Michael Shulman
duality involution

An important structure possessed by the 2-category CatCat is the duality involution () op:Cat coCat(-)^{op}:Cat^{co}\to Cat. Here we consider what the important properties of this involution are that should be generalized to other 2-categories.

Involutions

The most obvious property of () op(-)^{op} is that it is coherently self-inverse. To state this formally, let JJ be the walking isomorphism (01)(0 \overset{\cong}{\to} 1), considered as a 3-category with only identity 2-cells and 3-cells. Thus, a (pseudo) functor from JJ to any 3-category TT can be considered an “internal adjoint (bi)equivalence in TT.” If TT is a 3-category, let C 3opC^{3op} denote the 3-cell dual of TT; note that J 3opJJ^{3op}\cong J and that () co(-)^{co} is a functor 2Cat2Cat 3op2Cat\to 2Cat^{3op}. Finally, let τ:JJJ 3op\tau:J\to J\cong J^{3op} be the evident involution that switches 00 and 11.

Definition

A 2-contravariant involution (hereafter merely an involution) on a 2-category KK is functor W:J2CatW:J\to 2Cat such that W(0)=KW(0)=K and the square

J τ J 3op W W 2Cat () co 2Cat 3op\array{J & \overset{\tau}{\to} & J^{3op}\\ ^W \downarrow && \downarrow^W\\ 2Cat & \underset{(-)^{co}}{\to} & 2Cat^{3op}}

commutes (on the nose).

We write () o:KK co(-)^o:K\to K^{co} (or equivalently K coKK^{co}\to K) for the functor W(01)W(0\to 1); the rest of the data of WW simply says that (() o) oId((-)^o)^o\cong Id in a coherent way.

Example

Any (2,1)-category KK admits a canonical involution that is the identity on objects and morphisms and sends each 2-cell to its inverse.

Example

Let CC be a 2-category equipped with an involution and let K=[C,Cat]K=[C,Cat]. Then KK has a canonical induced involution defined by

[C,Cat] co[C co,Cat co][C,Cat co]() op[C,Cat].[C,Cat]^{co} \overset{\simeq}{\to} [C^{co},Cat^{co}] \overset{\simeq}{\to} [C, Cat^{co}] \overset{(-)^{op}\circ -}{\to} [C,Cat].

Conversely, if CC is Cauchy-complete, then C opC^{op} can be identified with the full subcategory of indecomposable projectives in KK, and thus an involution on KK induces an involution on CC.

Thus, in particular, any (2,1)-truncated 2-presheaf 2-topos admits a canonical involution. However, in general a 2-category that admits an involution will admit many involutions.

Proposition

For any 2-category KK, the 2-category Inv(K)Inv(K) of involutions on KK, if nonempty, is a torsor for the 3-group Aut(K)Aut(K) of (covariant) automorphisms of KK.

Proof

Easy.

In particular, any automorphism of a (2,1)-category CC induces an involution on [C,Cat][C,Cat].

Duality Involutions

As observed by WeberYS2T, experience suggests that the most important additional property of the involution () op(-)^{op} on CatCat is that fibrations over AA can be identified with opfibrations over A opA^{op}, since both are equivalent to functors A opCatA^{op}\to Cat. So it is reasonable to consider, together with an involution () o(-)^o on a 2-category KK, a family of equivalences Fib(X)Opf(X o)Fib(X)\simeq Opf(X^o).

The next question is what additional properties should be required of such a family. The following definition is provisional, but I believe whatever eventual definition we settle on should allow the construction of all the data given below.

Definition

A duality involution on a 2-category KK is an involution () o(-)^o together with the following data.

  1. Equivalences of 2-categories
    V X:Fib(X)Opf(X o)V_X:Fib(X) \simeq Opf(X^o)

    which are natural in XX (that is, VV is a natural equivalence between functors K coop2CatK^{coop}\to 2Cat),

  2. An equivalence between V 1:Fib(1)Opf(1 o)V_1:Fib(1)\simeq Opf(1^o) and the composite Fib(1)KOpf(1)Opf(1 o)Fib(1)\simeq K \simeq Opf(1)\simeq Opf(1^o) (note 1 o11^o\simeq 1 since () o(-)^o is an equivalence).
  3. For any pullback square
    P A B X\array{P & \to & A\\ \downarrow && \downarrow\\ B & \to & X}

    in which all the displayed maps are fibrations, an invertible modification between the two composites

