of all homotopy types
Subsets of a set correspond precisely to maps from to the set of truth values of classical logic via their characteristic function . The concept of a subobject classifier generalizes this situation to toposes other than Set:
A subobject classifier in a topos is a morphism such that every monomorphism in the topos (hence every subobject) is the pullback of this morphism along a unique morphism (the characteristic morphism of ) .
In this sense is the classifying object for subobjects and the generic subobject.
That the existence of a subobject classifier in a category is a very powerful property that induces much other structure lies at the heart of topos theory.
By restricting the class of monomorphism appropriately, the concept can be relativized to the concept of an M-subobject classifier1: e.g. demanding only classification of strong monomorphisms leads to quasitoposes.
In a category with finite limits, a subobject classifier is a monomorphism out of the terminal object, such that for every monomorphism in there is a unique morphism such that there is a pullback diagram
See for instance (MacLane-Moerdijk, p. 32).
If it exists, the object is also called the object of truth values, a global element is called a truth value and the element is the truth value true, where all these terms allude to the internal logic of the category .
Note that the subobjects classified by the truth values are subterminal objects.
Moreover, in this case is well powered.
This appears for instance as (MacLane-Moerdijk, prop. I.3.1).
In more detail: given a morphism in , the function
The representability of this functor means there is an object together with a subobject which is universal, meaning that given any subobject , there is a unique morphism such that is obtained as the pullback of along .
whose commutativity says that every element of is the pullback along some of the subobject of corresponding under the natural isomorphism to .
By further playing around with this one finds that this latter subobject of has to be a terminal object.
The corresponding morphism of presheaves is the natural transformation that picks over each object the maximal sieve .
As a special case of presheaf toposes, for a discrete group and the topos of permutation representations, there are precisely two sieves on the single object of the delooping groupoid : the trivial one and the empty one. Hence the subobject classifier here is the 2-element set as in Set, but now regarded as a -set with trivial -action.
An example of a non-Boolean topos is the category of sheaves over a “typical” topological space such as the real line in its usual topology. In this case, is the sheaf where the set of sections over an open subset is the set of open subsets of , with the obvious restriction maps; the sheaf topos in this case is guaranteed to be non-Boolean provided there are some non-regular open sets in (a open set is regular if it is the interior of its closure). The “internal logic” of such a topos is intuitionistic.
The subobject classifier of is .
This follows for instance from the statement that the inverse image of any base change geometric morphism is a logical functor and hence preserves subobject classifiers: Here we are looking at the base change along and hence .
But the statement is also easily directly checked.
The category of pointed sets has a subobject classifier (specified up to unique isomorphism as the pointed set with two elements).
Suppose a category has a subobject classifier; this entails some striking structural consequences for . We list a few here:
For the characteristic map of a mono , we find that is the equalizer of a pair of maps :
“Only if” is trivial. The “if” comes from the fact that an epic (epimorphic) equalizer must be an isomorphism, for if is the equalizer of and is epic, then , whence is their equalizer, so must have been an isomorphism.
Any two epi-mono factorizations of a map in are canonically isomorphic.
Suppose where are epic and are monic. Since is regular, it is the equalizer of some parallel pair as in the diagram
so that , whence since is epic, whence factors through as is the equalizer: for some . Then also since and is monic. We have that is monic since is, and is epic since is. Thus is an isomorphism.
Already these results impose some tight restrictions on . We get some more by exploiting the internal structure of .
The subobject classifier always comes with the structure of an internal poset; that is, a relation which is internally reflexive, antisymmetric, and transitive. This can be constructed directly (see Proposition 5 below), or obtained via the Yoneda lemma since the collection of subobjects of any object is an external poset.
Similarly, since we assume that is finitely complete, each subobject poset has intersections (gotten as pullbacks or fiber products of pairs of monics ), and the intersection operation
is natural in . Hence we have a family of maps
natural in ; by the Yoneda lemma, we infer the presence of an internal intersection map
making an internal meet-semilattice.
There is an internal implication operator
uniquely specified by the internal condition
Construct as the equalizer of the pair of maps
and then define to be the characteristic map of . Now if are two maps , one calculates that is contained in the subobject classified by iff , which is just a way of saying .
In every subobject poset , meets distribute over any joins that exist.
Because is left adjoint to the external operator on , it preserves any joins that happen to exist in .
Normally these results are proved in the context of toposes, where we may say for example that is an internal Boolean algebra if and only if the topos is Boolean. But as the proofs above indicate, we need only exploit the definition of subobject classifier making reference only to finite limit structure.
In a topos, the subobject classifier is always injective, and, so is the power object for every object . In particular, every object embeds into an injective object by the singleton monomorphism : ‘A topos has enough injective objects!’. More generally, injective objects in a topos are percisely the ones that are retracts of some (Cf. Borceux 1994, p.315; Moerdijk-MacLane 1994, p.210).
An online proof may be found here.
As the previous section indicates, having a subobject classifier is a very strong property of a category and “most” categories with finite limits don’t have one.
For example, there is an easy condition ensuring a category2 with a terminal object can’t have a subobject classifier: if there are no nonidentity morphisms out of the terminal object. This includes the following examples.
Any top bounded partial order.
Here’s another obstacle:
But a real killer is the fact that all monos are regular, or its consequences of the category being balanced and uniqueness of epi-mono factorizations:
Even though all monos in Grp are regular, we can kill off by observing that if were a subobject classifier, the proof of Proposition 3 indicates that every mono would have to be the kernel of . But not all monos in are kernels.
Perhaps an even more decisive killer is the observation that meets distribute over (arbitrary) joins in subobject orders. This eliminates many categories from consideration:
Lattices of subobjects in or are rarely distributive.
For any nontrivial category with biproducts, there are non-distributive subobject lattices. Take any object , so that we have three subobjects , , and . Then , whereas . Under distributivity we have
but forces . So the only such category that can have a subobject classifier is trivial.
In higher topoi the the subobject classifiers are the universal fibrations:
from the -category of pointed -categories to that of -categories, which forgets the point.
Whereas for 1-toposes the subobject classifier is the key structural ingredient (besides the exactness properties), in higher topos theory this role is taken over by the object classifier, as pointed out in Lurie (2009).
Francis Borceux, Handbook of Categorical Algebra 3 , Cambridge UP 1994.
Peter Johnstone, Sketches of an Elephant I , Oxford UP 2002.