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holographic principle of higher category theory

Context

Higher category theory

higher category theory

Basic concepts

Basic theorems

Applications

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Morphisms

Functors

Universal constructions

Extra properties and structure

1-categorical presentations

Functorial quantum field theory

Contents

The general abstract principle

In higher category theory it is easy to verify that a (strict) 11-transformation λ\lambda between n-functors F 1,F 2:CDF_1,F_2 : C \to D between strict n-categories CC and DD

λ:F 1F 2 \lambda : F_1 \Rightarrow F_2

is determined uniquely by an (n1)(n-1)-functor

η:C (n1)D Δ 1 \eta : C_{(n-1)} \to D^{\Delta^1}

on the strict (n1)(n-1)-category obtained from CC by discarding the n-morphisms. (Of course, not every such (n1)(n-1)-functor determines such a transformation; the missing condition is “naturality” at the top level.)

Analogous statements hold for general (weak) n-categories, although they are more complicated to formulate; see below.

As with various other easy facts about category theory, these become interesting statements when realized in a concrete context where certain structures are modeled by nn-functor categories for all nn.

Examples in low dimension

We spell out explicitly the (n1)(n-1)-functorial nature of transformation for low values of nn.

  • (n=1)(n=1) – A natural transformation η\eta between functors F 1,F 2:CDF_1,F_2 : C \to D between ordinary categories consists of components which are given by a function

    η:Obj(C)Mor(D) \eta : Obj(C) \to Mor(D)

    that sends objects of CC to morphisms in DD

    η:x(F 1(x)η(x)F 2(x)). \eta : x \mapsto ( F_1(x) \stackrel{\eta(x)}{\to} F_2(x)) \,.

    Saying that such a function extends to a functor CArr(D)C \to Arr(D):

    η:(x γ 1 y γ 2 z)(F 1(x) η(x) F 2(x) F 1(f) = F 2(f) F 1(y) η(y) F 2(y) F 1(g) = F 2(g) F 1(z) η(z) F 2(z)). \eta : \left( \array{ x \\ \downarrow^{\mathrlap{\gamma_1}} \\ y \\ \downarrow^{\mathrlap{\gamma_2}} \\ z } \right) \;\;\; \mapsto \;\;\; \left( \array{ F_1(x) &\stackrel{\eta(x)}{\to}& F_2(x) \\ {}^{\mathllap{F_1(f)}}\downarrow &=& \downarrow^{\mathrlap{F_2(f)}} \\ F_1(y) &\stackrel{\eta(y)}{\to}& F_2(y) \\ {}^{\mathllap{F_1(g)}}\downarrow &=& \downarrow^{\mathrlap{F_2(g)}} \\ F_1(z) &\underset{\eta(z)}{\to}& F_2(z) } \right) \,.

    is equivalent to saying that these components form a natural transformation. Since there are no nontrivial 2-morphisms in DD—in other words, the forgetful functor Arr(D)D×DArr(D) \to D\times D is faithful—such an extension to a functor is necessarily unique.

    So we may regard the component function of η\eta as a 0-functor

    η:sk 0C=Obj(C)D Δ[1]=Arr(D) \eta : \mathbf{sk}_0 C = Obj(C) \to D^{\Delta[1]} = Arr(D)

    from the discrete category on the set of objects of CC to the arrow category of DD.

  • (n=2)(n=2) A pseudonatural transformation η\eta between (strict, say, for ease of of notation) 2-functors F 1,F 2:CDF_1,F_2 : C \to D between (strict, for simplicity) 2-categories is in components a 1-functor that functorially assigns pseudonaturality squares:

    η:(x γ 1 y γ 2 z)(F 1(x) η(x) F 2(x) F 1(f) η(f) F 2(f) F 1(y) η(y) F 2(y) F 1(g) η(g) F 2(g) F 1(z) η(z) F 2(z)) \eta : \left( \array{ x \\ \downarrow^{\mathrlap{\gamma_1}} \\ y \\ \downarrow^{\mathrlap{\gamma_2}} \\ z } \right) \;\;\; \mapsto \;\;\; \left( \array{ F_1(x) &\stackrel{\eta(x)}{\to}& F_2(x) \\ {}^{\mathllap{F_1(f)}}\downarrow &\swArrow_{\eta(f)}& \downarrow^{\mathrlap{F_2(f)}} \\ F_1(y) &\stackrel{\eta(y)}{\to}& F_2(y) \\ {}^{\mathllap{F_1(g)}}\downarrow &\swArrow_{\eta(g)}& \downarrow^{\mathrlap{F_2(g)}} \\ F_1(z) &\underset{\eta(z)}{\to}& F_2(z) } \right)

    We may regard this as a 1-functor

    η:sk 1CArr(D) \eta : \mathbf{sk}_1 C \to Arr(D)

    from the underlying 1-category of CC to the arrow category of DD, whose objects are morphisms in DD, whose morphisms are squares in DD, and whose composition is pasting of such squares (see double category for details).

    Again, saying that this 1-functor extends to a 2-functor from CC to the arrow 2-category of DD says precisely that these components form a pseudonatural transformation, and any such extension is unique when it exists since the forgetful 2-functor Arr(D)D×DArr(D)\to D\times D is locally faithful.