    Fib(P)V POpf(P o)Opf Opf(A o)(P oA o)V A 1Opf Fib(A)(V AP oA) Opf Fib Fib(X)(AX)(V AP oA)Fib Fib(X)(V X 1B o,A)\begin{gathered} Fib(P) \overset{V_P}{\to} Opf(P^o) \simeq Opf_{Opf(A^o)}(P^o\to A^o) \overset{V_A^{-1}}{\to} Opf_{Fib(A)}(V_A P^o \to A)\\ \simeq Opf_{Fib_{Fib(X)}(A\to X)}(V_A P^o\to A) \simeq Fib_{Fib(X)}(V_X^{-1} B^o, A) \end{gathered}

    and

    Fib(P)Fib Fib(B)(PB)V BFib Opf(B o)(V BPB o)Fib Opf Opf(X o)(B oX o)(V BPB o) V X 1Fib Opf Fib(X)(V X 1B oX)(V X 1V BPV X 1B o)Fib Fib(X)(V X 1B o,A).\begin{gathered} Fib(P) \simeq Fib_{Fib(B)}(P\to B) \overset{V_B}{\to} Fib_{Opf(B^o)}(V_B P\to B^o) \simeq Fib_{Opf_{Opf(X^o)}(B^o\to X^o)}(V_B P\to B^o)\\ \overset{V_X^{-1}}{\to} Fib_{Opf_{Fib(X)}(V_X^{-1} B^o\to X)}(V_X^{-1} V_B P\to V_X^{-1} B^o) \simeq Fib_{Fib(X)}(V_X^{-1} B^o, A). \end{gathered}
  4. An invertible modification filling the square
    Fib(X) V X Opf(X o) () o () o Opf(X o) co V X 1 Fib(X) co.\array{Fib(X) & \overset{V_X}{\to} & Opf(X^o)\\ (-)^o\downarrow && \downarrow (-)^o\\ Opf(X^o)^{co}& \underset{V_X^{-1}}{\to} & Fib(X)^{co}.}

    which commutes with the previous two equivalences.

Perhaps there should also be some coherence data and axioms relating these data to each other.

The first two data should be fairly self-explanatory, but the third datum may seem quite complicated. In fact, it requires some thought to see that the two displayed composites are even well-defined. Note first that since P=A× XBP=A\times_X B, by naturality of VV we have V BPB o× X oV XAV_B P \simeq B^o\times_{X^o} V_X A, and thus V X 1V BPV X 1B o× XAV_X^{-1} V_B P \simeq V_X^{-1} B^o\times_X A. Also, P oB o× X oA oP^o \simeq B^o\times_{X^o} A^o, so again by naturality V A 1P oV X 1B o× XAV_A^{-1} P^o \simeq V_X^{-1} B^o \times_X A. Thus V X 1V BPV A 1P oV_X^{-1} V_B P\simeq V_A^{-1} P^o, and the final two equivalences in each displayed composite are instances of the theorem on iterated fibrations. One way to think about this datum is as sort of an “inclusion-exclusion” property for reversal of arrows.

We call the final datum commutation of opposites. To explain it, observe that because () o(-)^o is a 2-contravariant involution, we automatically have equivalences Fib K(X)Fib K co(X o)=Opf K(X o) coFib_K(X) \simeq Fib_{K^{co}}(X^o) = Opf_K(X^o)^{co} (these are the vertical maps in the displayed square). In CatCat, this equivalence corresponds to composing a functor A opCatA^{op}\to Cat with () op:CatCat(-)^{op}:Cat\to Cat as well as reinterpreting it as an opfibration. Commutation of opposites then says that this operation commutes with VV.

One immediate application of commutation of opposites is to deduce a version of the third datum for a pullback square of opfibrations, by passage along the equivalences Fib(X)Opf(X o) coFib(X)\simeq Opf(X^o)^{co}. We will see others below.

Example

If KK is a (2,1)-category, then its canonical involution extends to a canonical duality involution, since Fib(X)K/XOpf(X)Fib(X)\simeq K/X\simeq Opf(X).

Example

If CC is a (2,1)-category, then the canonical involution on K=[C,Cat]K=[C,Cat] extends in a natural way to a duality involution. This does not seem to be the case if CC is merely a 2-category equipped with an involution.

The 3-group Aut(K)Aut(K) also acts on the 2-category DualInv(K)DualInv(K) of duality involutions on KK; the action is free but (seemingly) no longer necessarily transitive. However, I do not know an example of two inequivalent duality involutions having the same underlying involution (as would be necessary for non-transitivity).

The differences between Definition 2 and the definition of WeberYS2T are threefold.

