# nLab holographic principle of higher category theory

### Context

#### Higher category theory

higher category theory

# Contents

## The general abstract principle

In higher category theory it is easy to verify that a (strict) $1$-transformation $\lambda$ between n-functors ${F}_{1},{F}_{2}:C\to D$ between strict n-categories $C$ and $D$

$\lambda :{F}_{1}⇒{F}_{2}$\lambda : F_1 \Rightarrow F_2

is determined uniquely by an $\left(n-1\right)$-functor

$\eta :{C}_{\left(n-1\right)}\to {D}^{{\Delta }^{1}}$\eta : C_{(n-1)} \to D^{\Delta^1}

on the strict $\left(n-1\right)$-category obtained from $C$ by discarding the n-morphisms. (Of course, not every such $\left(n-1\right)$-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 $n$-functor categories for all $n$.

### Examples in low dimension

We spell out explicitly the $\left(n-1\right)$-functorial nature of transformation for low values of $n$.

• $\left(n=1\right)$ – A natural transformation $\eta$ between functors ${F}_{1},{F}_{2}:C\to D$ between ordinary categories consists of components which are given by a function

$\eta :\mathrm{Obj}\left(C\right)\to \mathrm{Mor}\left(D\right)$\eta : Obj(C) \to Mor(D)

that sends objects of $C$ to morphisms in $D$

$\eta :x↦\left({F}_{1}\left(x\right)\stackrel{\eta \left(x\right)}{\to }{F}_{2}\left(x\right)\right)\phantom{\rule{thinmathspace}{0ex}}.$\eta : x \mapsto ( F_1(x) \stackrel{\eta(x)}{\to} F_2(x)) \,.

Saying that such a function extends to a functor $C\to \mathrm{Arr}\left(D\right)$:

$\eta :\left(\begin{array}{c}x\\ {↓}^{{\gamma }_{1}}\\ y\\ {↓}^{{\gamma }_{2}}\\ z\end{array}\right)\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}↦\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\left(\begin{array}{ccc}{F}_{1}\left(x\right)& \stackrel{\eta \left(x\right)}{\to }& {F}_{2}\left(x\right)\\ {}^{{F}_{1}\left(f\right)}↓& =& {↓}^{{F}_{2}\left(f\right)}\\ {F}_{1}\left(y\right)& \stackrel{\eta \left(y\right)}{\to }& {F}_{2}\left(y\right)\\ {}^{{F}_{1}\left(g\right)}↓& =& {↓}^{{F}_{2}\left(g\right)}\\ {F}_{1}\left(z\right)& \underset{\eta \left(z\right)}{\to }& {F}_{2}\left(z\right)\end{array}\right)\phantom{\rule{thinmathspace}{0ex}}.$\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 $D$—in other words, the forgetful functor $\mathrm{Arr}\left(D\right)\to D×D$ is faithful—such an extension to a functor is necessarily unique.

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

$\eta :{\mathrm{sk}}_{0}C=\mathrm{Obj}\left(C\right)\to {D}^{\Delta \left[1\right]}=\mathrm{Arr}\left(D\right)$\eta : \mathbf{sk}_0 C = Obj(C) \to D^{\Delta[1]} = Arr(D)

from the discrete category on the set of objects of $C$ to the arrow category of $D$.

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

$\eta :\left(\begin{array}{c}x\\ {↓}^{{\gamma }_{1}}\\ y\\ {↓}^{{\gamma }_{2}}\\ z\end{array}\right)\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}↦\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\left(\begin{array}{ccc}{F}_{1}\left(x\right)& \stackrel{\eta \left(x\right)}{\to }& {F}_{2}\left(x\right)\\ {}^{{F}_{1}\left(f\right)}↓& {⇙}_{\eta \left(f\right)}& {↓}^{{F}_{2}\left(f\right)}\\ {F}_{1}\left(y\right)& \stackrel{\eta \left(y\right)}{\to }& {F}_{2}\left(y\right)\\ {}^{{F}_{1}\left(g\right)}↓& {⇙}_{\eta \left(g\right)}& {↓}^{{F}_{2}\left(g\right)}\\ {F}_{1}\left(z\right)& \underset{\eta \left(z\right)}{\to }& {F}_{2}\left(z\right)\end{array}\right)$\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

$\eta :{\mathrm{sk}}_{1}C\to \mathrm{Arr}\left(D\right)$\eta : \mathbf{sk}_1 C \to Arr(D)

from the underlying 1-category of $C$ to the arrow category of $D$, whose objects are morphisms in $D$, whose morphisms are squares in $D$, 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 $C$ to the arrow 2-category of $D$ says precisely that these components form a pseudonatural transformation, and any such extension is unique when it exists since the forgetful 2-functor $\mathrm{Arr}\left(D\right)\to D×D$ is locally faithful.

• $\left(n=3\right)$ – A transformation between 3-functors is in components a 2-functor that sends 2-morphisms in $C$ to cyclinders in $D$. This is shown in the $\left(n=3\right)$-row of the following diagram

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

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

$\eta :C\to \mathrm{Arr}\left(D\right)\phantom{\rule{thinmathspace}{0ex}},$\eta : C \to Arr(D) \,,

where on the right we have the 2-category whose objects are morphisms in $D$, whose morphisms are squares in $D$ 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 $C,D\in \mathrm{Str}n\mathrm{Cat}$ and ${F}_{1},{F}_{2}:C\to D$ two strict $n$-functors, transformations $\eta :{F}_{1}⇒{F}_{2}$ which are in components given by $n$-functors

$\eta :C\to {D}^{{G}_{1}}$\eta : C \to D^{G_1}

are entirely specified by their underlying $\left(n-1\right)$-functors

$\eta :{C}_{n-1}\to {D}^{{G}_{1}}\phantom{\rule{thinmathspace}{0ex}}.$\eta : C_{n-1} \to D^{G_1} \,.

For weak $n$-categories analogous statements hold, but may have a less straightforward formulation. What is always true is that the transformation $\eta$ is specified by its values on $\left(n-1\right)$-morphisms (and below) and will be functorial in a weak sense on these, but these $\left(n-1\right)$-morphisms and below will usually not form an $\left(n-1\right)$-category themselves, since they will compose coherently only up to $n$-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 $C$ is such a semistrict model of an $n$-category, then ${C}_{n-1}$ is an $\left(n-1\right)$-semicategory and the transformation

$\eta :{C}_{n-1}\to {D}^{\Delta \left[1\right]}$\eta : C_{n-1} \to D^{\Delta[1]}

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

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

## Application in functorial QFT

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

$Z:{\mathrm{Bord}}_{n}^{S}\to 𝒞$Z : Bord_{n}^S \to \mathcal{C}

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

It follows that with ${Z}_{1},{Z}_{2}$ two such $n$-dimensional QFTs, a transformation $B:{Z}_{1}⇒{Z}_{2}$ does look in components itself like an QFT – which is twisted by ${Z}_{1}$ and ${Z}_{2}$ in some sense (see below) – , but in dimension $\left(n-1\right)$.

More specifically, if $𝒞$ is a symmetric monoidal (∞,n)-category with tensor unit $1$ there is the trivial FQFT $1$ given by the constant $\left(\infty ,n\right)$-functor $1:{\mathrm{Bord}}_{n}\to 𝒞$.

One can see in examples that the transformations

$B:Z⇒1$B : Z \Rightarrow \mathbf{1}

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

For the case of $n=2$ and 2-dimensional cobordisms without any extra structure, a detailed version of these statements are given in (Schommer-Pries). For $n=3$ 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 $n$-dimensional QFTs and $\left(n-1\right)$-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

## 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)