classifying discrete opfibration

As in WeberYS2T, a **classifying discrete opfibration** in a finitely complete 2-category $K$ is a discrete opfibration $p:E\to S$ such that for any $X$, the functor

$$K(X,S)\to \mathrm{DOpf}(X),$$

given by pullback of $p$, is full and faithful.

The canonical example is when $K$ is some 2-category of “large categories” and $S$ is the category of (small) sets. More generally, we could take $K$ to be the 2-category $\mathrm{Cat}$ and $S$ the category of sets of cardinality bounded by some cardinal $\lambda $.

When $K$ is equipped with a classifying discrete opfibration, we make the following definitions.

- A discrete opfibration is
**small**if it is in the image of the above functor. - A discrete object $X$ is
**small**if $X\to 1$ is a small discrete opfibration. - If $K$ has a duality involution, we say an object $A$ is
**locally small**or**arrow-small**if the discrete opfibration $X\to A\times {A}^{op}$ corresponding to the 2-sided fibration $A\leftarrow {A}^{2}\to A$ is small. - A not-necessarily-discrete object $A$ is
**small**if it is locally small and admits an eso from a small discrete object.

If $K$ lacks a duality involution, then merely giving the discrete opfibration $E\to S$ doesn’t suffice to characterize the notion of “smallness,” since we can’t define what it means to be locally small. This suggests the definition of a size structure.

We may impose additional axiom on a classifying discrete opfibration, many of which assert closure conditions of the class of small maps.

- Identity maps are small.
- The composite of small discrete opfibrations is small (the “axiom of replacement”).
- All cosieves are small (the “axiom of separation”).
- If $qr$ and $q$ are small discrete opfibrations, then so is $r$ (“full slices”).
- If $q$ is a discrete opfibration, $qr$ is a small discrete opfibration and $r$ is eso, then $q$ is a small discrete opfibration (“quotients”).
- If $q$ is a discrete opfibration whose pullback along an eso is a small discrete opfibration, then $q$ is small (“descent”).
- The map $p:E\to S$ is exponentiable.

Some immediate consequences are:

- If identity maps are small, then each category $K(X,S)$ has a natural terminal object, and so $S$ “has a terminal object” internally, i.e. $S\to 1$ has a right adjoint.
- If small discrete opfibrations are closed under composition, then if $Y\to X$ and $Z\to X$ are small discrete opfibrations, so is $Y{\times}_{X}Z\to Y$, and thus so is the composite $Y{\times}_{X}Z\to X$. Hence $K(X,S)$ has binary products naturally, and thus $S$ “has binary products” internally, i.e. the diagonal $S\to S\times S$ has a right adjoint.
- In particular, if small discrete opfibrations form a subcategory, then $S$ is a cartesian object.
- One can also show that the “axiom of replacement” implies that any small object is the quotient of an internal category (2-congruence) in $\mathrm{sm}(\mathrm{disc}(K))$, the category of small discrete objects.

We will see some further consequences of these axioms below.

Suppose that $p:E\to S$ is a classifying discrete opfibration for which identity maps are small. Then there is a comma square

$$\begin{array}{ccc}E& \stackrel{}{\to}& 1\\ {}^{p}\downarrow & \Downarrow & {\downarrow}^{1}\\ S& \underset{\mathrm{id}}{\to}& S\end{array}$$

in which the map $1\to S$ classifies the discrete opfibration ${\mathrm{id}}_{1}:1\to 1$.

A map $X\to E$ is the same as a discrete opfibration $q:Y\to X$ equipped with a section $s:X\to Y$ with $ps\cong 1$. Thus, it suffices to show that giving such data is equivalent to giving a square

$$\begin{array}{ccc}X& \stackrel{}{\to}& 1\\ \downarrow & \Downarrow & \downarrow \\ S& \underset{\mathrm{id}}{\to}& S\end{array}$$

in which the left-hand vertical map $X\to S$ classifies $q$. But the composite $X\to 1\to S$ classifies $\mathrm{id}:X\to X$, so since pullback of $p$ is full and faithful into discrete opfibrations, giving such a 2-cell is the same as giving a map $\mathrm{id}\to q$ of discrete opfibrations over $X$, which is precisely to give a section of $q$.