  1. Our definition is less strict; () o(-)^o is only required to be self-inverse in a 2-categorically non-evil way.
  2. Our definition refers to arbitrary fibrations and opfibrations, rather than merely discrete ones.
  3. Our definition refers only to one-sided fibrations, whereas WeberYS2T requires an equivalence DFib(A×B,C)DFib(A,B o×C)DFib(A\times B, C)\simeq DFib(A,B^o\times C).

Of course, since any equivalence of 2-categories preserves discrete objects, our definition implies an equivalence DFib(X)DOpf(X o)DFib(X)\simeq DOpf(X^o). More surprisingly, it turns out that the seemingly stronger two-sided version is a consequence of our definition.

Theorem

If KK is equipped with a duality involution, then we have natural equivalences Fib(A×B,C)Fib(A,B o×C)Fib(A\times B,C)\simeq Fib(A, B^o\times C).

Proof

We first construct an equivalence Opf(A×B)Fib(A,B o)Opf(A\times B)\simeq Fib(A,B^o). Note that since Fib(1)Opf(1 o)Fib(1)\simeq Opf(1^o) is the identity, naturality implies that V A(A×B)A o×BV_A(A\times B) \simeq A^o\times B. Also, since () o(-)^o is an equivalence, and products in KK are the same as products in K coK^{co}, we have (A×B) oA o×B o(A\times B)^o \simeq A^o\times B^o. Now we have the composite equivalence (using the theorem on iterated fibrations):

Opf(A×B)Opf Opf(B)(A×BB)V BOpf Fib(B o)(A×B oB o)Fib(A,B o).Opf(A\times B) \simeq Opf_{Opf(B)}(A\times B \to B) \overset{V_B}{\to} Opf_{Fib(B^o)}(A\times B^o \to B^o) \simeq Fib(A,B^o).

Note that there is also another equivalence

Opf(A×B)V A×BFib((A×B) o)Fib Fib(A o)(A o×B oA o)V AFib Opf(A)(A×B oA)Fib(A,B o),Opf(A\times B) \overset{V_{A\times B}}{\to} Fib((A\times B)^o) \simeq Fib_{Fib(A^o)}(A^o\times B^o\to A^o) \overset{V_A}{\to} Fib_{Opf(A)}(A\times B^o\to A) \simeq Fib(A,B^o),

but the second and third data in the definition of a duality involution supply a canonical equivalence between these composites, so it doesn’t matter which we use. Finally, for the general case, we simply apply this twice:

Fib(A×B,C)Opf(A×B×C o)Opf(A×(B o×C) o)Fib(A,B o×C).Fib(A\times B, C) \simeq Opf(A\times B\times C^o) \simeq Opf(A\times (B^o\times C)^o) \simeq Fib(A, B^o\times C).

This gives our desired equivalence.

Note, though, that this proof crucially requires that the duality involution come with an equivalence Fib(X)Opf(X o)Fib(X)\simeq Opf(X^o), and not merely DFib(X)DOpf(X o)DFib(X)\simeq DOpf(X^o).

Commutation of opposites, which we have not yet used, is necessary for the following important, and perhaps also surprising, result: duality involutions are preserved by (fibrational) slicing.

Theorem

Any duality involution on KK induces a duality involution on each fibrational slice Fib K(X)Fib_K(X) and Opf K(X)Opf_K(X).

Proof

We define () o X:Fib(X) coFib(X)(-)^{o_X}:Fib(X)^{co}\to Fib(X) to be the composite

Fib(X)() oOpf(X o) coV X 1Fib(X) co.Fib(X) \overset{(-)^o}{\to} Opf(X^o)^{co} \overset{V_{X}^{-1}}{\to} Fib(X)^{co}.

To show that () o X(-)^{o_X} is an involution, we verify

(A o X) o X=V X 1((V X 1(A o)) o)V X 1(V X((A o) o))(A o) oA.(A^{o_X})^{o_X} = V_X^{-1}((V_X^{-1}(A^o))^o) \simeq V_X^{-1}(V_X((A^o)^o)) \simeq (A^o)^o \simeq A.

Here we use commutation of opposites in KK to commute () o(-)^o past V XV_X. We now define

V A X:Fib Fib(X)(A)Opf Fib(X)(A o X)V^X_A: Fib_{Fib(X)}(A) \simeq Opf_{Fib(X)}(A^{o_X})

to be the composite

Fib Fib(X)(A)Fib(A)V AOpf(A o)Opf Opf(X o)(A o)V X 1Opf Fib(X)(V X(A o))=Opf Fib(X)(A o X).Fib_{Fib(X)}(A) \simeq Fib(A) \overset{V_A}{\to} Opf(A^o) \simeq Opf_{Opf(X^o)}(A^o) \overset{V_X^{-1}}{\to} Opf_{Fib(X)}(V_X(A^o)) = Opf_{Fib(X)}(A^{o_X}).

