Michael Shulman classifying discrete opfibration

Definition

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

$$K(X,S) \to DOpf(X),$$

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 $Cat$ 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 $A \leftarrow A ^{\mathbf{2}} \to A$ 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.

Additional axioms

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 $q r$ and $q$ are small discrete opfibrations, then so is $r$ (“full slices”).
• If $q$ is a discrete opfibration, $q r$ 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 $p\colon E\to S$ 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 $sm(disc(K))$, the category of small discrete objects.

We will see some further consequences of these axioms below.

The universal map as a comma object

Theorem

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

$$\array{E & \overset{}{\to} & 1\\ ^p\downarrow &\Downarrow& \downarrow^1\\ S & \underset{id}{\to} & S}$$

in which the map $1\to S$ classifies the discrete opfibration $id_1\colon 1\to 1$.

Proof

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

$$\array{X & \overset{}{\to} & 1\\ \downarrow & \Downarrow& \downarrow\\ S & \underset{id}{\to} & S}$$

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

Families

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

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

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 $Fam(A)\to S$ is also a fibration, as we would expect.

$Fam(A)$ has a universal property that can be directly expressed as follows. Evidently, to give a morphism $X\to Fam(A)$ 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 $X\to Fam(A)$ 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.)

Families of sets as a codomain fibration

Now consider the special case when $A=S$. As above, to give a map $X\to Fam(S)$ 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 $K(X,Fam(S))$ 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 $DOpf(K/X)^{\mathbf{2}}$ consists of composable pairs $Z\to Y\to X$ of discrete opfibrations; thus $K(X,Fam(S))$ is naturally a full subcategory of this category.

On the other hand, we can consider another full subcategory of $DOpf(K/X)^{\mathbf{2}}$ 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 $SDOpf(K/X)^{\mathbf{2}}$, where $SDOpf(K/X)$ consists of small discrete opfibrations, and is thus equivalent to $K(X,S)$. It follows that this second full subcategory of $DOpf(K/X)^{\mathbf{2}}$ is equivalent to $K(X,S^{\mathbf{2}})$. 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:

Theorem

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

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

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

(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” $Fam(S)$ of $S$ over itself is equivalent to its “self-indexing” $S^{\mathbf{2}}$. 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, $Fam(S)$ 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.

Constructing familiar categories

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 $id_S$ 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 $magma\to set$, and also a functor $magma \to set$ 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 $ring(K(X,S))$ of rings internal to it. Then the category $ring$ is equipped with a forgetful functor $ring\to set$ which has the structure of a ring in $K(ring,set)$, and which is universal in the sense that we have a natural equivalence $ring(K(X,S))\simeq K(X,ring)$. The above construction then just shows that such a representing object exists.

Preservation

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 $K^{co}$.

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 $Opf(X)$ can be identified with morphisms in $Opf(X)$ 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.

Theorem

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

Proof

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

$$SOpf(A) \simeq K(A,S) \simeq K(\Sigma_X (A\to X), S) \simeq K/X(A, X\times S) \simeq Opf(X)(A,G_X(X\times S)).$$

Therefore, defining $S_X = G_X(X\times S\to X)$ we obtain a classifying discrete opfibration in $Opf(X)$ that classifies the desired small opfibrations.

Corollary

If $K$ has exponentials, a duality involution, and a classifying discrete opfibration, then each 2-category $Fib_K(X)$ also has a classifying discrete opfibration.

Proof

This follows from Theorem since $Fib(X) \simeq Opf(X^o)$. 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.