Michael Shulman
classifying cosieve

A cosieve is a morphism AX in a 2-category that is both ff and a discrete opfibration. Equivalently, it is a subterminal object in Opf(X). It is easy to check that in Cat, this is equivalent to saying that A is a full subcategory of X such that if aA and f:ab, then bA.

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 ζΩ for a classifying cosieve. Clearly any such Ω 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 ζ=1. In Cat, the “walking arrow” 2 is a cosieve classifier.

Construction from classifying discrete opfibrations

If ES is any classifying discrete opfibration in a Heyting 2-pretopos K, then the subobject of S described in the internal logic by

{x:S(a:E(x))(b:E(x))(f:hom E(a,b))}\{x:S | (\forall a:E(x))(\forall b:E(x))(\exists f:hom_E(a,b))\top \}

is the largest subobject ΩS such that the pullback of E to Ω 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 ES.

In Cat, the cosieve classifier 2 arises from Set (or any full subcategory of it containing 0 and 1) in this way.

From cosieve classifiers to subobject classifiers

If X is groupoidal, then every ff into X is a cosieve. Therefore, maps from a groupoidal X into a cosieve classifier Ω classify all subobjects of X. Since subobjects of X are the same as subobjects of its core J(X) if that exists, subobjects of X can be classified by maps J(X)Ω.

Moreover, if a cosieve classifier Ω itself has a core, then since J(Ω) is a coreflection of Ω into gpd(K), it is a subobject classifier in gpd(K) in a suitable (2,1)-categorical sense. Moreover, since Ω is posetal, its core (if it exists) is discrete. Thus:

Theorem

If K is a 2-category having a cosieve classifier and enough groupoids, then disc(K) has a subobject classifier.

In particular, if K also has (discrete) exponentials, then disc(K) is a topos.

However, disc(K) 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 K=[C,Cat], the category disc(K) is the 1-topos of 1-sheaves on the homwise-discrete reflection of C, 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 Ω is a cosieve classifier, Ω o is a sieve opclassifier, i.e. K(X,Ω o) is equivalent to the opposite of the poset of sieves on X. On Ω×Ω o we thus have both a sieve R and a cosieve S, pulled back from Ω and Ω o; let Ω d be the subobject of Ω×Ω o defined as RS in the Heyting algebra structure. Now maps into Ω d classify sieves and cosieves that are equal as subobjects, which is to say, subobjects that are both sieves and cosieves. And transformations between maps XΩ d correspond to both inclusions of cosieves and coinclusions of sieves, which is to say, identities; thus Ω d is discrete, and hence a subobject classifier in disc(K).

Theorem

If K is a Heyting 2-category having a cosieve classifier and a duality involution, then disc(K) has a subobject classifier.

Replacing subobject classifiers

If K has a cosieve classifier and (discrete) exponentials, but not enough groupoids, then disc(K) may not be a topos. But it retains many of the properties of a topos, because even though the “power object” PX=Ω X is not an object of disc(K), it can still be quantified over in the internal logic of K to define objects and properties in disc(K), and even in gpd(K), where all subobjects are cosieves.

For example, if K is also Heyting, then for any groupoidal X we can construct the “internally least” subobject of X with some property, as

{x:X(S:PX)(φ(S)xS)}.\{x:X | (\forall S:P X)(\varphi(S) \Rightarrow x\in S)\}.

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:

Proposition

If K is 1-exact and Heyting with exponentials and a cosieve classifier, then disc(K) is finitely cocomplete.

If X is not groupoidal, then the above technique only constructs cosieves in X rather than arbitrary subobjects of it. However, if there are enough groupoids, we can construct arbitrary subobjects of any object X in this way, since subobjects of X are bijective with subobjects of its core J(X). In particular:

Proposition

If K is 1-exact and Heyting with exponentials, a cosieve classifier, and either enough groupoids or a duality involution, then it has discrete reflections.

Proof

It suffices to be able to construct the equivalence relation generated by the image of A 2A×A, for any A. Note that these relations are not cosieves on A×A, but as remarked above, we can get around this if A×A has a core. Alternately, since the relations we care about are all “2-sided sieves” (subterminals in Fib(A,A)), if there is a duality involution we can turn them into cosieves on A op×A and perform the closure there.

In another vein, if K is a positive Heyting 2-category with a (necessarily discrete) natural numbers object N, we can of course construct the discrete object of rational numbers Q in the usual way, and then define the Dedekind real numbers as two-sided cuts. Thus, R is a subobject of PQ×PQ, and hence posetal, but since the order relation on R inherited from the two copies of PQ would go in different directions, in fact R 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 disc(K) 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.

Revised on December 22, 2009 07:23:28 by Mike Shulman (67.189.60.12)