It is easy to see that V 1 X XV X 1V X1V^X_{1_X}\simeq V_X^{-1}\circ V_X \simeq 1, and to construct the pullback-commutation equivalence. Finally, we construct commutation of opposites in Fib(X)Fib(X) as the composite

(V A X(B)) o XV X 1((V X 1(V A(B))) o)V X 1(V X((V A(B)) o))(V A(B)) oV A 1(B o)V A 1(V X(V X 1(B o)))=(V A X) 1(B o X).(V^X_A(B))^{o_X} \simeq V_X^{-1}((V_X^{-1}(V_A(B)))^o) \simeq V_X^{-1}(V_X((V_A(B))^o)) \simeq (V_A(B))^o\simeq V_A^{-1}(B^o)\simeq V_A^{-1}(V_X(V_X^{-1}(B^o))) = (V^X_A)^{-1}(B^{o_X}).

The case of Opf(X)Opf(X) is dual.

Fixing groupoids

We saw above that in general, a 2-category admitting an involution or duality involution will admit many. One way to get rid of some of these spurious involutions is to require that () o(-)^o fix groupoidal objects, as is clearly the case for the involution of CatCat and the “canonical” involutions of (2,1)-truncated 2-presheaf 2-toposes.

Note that any involution takes groupoidal objects to groupoidal objects.

Definition

An involution () o(-)^o on KK fixes groupoids it it restricts to the canonical involution on gpd(K)gpd(K), i.e. if X oXX^o\simeq X coherently whenever XX is groupoidal.

Lemma

If KK is a regular nn-category having enough groupoids, then it admits, up to equivalence, at most one involution that fixes groupoids. If KK is also nn-exact, then it admits exactly one such involution.

Proof

Suppose that () o(-)^o is an involution on KK that fixes groupoids. Then for any AA and any groupoidal XX, we have

K(X,A o)K co(X o,A)K co(X,A)K(X,A) op.K(X,A^o) \cong K^{co}(X^o,A) \cong K^{co}(X,A) \cong K(X,A)^{op}.

Now given BB as well, with an eso p:XBp:X\to B where XX is groupoidal, any morphism BA oB\to A^o is determined by a map XA oX\to A^o with an action by ker(p)ker(p). But ker(p)=(p/p)ker(p)=(p/p) is also groupoidal, so morphisms BA oB\to A^o are completely determined by morphisms to AA out of groupoidal objects. Thus, by the Yoneda lemma, any two values of A oA^o must be canonically isomorphic.

Now suppose KK is exact, and let p:XAp:X\to A be an eso where XX is groupoidal. Then its kernel (p/p)(p/p) is also groupoidal, and so ker(p)ker(p) is a homwise-discrete category in gpd(K)gpd(K). Thus we can consider its opposite, which is also a homwise-discrete category in gpd(K)gpd(K). In general, the opposite of a 2-congruence in a 2-category will not be a 2-congruence, since the condition of being a two-sided fibration is not preserved. But in a (2,1)-category such as gpd(K)gpd(K), this extra condition is automatic, so this opposite of is also a 2-congruence in gpd(K)gpd(K) (and, in fact, an nn-congruence when KK is an nn-category). Thus, since KK is nn-exact, this opposite has a quotient, which we call A oA^o. It is straightforward to verify that this defines an involution on KK that fixes groupoids.

Of course, the only interesting values of nn in the above lemma are 2 and (1,2), since any groupoid-fixing involution on a (2,1)-category is equivalent to the canonical one. Of particular note is that any (2,1)-truncated Grothendieck 2-topos admits a unique groupoid-fixing involution.

In fact, this groupoid-fixing involution should also extend to a duality involution. For when KK is nn-exact, the functor Opf:K op2CatOpf:K^{op}\to 2Cat ought to be a 3-sheaf? (a 2-stack). This means that if p:XAp:X\to A is an eso with XX groupoidal, then Opf(A)Opf(A) can be reconstructed from Opf(X)Opf(X) and Opf(p/p)Opf(p/p) with descent data. Similarly, Fib:K coop2CatFib:K^{coop}\to 2Cat should also be a 3-sheaf, and so Fib(A)Fib(A) can be recovered from Fib(X)Fib(X) and Fib(p/p)Fib(p/p) with “op-descent data.” But since XX is groupoidal, Opf(X)Fib(X)Opf(X)\simeq Fib(X), and likewise for (p/p)(p/p). Therefore, descent data for ker(p)ker(p) over Opf(X)Opf(X), which determines an object of Opf(A)Opf(A), is the same as op-descent data for the opposite of ker(p)ker(p) over Fib(X)Fib(X), which determines an object of Fib(A o)Fib(A^o). This gives the equivalence Opf(A)Fib(A o)Opf(A) \simeq Fib(A^{o}).

