Michael Shulman classifying cosieve

A cosieve is a morphism $A\to X$ 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 $a\in A$ and $f:a\to b$, then $b\in A$.

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 $\zeta\to \Omega$ for a classifying cosieve. Clearly any such $\Omega$ 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 $\zeta=1$. In $Cat$, the “walking arrow” $\mathbf{2}$ is a cosieve classifier.

Construction from classifying discrete opfibrations

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

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

is the largest subobject $\Omega\hookrightarrow S$ such that the pullback of $E$ to $\Omega$ 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 $E\to S$.

In $Cat$, the cosieve classifier $\mathbf{2}$ 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 $\Omega$ 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 $J(X) \to \Omega$.

Moreover, if a cosieve classifier $\Omega$ itself has a core, then since $J(\Omega)$ is a coreflection of $\Omega$ into $gpd(K)$, it is a subobject classifier in $gpd(K)$ in a suitable (2,1)-categorical sense. Moreover, since $\Omega$ 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 $\Omega$ is a cosieve classifier, $\Omega^o$ is a sieve opclassifier, i.e. $K(X,\Omega^o)$ is equivalent to the opposite of the poset of sieves on $X$. On $\Omega\times\Omega^o$ we thus have both a sieve $R$ and a cosieve $S$, pulled back from $\Omega$ and $\Omega^o$; let $\Omega_d$ be the subobject of $\Omega\times\Omega^o$ defined as $R\Leftrightarrow S$ in the Heyting algebra structure. Now maps into $\Omega_d$ classify sieves and cosieves that are equal as subobjects, which is to say, subobjects that are both sieves and cosieves. And transformations between maps $X\to \Omega_d$ correspond to both inclusions of cosieves and coinclusions of sieves, which is to say, identities; thus $\Omega_d$ 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” $P X = \Omega^X$ 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

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

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 $A ^{\mathbf{2}} \to A\times A$, for any $A$. Note that these relations are not cosieves on $A\times A$, but as remarked above, we can get around this if $A\times A$ 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 $A^{op}\times A$ 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 $P Q \times P Q$, and hence posetal, but since the order relation on $R$ inherited from the two copies of $P Q$ 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.