume that is a flat functor. Since (by the discussion there) this preserves all finite limits that exist in , equivalently that it sends the finite colimits that exist in to limits in , it in particular sends the gluings of intervals
in to isomorphisms
Moreover, since monomorphisms are characterized by pullbacks, being flat means that it sends jointly epimorphic families of morphisms in to monomorphisms in . In particular, the epimorphic family is sent to an injection
The category of elements is inhabited, hence the poset of which is the nerve is inhabited.
For every two elements there exist morphisms in and such that and . Since is the nerve of a poset, this means that there is a totally ordered set and and are among its elements , . Accordingly we have either or and hence is in fact the nerve of a total order.
If are elements in the total order with and , this means that in the nerve we have elements and with and .
By co-filtering, there exists a cone over this diagram in the category of elements, hence morphisms in and such that
Here the last condition in can only hold if , hence if .
Conversely, assume that is the nerve of a linear order. We show that then it is a flat functor .
Specifically a geometric morphism from to is an linear interval in Set, meaning a totally ordered set with distinct top and bottom elements. In general, a linear interval is a model for the one-sorted geometric theory whose signature consists of a binary relation and two constants? , , subject to the following axioms:
(Joyal calls this a strict linear interval; by removing the hypothesis of distinct top and bottom, one arrives at a weaker notion he calls “linear interval”. Linear intervals in this sense are classified by the topos , where , sometimes called the algebraist’s Delta or the augmented simplex category, is the category of all finite ordinals including the empty one.)
In a topos of sheaves over a sober space, a local ring is precisely what algebraic geometers usually call a “sheaf of local rings”: namely, a sheaf of rings all of whose stalks are local. See locally ringed topos. This is a special case of the case of Cover-preserving flat functors below.
See also this MO discussion
This is a weak homotopy equivalence of toposes, in that it induces isomorphisms on geometric homotopy groups of the terminal object.
A geometric theory whose models are -torsors can be described as follows. It has one sort, , and one unary operation for every element . It has algebraic axioms and , which make into a -set, and geometric axioms (inhabited-ness), for all (freeness), and (transitivity).
This is such that for a topological space, geometric morphisms classifies topological -principal bundles on .
At generalized universal bundle and principal ∞-bundle it is discussed that the principal bundle classified by a morphims into a classifying object is its homotopy fiber, and how the universal bundle is a replacement of the point such that its ordinary pullback models that homotopy pullback.
We can send this morphism in Grpd with
which sends a set to the -set equipped with the evident -action induced by that of on itself.
Because for any set with -action we have naturally
singled out this way in this way is the universal object in , namely equipped with the canonical -action on itself.
It ought to be true that the topos-incarnation of the -principal bundle on a topological space classified by a geometric morphism is the -pullback
needs more discussion…
In fact, any Grothendieck topos can be thought of as a classifying topos for some localic groupoid. This is related to the discussion above, since Joyal and Tierney showed that any Grothendieck topos is equivalent to the for some localic groupoid . A useful discussion of this idea starts here.
As a special case of the above, any presheaf topos, i.e. any topos of the form , is the classifying topos for flat functors from (sometimes also called “-torsors”). In other words, geometric morphisms are the same as flat functors . This is Diaconescu's theorem. If has finite limits, then a flat functor is the same as a functor that preserves finite limits.
Another way, apart from that above, of viewing any Grothendieck topos as a classifying topos is to start with a small site of definition for it. Any such site gives rise to a geometric theory called the theory of cover-preserving flat functors on that site. The classifying topos of this theory is again .
Moreover, for any object of , there is a small site of definition for which includes , and thus for which is (part of) the universal object.
This is the classifying topos for cover-preserving flat functors out of .
Every category of such functors is the category of models of some geometric theory, and for every geometric theory there is such a cartesian site.
This appears as (Johnstone, remark D3.1.13).
As a special case or rather re-interpretation of the above, let be any essentially algebraic theory and equip its syntactic category with some coverage . Then the sheaf topos is the classifying topos for local -algebras :
such that this sheaf of algebras is local as seen by the respective topologies.
See locally algebra-ed topos for more on this.
we should expect there to be a topos analog of the total space, , for the classifying space. This analog is the generic G-torsor, which is an internal -torsor in the topos . The important aspect of the space is that as a principal -bundle over , it is a universal element, i.e. the natural transformation that it induces (by the Yoneda lemma) is the isomorphism which exhibits as the object representing the functor . For the same Yoneda reasons, the classifying topos of any geometric theory comes with a generic -model, which is a -model in which represents the functor in the same way. For = the theory of -torsors, this generic model is the generic -torsor.
Standard textbooks reference for classifying topoi of theories
The study of classifying spaces of topological categories is described in the monograph
The original theory for a general algebraic theory is developed in
The results for the continuous groupoids include
Ieke Moerdijk, The classifying topos of a continuous groupoid I, Trans. A.M.S. 310 (1988), 629-668.
Ieke Moerdijk, Classifying spaces and classifying topoi, Lecture Notes in Math. 1616, Springer-Verlag, New York, 1995.
Classifying toposes as locally algebra-ed (infinity,1)-toposes are discussed in section 1.4 of