  • (n=3)(n=3) – A transformation between 3-functors is in components a 2-functor that sends 2-morphisms in CC to cyclinders in DD. This is shown in the (n=3)(n=3)-row of the following diagram

    components of transformations

    The pseudonaturality condition on η\eta, which is componentwise the equation

    2-pseudonaturality

    and the fact that there are only identity 3-morphisms in DD implies that this already uniquely extends to a 2-functor

    η:CArr(D), \eta : C \to Arr(D) \,,

    where on the right we have the 2-category whose objects are morphisms in DD, whose morphisms are squares in DD and whose 2-morphisms are cylinders bounded by these squares.

Formalizations

For strict ∞-categories modeled as globular strict ω-categories we have the following simple statement of the general principle.

Observation

For C,DStrnCatC,D \in Str n Cat and F 1,F 2:CDF_1, F_2 : C \to D two strict nn-functors, transformations η:F 1F 2\eta : F_1 \Rightarrow F_2 which are in components given by nn-functors

η:CD G 1 \eta : C \to D^{G_1}

are entirely specified by their underlying (n1)(n-1)-functors

η:C n1D G 1. \eta : C_{n-1} \to D^{G_1} \,.

For weak nn-categories analogous statements hold, but may have a less straightforward formulation. What is always true is that the transformation η\eta is specified by its values on (n1)(n-1)-morphisms (and below) and will be functorial in a weak sense on these, but these (n1)(n-1)-morphisms and below will usually not form an (n1)(n-1)-category themselves, since they will compose coherently only up to nn-morphisms.

One way to bring the general case into the above simple form is to invoke models by semi-strict ∞-categories. By Simpson's conjecture, every ∞-category has a model in which everything is strict except possibly the identities and their unitalness coherence laws. This means that if CC is such a semistrict model of an nn-category, then C n1C_{n-1} is an (n1)(n-1)-semicategory and the transformation

η:C n1D Δ[1] \eta : C_{n-1} \to D^{\Delta[1]}

is an nn-functor on that. (By naturalness we have that η\eta is guaranteed also to respect the weak identities in CC in some way, but that way is not so easy to formalize.)

More generally, for any algebraic notion of weak nn-category, there is a corresponding algebraic “(n1)(n-1)-dimensional” structure containing only the operations on (n1)(n-1)-dimensional cells and below in the given notion of weak nn-category. This is not in general a notion of weak (n1)(n-1)-category, but it may suffice to formulate the above principle precisely. If the starting notion of nn-category had strict associativity and interchange, then the resulting (n1)(n-1)-dimensional structure will be a notion of (n1)(n-1)-semicategory.

Application in functorial QFT

For instance in FQFT one models nn-dimensional topological quantum field theories as (∞,n)-functors on a flavor of an (∞,n)-category of cobordisms

Z:Bord n S𝒞 Z : Bord_{n}^S \to \mathcal{C}

(where the superscript SS is to remind us that this may be cobordisms equipped with some extra structure).

It follows that with Z 1,Z 2Z_1, Z_2 two such nn-dimensional QFTs, a transformation B:Z 1Z 2B : Z_1 \Rightarrow Z_2 does look in components itself like an QFT – which is twisted by Z 1Z_1 and Z 2Z_2 in some sense (see below) – , but in dimension (n1)(n-1).

More specifically, if 𝒞\mathcal{C} is a symmetric monoidal (∞,n)-category with tensor unit 11 there is the trivial FQFT 1\mathbf{1} given by the constant (,n)(\infty,n)-functor 1:Bord n𝒞\mathbf{1} : Bord_n \to \mathcal{C}.

One can see in examples that the transformations

B:Z1 B : Z \Rightarrow \mathbf{1}

encode boundary conditions on cobordisms with boundary for the theory ZZ. Conversely, this means that one discovers on the boundary of the nn-dimensional QFT ZZ the (n1)(n-1)-dimensional QFT BB. Or rather, this is the case if instead of natural transformations η\eta one uses canonical transformations: those component maps η:C n1D I\eta : C_{n-1} \to D^{I} that are required to be natural only with respect to the invertible (n1)(n-1)-morphisms in CC.

For the case of n=2n=2 and 2-dimensional cobordisms without any extra structure, a detailed version of these statements are given in (Schommer-Pries). For n=3n=3 and the holographic relation between Reshetikhin–Turaev model and rational 2d CFT in FFRS-formalism some remarks are in (Schreiber).

In the study of quantum field theory and string theory such kinds of relations between nn-dimensional QFTs and (n1)(n-1)-dimensional QFTs on their boundary have been called the holographic principle . The degree to which this principle has been formalized and the degree to which this formalization has been verified varies greatly. Examples include

See also higher category theory and physics.

References

The discussion of transformations between 2d FQFTs and how these encode boundary 1-branes and defect 1-bi-branes is in

  • Chris Schommer-Pries, Topological defects and classifying local topological field theories in low dimension (pdf)

from slide 65 on.

A formally comparatively well understood case of QFT holography is the relation between 3-dimensional Chern-Simons theory and the 2-dimensional WZW-model. This is formalized by the Reshetikhin–Turaev model on the 3-dimensional side and the Fuchs-Runkel-Schweigert construction on the 2-dimensional side.

Remarks on how the relation between Reshitikhin-Turaev and FSR seem to have an interpretation in terms transformations between 3-functors are at

There is it discussed how the basic string diagram that in FSR formalism encodes a field insertion on, possibly, a defect line and encodes the disk amplittudes of the CFT is the string diagram Poincaré-dual to the cylinder in a 3-category of 3-vector spaces.

For references on the holographic principle in QFT, see there.

Revised on July 21, 2011 10:19:29 by Urs Schreiber (89.204.153.85)