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 commuativity 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 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, or obtained via the Yoneda lemma since the collection of subobjects of any object is an external poset.
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).
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.
For M the class of strong monomorphisms, this is called a weak subobject classifier in Johnstone (2002, p.120). ↩