A cosieve is a morphism in a 2-category that is both ff and a discrete opfibration. Equivalently, it is a subterminal object in . It is easy to check that in , this is equivalent to saying that is a full subcategory of such that if and , then .
A classifying cosieve is a classifying discrete opfibration which is a cosieve—and hence classifies only cosieves, since cosieves are stable under pullback. We write for a classifying cosieve. Clearly any such is posetal.
A cosieve classifier is a classifying cosieve which classifies all cosieves. In this case one can show, just as for the subobject classifier in a topos, that . In , the “walking arrow” is a cosieve classifier.
is the largest subobject such that the pullback of to is a cosieve. (Verifying this is a straightforward argument using the Kripke-Joyal semantics?.) It is thus a classifying cosieve, which is canonically associated to .
In , the cosieve classifier arises from (or any full subcategory of it containing and ) in this way.
If is groupoidal, then every ff into is a cosieve. Therefore, maps from a groupoidal into a cosieve classifier classify all subobjects of . Since subobjects of are the same as subobjects of its core if that exists, subobjects of can be classified by maps .
Moreover, if a cosieve classifier itself has a core, then since is a coreflection of into , it is a subobject classifier in in a suitable (2,1)-categorical sense. Moreover, since is posetal, its core (if it exists) is discrete. Thus:
If is a 2-category having a cosieve classifier and enough groupoids, then has a subobject classifier.
In particular, if also has (discrete) exponentials, then is a topos.
However, can have both a subobject classifier and a cosieve classifier without the former being a core of the latter. For instance, in the 2-presheaf 2-topos , the category is the 1-topos of 1-sheaves on the homwise-discrete reflection of , but there will not in general be a map in either direction relating its subobject classifier to the cosieve classifier.
A subobject classifier can also be constructed from a cosieve classifier in a Heyting 2-category with a duality involution. For then if is a cosieve classifier, is a sieve opclassifier, i.e. is equivalent to the opposite of the poset of sieves on . On we thus have both a sieve and a cosieve , pulled back from and ; let be the subobject of defined as in the Heyting algebra structure. Now maps into classify sieves and cosieves that are equal as subobjects, which is to say, subobjects that are both sieves and cosieves. And transformations between maps correspond to both inclusions of cosieves and coinclusions of sieves, which is to say, identities; thus is discrete, and hence a subobject classifier in .
If is a Heyting 2-category having a cosieve classifier and a duality involution, then has a subobject classifier.
If has a cosieve classifier and (discrete) exponentials, but not enough groupoids, then may not be a topos. But it retains many of the properties of a topos, because even though the “power object” is not an object of , it can still be quantified over in the internal logic of to define objects and properties in , and even in , where all subobjects are cosieves.
For example, if is also Heyting, then for any groupoidal we can construct the “internally least” subobject of with some property, as
This allows the construction of all sorts of “closure” operations that exist in a topos, such as the equivalence relation generated by any given relation on a groupoidal object. In particular:
If is 1-exact and Heyting with exponentials and a cosieve classifier, then is finitely cocomplete.
If is not groupoidal, then the above technique only constructs cosieves in rather than arbitrary subobjects of it. However, if there are enough groupoids, we can construct arbitrary subobjects of any object in this way, since subobjects of are bijective with subobjects of its core . In particular:
It suffices to be able to construct the equivalence relation generated by the image of , for any . Note that these relations are not cosieves on , but as remarked above, we can get around this if has a core. Alternately, since the relations we care about are all “2-sided sieves” (subterminals in ), if there is a duality involution we can turn them into cosieves on and perform the closure there.
In another vein, if is a positive Heyting 2-category with a (necessarily discrete) natural numbers object , we can of course construct the discrete object of rational numbers in the usual way, and then define the Dedekind real numbers as two-sided cuts. Thus, is a subobject of , and hence posetal, but since the order relation on inherited from the two copies of would go in different directions, in fact is discrete.
I haven’t made a concerted effort yet, but I haven’t yet thought of any really important aspect of topos-ness for that isn’t almost as well-served by having a posetal power-object rather than a discrete one. Mathieu Dupont was the one who originally pointed out to me that 2-categorically, power-objects are naturally posets rather than sets.