Suppose that $p:E\to S$ is a CDO which is exponentiable. Then for any object $A$, we call the exponential

$$\mathrm{Fam}(A)\u2254(S\times A\to S{)}^{(E\to S)}$$

the object of *(small) families in $A$*. It comes with a projection to $S$ which “assigns to each family its indexing set.” Moreover, as observed in our study of exponentials, since $p$ is an opfibration and $S\times A\to S$ is a fibration, then $\mathrm{Fam}(A)\to S$ is also a fibration, as we would expect.

$\mathrm{Fam}(A)$ has a universal property that can be directly expressed as follows. Evidently, to give a morphism $X\to \mathrm{Fam}(A)$ is equivalent to giving a map $X\to S$ together with a map $X{\times}_{S}E\to S\times A$ over $S$. But $X{\times}_{S}E$ is simply the discrete opfibration classified by $X\to S$, and a map to $S\times A$ over $S$ is just an arbitrary map to $A$. Thus to give a map $X\to \mathrm{Fam}(A)$ is the same as to give a small discrete opfibration $Y\to X$ together with a map $Y\to A$: in other words, an $X$-indexed family of small sets, each of which indexes a family of objects of $A$.

(This sort of thing is closely related to the construction of generic families in Algebraic Set Theory.)

Now consider the special case when $A=S$. As above, to give a map $X\to \mathrm{Fam}(S)$ is to give a small discrete opfibration $Y\to X$ and a map $Y\to S$. But a map $Y\to S$ is in turn equivalent to a small discrete opfibration $Z\to Y$. Thus $K(X,\mathrm{Fam}(S))$ is naturally equivalent the category of composable pairs $Z\to Y\to X$ of small discrete opfibrations. Recalling that any map between discrete opfibrations over $X$ is again a discrete opfibration, we observe that the 1-category $\mathrm{DOpf}(K/X{)}^{2}$ consists of composable pairs $Z\to Y\to X$ of discrete opfibrations; thus $K(X,\mathrm{Fam}(S))$ is naturally a full subcategory of this category.

On the other hand, we can consider another full subcategory of $\mathrm{DOpf}(K/X{)}^{2}$ determined by those composable pairs $Z\to Y\to X$ in which $Y\to X$ and the composite $Z\to X$ are small. This is precisely the subcategory $\mathrm{SDOpf}(K/X{)}^{2}$, where $\mathrm{SDOpf}(K/X)$ consists of small discrete opfibrations, and is thus equivalent to $K(X,S)$. It follows that this second full subcategory of $\mathrm{DOpf}(K/X{)}^{2}$ is equivalent to $K(X,{S}^{2})$. Clearly these two subcategories agree if and only if small discrete opfibrations are closed under composition and have full slices, in the terminology defined above. Thus we have proven:

Let $p:E\to S$ be an exponentiable classifying discrete opfibration; the following are equivalent.

- Small discrete opfibrations are closed under composition and have full slices.
- For each $X$, the categories $K(X,\mathrm{Fam}(S))$ and $K(X,{S}^{2})$ are equivalent as full subcategories of $\mathrm{DOpf}(K/X{)}^{2}$.

When these conditions hold, we clearly have a natural equivalence $K(X,\mathrm{Fam}(S))\simeq K(X,{S}^{2})$ lying over $K(X,S)$, and therefore, by the Yoneda lemma, an equivalence over $S$ between $\mathrm{Fam}(S)\to S$ and the codomain fibration ${S}^{2}\to S$.

(Note the slight peculiarity of this result: it is more common, when showing that two things are equivalent under certain conditions, to construct a canonical map between them which always exists and happens to be an equivalence in the cases of interest. Here instead we have constructed a canonical cospan which *induces* an equivalence in the cases of interest.)

Colloquially speaking, this theorem says that the category $S$ of sets satisfies the “replacement axiom” if and only if the “naive indexing” $\mathrm{Fam}(S)$ of $S$ over itself is equivalent to its “self-indexing” ${S}^{2}$. In classical material set theory, this is well-known to be equivalent to the usual axiom of replacement.

Note, though, that depending on what $K$ is, $\mathrm{Fam}(S)$ may not be the “naive indexing” at all. For instance, if $K$ is the category of stacks on a topos $E$, then the self-indexing of $E$ is a classifying discrete opfibration in $K$, which *always* satisfies the “axiom of replacement,” essentially by construction. Analogous facts are well-known in algebraic set theory, and are one reason why “the axiom of replacement” is a bit slippery in structural set theory. Generally, in intuitionistic logic, the axioms of replacement and collection add no extra proof-theoretic strength because they can be made to hold internally in suitable (2-)topoi of sheaves/stacks; it is the axiom of *separation* which carries all the power distinguishing IZF from IETCS.

