Michael Shulman
classifying discrete opfibration

Definition

As in WeberYS2T, a classifying discrete opfibration in a finitely complete 2-category KK is a discrete opfibration p:ESp:E\to S such that for any XX, the functor

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

given by pullback of pp, is full and faithful.

The canonical example is when KK is some 2-category of “large categories” and SS is the category of (small) sets. More generally, we could take KK to be the 2-category CatCat and SS the category of sets of cardinality bounded by some cardinal λ\lambda.

When KK 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 XX is small if X1X\to 1 is a small discrete opfibration. * If KK has a duality involution, we say an object AA is locally small or arrow-small if the discrete opfibration XA×A opX\to A\times A^{\op} corresponding to the 2-sided fibration AA 2AA \leftarrow A ^{\mathbf{2}} \to A is small. * A not-necessarily-discrete object AA is small if it is locally small and admits an eso from a small discrete object.

If KK lacks a duality involution, then merely giving the discrete opfibration ESE\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 qrq r and qq are small discrete opfibrations, then so is rr (“full slices”).
  • If qq is a discrete opfibration, qrq r is a small discrete opfibration and rr is eso, then qq is a small discrete opfibration (“quotients”).
  • If qq is a discrete opfibration whose pullback along an eso is a small discrete opfibration, then qq is small (“descent”).
  • The map p:ESp\colon E\to S is exponentiable.

Some immediate consequences are:

  • If identity maps are small, then each category K(X,S)K(X,S) has a natural terminal object, and so SS “has a terminal object” internally, i.e. S1S\to 1 has a right adjoint.
  • If small discrete opfibrations are closed under composition, then if YXY\to X and ZXZ\to X are small discrete opfibrations, so is Y× XZYY\times_X Z \to Y, and thus so is the composite Y× XZXY\times_X Z\to X. Hence K(X,S)K(X,S) has binary products naturally, and thus SS “has binary products” internally, i.e. the diagonal SS×SS\to S\times S has a right adjoint.
  • In particular, if small discrete opfibrations form a subcategory, then SS 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))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:ESp\colon E\to S is a classifying discrete opfibration for which identity maps are small. Then there is a comma square

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

in which the map 1S1\to S classifies the discrete opfibration id 1:11id_1\colon 1\to 1.

Proof

A map XEX\to E is the same as a discrete opfibration q:YXq\colon Y\to X equipped with a section s:XYs\colon X\to Y with ps1p s \cong 1. Thus, it suffices to show that giving such data is equivalent to giving a square

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

in which the left-hand vertical map XSX\to S classifies qq. But the composite X1SX\to 1 \to S classifies id:XXid\colon X\to X, so since pullback of pp is full and faithful into discrete opfibrations, giving such a 2-cell is the same as giving a map idqid \to q of discrete opfibrations over XX, which is precisely to give a section of qq.

Families

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

Fam(A)(S×AS) (ES) Fam(A) \coloneqq (S\times A \to S)^{(E\to S)}

the object of (small) families in AA. It comes with a projection to SS which “assigns to each family its indexing set.” Moreover, as observed in our study of exponentials, since pp is an opfibration and S×ASS\times A\to S is a fibration, then Fam(A)SFam(A)\to S is also a fibration, as we would expect.

Fam(A)Fam(A) has a universal property that can be directly expressed as follows. Evidently, to give a morphism XFam(A)X\to Fam(A) is equivalent to giving a map XSX\to S together with a map X× SES×AX\times_S E \to S\times A over SS. But X× SEX\times_S E is simply the discrete opfibration classified by XSX\to S, and a map to S×AS\times A over SS is just an arbitrary map to AA. Thus to give a map XFam(A)X\to Fam(A) is the same as to give a small discrete opfibration YXY\to X together with a map YAY\to A: in other words, an XX-indexed family of small sets, each of which indexes a family of objects of AA.

(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=SA=S. As above, to give a map XFam(S)X\to Fam(S) is to give a small discrete opfibration YXY\to X and a map YSY\to S. But a map YSY\to S is in turn equivalent to a small discrete opfibration ZYZ\to Y. Thus K(X,Fam(S))K(X,Fam(S)) is naturally equivalent the category of composable pairs ZYXZ\to Y\to X of small discrete opfibrations. Recalling that any map between discrete opfibrations over XX is again a discrete opfibration, we observe that the 1-category DOpf(K/X) 2DOpf(K/X)^{\mathbf{2}} consists of composable pairs ZYXZ\to Y\to X of discrete opfibrations; thus K(X,Fam(S))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) 2DOpf(K/X)^{\mathbf{2}} determined by those composable pairs ZYXZ\to Y\to X in which YXY\to X and the composite ZXZ\to X are small. This is precisely the subcategory SDOpf(K/X) 2SDOpf(K/X)^{\mathbf{2}}, where SDOpf(K/X)SDOpf(K/X) consists of small discrete opfibrations, and is thus equivalent to K(X,S)K(X,S). It follows that this second full subcategory of DOpf(K/X) 2DOpf(K/X)^{\mathbf{2}} is equivalent to K(X,S 2)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:ESp\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 XX, the categories K(X,Fam(S))K(X,Fam(S)) and K(X,S 2)K(X,S^{\mathbf{2}}) are equivalent as full subcategories of DOpf(K/X) 2DOpf(K/X)^{\mathbf{2}}.

When these conditions hold, we clearly have a natural equivalence K(X,Fam(S))K(X,S 2)K(X,Fam(S))\simeq K(X,S^{\mathbf{2}}) lying over K(X,S)K(X,S), and therefore, by the Yoneda lemma, an equivalence over SS between Fam(S)SFam(S)\to S and the codomain fibration S 2SS^{\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 SS of sets satisfies the “replacement axiom” if and only if the “naive indexing” Fam(S)Fam(S) of SS over itself is equivalent to its “self-indexing” S 2S^{\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 KK is, Fam(S)Fam(S) may not be the “naive indexing” at all. For instance, if KK is the category of stacks on a topos EE, then the self-indexing of EE is a classifying discrete opfibration in KK, 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 SS. For instance, if small discrete opfibrations are a subcategory, so that SS is a cartesian object, then we have the composite SS×SSS \to S\times S \to S of the diagonal with the “binary products” morphism which, intuitively, takes a set AA to the set A×AA\times A. Now the inserter of this composite and id Sid_S can be considered “the category of sets AA equipped with a function A×AAA\times A\to A,” i.e. the category of magmas.

Now we have a forgetful morphism magmasetmagma\to set, and also a functor magmasetmagma \to set which takes a magma to the triple product A×A×AA\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 SS is a cartesian object, each hom-category K(X,S)K(X,S) has finite products, so we can define the category ring(K(X,S))ring(K(X,S)) of rings internal to it. Then the category ringring is equipped with a forgetful functor ringsetring\to set which has the structure of a ring in K(ring,set)K(ring,set), and which is universal in the sense that we have a natural equivalence ring(K(X,S))K(X,ring)ring(K(X,S))\simeq K(X,ring). The above construction then just shows that such a representing object exists.

Preservation

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

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

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

Theorem

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

Proof

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

SOpf(A)K(A,S)K(Σ X(AX),S)K/X(A,X×S)Opf(X)(A,G X(X×S)).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×SX)S_X = G_X(X\times S\to X) we obtain a classifying discrete opfibration in Opf(X)Opf(X) that classifies the desired small opfibrations.

Corollary

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

Proof

This follows from Theorem 2 since Fib(X)Opf(X o)Fib(X) \simeq Opf(X^o). The classifying object is explicitly given by V X(G X o(X o×SX o))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)