Alternately, there is also a stronger theorem that if KK is 2-exact with enough groupoids, then it is equivalent to a particular 2-category of internal categories and anafunctors in the (2,1)-category gpd(K)gpd(K) (more precisely, it is the 2-exact completion of gpd(K)gpd(K)). And it is fairly easy to see, by its construction, that this 2-category has a duality involution, just as the 2-category of internal categories in any 1-category does.

Modulo the action of automorphisms, these are the only examples of duality involutions that I know. I do not know an example of a duality involution on a 2-category that does not have enough groupoids.

Fixing groupoids locally

Now, since a duality involution on KK extends to a duality involution on Fib K(X)Fib_K(X) and Opf K(X)Opf_K(X), it is natural to strengthen the notion of groupoid-fixing so that it carries over to fibrational slices. We write GFib(X)=gpd(Fib(X))GFib(X) = gpd(Fib(X)) and similarly GOpf(X)GOpf(X).

Definition

A duality involution () o(-)^o on KK fixes groupoids locally if we have a equivalence modification between V X:GFib(X)GOpf(X o)V_X:GFib(X)\to GOpf(X^o) and the composite GFib(X)() oGOpf(X o) coGOpf(X o)GFib(X) \overset{(-)^o}{\to} GOpf(X^o)^{co} \simeq GOpf(X^o).

The latter equivalence GOpf(X o) coGOpf(X o)GOpf(X^o)^{co} \simeq GOpf(X^o) is the canonical involution on the (2,1)-category GOpf(X o)GOpf(X^o). Taking X=1X=1, together with the second condition in the definition of a duality involution, this implies that () o(-)^o fixes groupoids in the previous sense. Moreover, we have:

Theorem

If () o(-)^o is a duality involution that fixes groupoids locally, then the induced duality involutions on Fib(X)Fib(X) and Opf(X)Opf(X) also fix groupoids locally.

Proof

For a fibration AXA\to X, on Fib Fib(X)(A)Fib_{Fib(X)}(A) we have () o X=V X 1() o(-)^{o_X} = V_X^{-1}\circ (-)^o and V A X=V X 1V AV^X_A = V_X^{-1}\circ V_A, so the equivalence between () o(-)^o and V AV_A in KK induces one between () o X(-)^{o_X} and V A XV^X_A in Fib(X)Fib(X).

Now, suppose that () o(-)^o is a duality involution that fixes groupoids locally. Then for any AA and BB, the pullback-commutation datum shows that the composite equivalence

Fib(A×B o)Fib(B,A)Opf(A o×B)Opf((A×B o) o)Fib(A\times B^o) \simeq Fib(B,A) \simeq Opf(A^o\times B)\simeq Opf((A\times B^o)^o)

derived from Theorem 1 is equivalent to V A×B oV_{A\times B^o}. But if we restrict to groupoidal fibrations, then

V A×B o:GFib(A×B o)GOpf(A o×B)V_{A\times B^o}:GFib(A\times B^o) \simeq GOpf(A^o\times B)

is equivalent to () o(-)^o. Together with commutation of opposites, this implies that the canonical equivalence

GFib(B,A)GFib(A o,B o)GFib(B,A)\simeq GFib(A^o,B^o)

is, up to equivalence, simply given by () o(-)^o. In particular, since () o(-)^o preserves powers, the canonical equivalence

DFib(A,A)DFib(A o,A o)DFib(A,A)\simeq DFib(A^o,A^o)

takes A 2A^{\mathbf{2}} to (A o) 2(A^o)^{\mathbf{2}}. In the 2-internal logic, this corresponds to the statement that “hom A(x,y)hom A o(y,x)hom_A(x,y) \cong hom_{A^o}(y,x),” which is certainly an expected part of the behavior of opposite categories. Of course, for this it suffices that () o(-)^o fix discretes locally, which has the evident definition.

Further axioms

Another natural requirement is that when XX is groupoidal, V XV_X should be equivalent to the composite

Fib(X)K/XOpf(X)Opf(X o).Fib(X)\simeq K/X \simeq Opf(X)\simeq Opf(X^o).

Note that this includes the second datum in the definition of a duality involution as a special case, since 11 is groupoidal. There seems no reason for this condition to be stable under slicing, but it could be stabilized.

Revised on June 12, 2012 11:10:00 by Andrew Stacey? (129.241.15.200)