Given a classifying discrete opfibration, we can use finite 2-categorical limits and the “internal logic” to construct all the usual concrete categories out of the object $S$. For instance, if small discrete opfibrations are a subcategory, so that $S$ is a cartesian object, then we have the composite $S\to S\times S\to S$ of the diagonal with the “binary products” morphism which, intuitively, takes a set $A$ to the set $A\times A$. Now the inserter of this composite and ${\mathrm{id}}_{S}$ can be considered “the category of sets $A$ equipped with a function $A\times A\to A$,” i.e. the category of magmas.

Now we have a forgetful morphism $\mathrm{magma}\to \mathrm{set}$, and also a functor $\mathrm{magma}\to \mathrm{set}$ which takes a magma to the triple product $A\times A\times A$, and there are two 2-cells relating these constructed from two different composites of the inserter 2-cell defining the category of magmas. It makes sense to call the equifier of these 2-cells “the category of semigroups” (sets with an associative binary operation). Proceeding in this way we can construct the categories of monoids, groups, abelian groups, and eventually rings.

A more direct way to describe these categories with a universal property is as follows. Since $S$ is a cartesian object, each hom-category $K(X,S)$ has finite products, so we can define the category $\mathrm{ring}(K(X,S))$ of rings internal to it. Then the category $\mathrm{ring}$ is equipped with a forgetful functor $\mathrm{ring}\to \mathrm{set}$ which has the structure of a ring in $K(\mathrm{ring},\mathrm{set})$, and which is universal in the sense that we have a natural equivalence $\mathrm{ring}(K(X,S))\simeq K(X,\mathrm{ring})$. The above construction then just shows that such a representing object exists.

If $K$ has a duality involution and $p:E\to S$ is a classifying discrete opfibration, then ${p}^{o}:{E}^{o}\to {S}^{o}$ is a “classifying discrete fibration,” and therefore also a classifying discrete opfibration in ${K}^{\mathrm{co}}$.

A classifying discrete opfibration in $K$ is not inherited by any category of truncated objects in $K$, since $E$ and $S$ will generally not be truncated. However, it is inherited by (op)fibrational slices.

Recall first that opfibrations in $\mathrm{Opf}(X)$ can be identified with morphisms in $\mathrm{Opf}(X)$ whose underlying morphism in $K$ is an opfibration. Moreover, such an opfibration is discrete if and only if its underlying morphism in $K$ is so. Thus, it is natural to hope for the following.

If $K$ has exponentials and a classifying discrete opfibration, then each 2-category ${\mathrm{Opf}}_{K}(X)$ has a classifying discrete opfibration, and the small opfibrations in $\mathrm{Opf}(X)$ are those whose underlying opfibrations in $K$ are small.

Recall that when $K$ has exponentials, $\mathrm{Opf}(X)$ is comonadic over $K/X$; let ${G}_{X}$ denote the comonad. For any $A$, write $\mathrm{SOpf}(A)$ for the category of small discrete opfibrations over $A$. Then our goal is for ${\mathrm{SOpf}}_{\mathrm{Opf}(X)}(A\to X):=\mathrm{SOpf}(A)$ to be representable. But we have

$$\mathrm{SOpf}(A)\simeq K(A,S)\simeq K({\Sigma}_{X}(A\to X),S)\simeq K/X(A,X\times S)\simeq \mathrm{Opf}(X)(A,{G}_{X}(X\times S)).$$

Therefore, defining ${S}_{X}={G}_{X}(X\times S\to X)$ we obtain a classifying discrete opfibration in $\mathrm{Opf}(X)$ that classifies the desired small opfibrations.

If $K$ has exponentials, a duality involution, and a classifying discrete opfibration, then each 2-category ${\mathrm{Fib}}_{K}(X)$ also has a classifying discrete opfibration.

This follows from Theorem 2 since $\mathrm{Fib}(X)\simeq \mathrm{Opf}({X}^{o})$. The classifying object is explicitly given by ${V}_{X}({G}_{{X}^{o}}({X}^{o}\times S\to {X}^{o}))$, although the small maps are not as explicitly characterized.

Revised on March 11, 2010 02:32:11
by Mike Shulman
(75.3.151.132)