# nLab Isbell duality

Contents

### Context

#### Higher algebra

higher algebra

universal algebra

## Theorems

Ingredients

Concepts

Constructions

Examples

Theorems

#### Duality

duality

Examples

In QFT and String theory

higher geometry$\leftarrow$ Isbell duality $\rightarrow$ higher algebra

# Contents

## Idea

$(\mathcal{O} \dashv Spec) : CoPresheaves \underoverset{Spec}{\mathcal{O}}{\leftrightarrows} Presheaves$

relates (higher) presheaves with (higher) copresheaves on a given (higher) category $C$: this is called Isbell conjugation or Isbell duality (after John Isbell).

To the extent that this adjunction descends to presheaves that are (higher) sheaves and copresheaves that are (higher) algebras this duality relates higher geometry with higher algebra.

Objects preserved by the monad of this adjunction are called Isbell self-dual.

Under the interpretation of presheaves as generalized spaces and copresheaves as generalized quantities modeled on $C$ (Lawvere 86, see at space and quantity), Isbell duality is the archetype of the duality between geometry and algebra that permeates mathematics (such as Gelfand duality, Stone duality, or the embedding of smooth manifolds into formal duals of R-algebras).

## Definition

Let $\mathcal{V}$ be a good enriching category (a cosmos, i.e. a complete and cocomplete closed symmetric monoidal category).

Let $\mathcal{C}$ be a small $\mathcal{V}$-enriched category.

Write $[\mathcal{C}^{op}, \mathcal{V}]$ and $[\mathcal{C}, \mathcal{V}]$ for the enriched functor categories.

###### Proposition

There is a $V$-adjunction

$(\mathcal{O} \dashv Spec) \colon [C, \mathcal{V}]^{op} \underoverset{Spec}{\mathcal{O}}{\leftrightarrows} [C^{op}, \mathcal{V}]$

where

$\mathcal{O}(X) \colon c \mapsto [C^{op}, \mathcal{V}](X, C(-,c)) \,,$

and

$Spec(A) \colon c \mapsto [C, \mathcal{V}]^{op}(C(c,-),A) \,.$
###### Remark

This is also called Isbell duality. Objects which are preserved by $\mathcal{O} \circ Spec$ or $Spec \mathcal{O}$ are called Isbell self-dual.

The proof is mostly a tautology after the notation is unwound. The mechanism of the proof may still be of interest and be relevant for generalizations and for less tautological variations of the setup. We therefore spell out several proofs.

###### Proof A

Use the end-expression for the hom-objects of the enriched functor categories to compute

\begin{aligned} [C,\mathcal{V}]^{op}(\mathcal{O}(X), A) & := \int_{c \in C} \mathcal{V}(A(c), \mathcal{O}(X)(c)) \\ & := \int_{c \in C} \mathcal{V}(A(c), [C^{op}, \mathcal{V}](X, C(-,c))) \\ & := \int_{c \in C} \int_{d \in C} \mathcal{V}(A(c), \mathcal{V}(X(d), C(d,c))) \\ & \simeq \int_{d \in C} \int_{c \in C} \mathcal{V}(X(d), \mathcal{V}(A(c), C(d,c))) \\ & =: \int_{d \in C} \mathcal{V}(X(d), [C,\mathcal{V}]^{op}(C(d,-),A)) \\ & =: \int_{d \in C} \mathcal{V}(X(d), Spec(A)(d)) \\ & =: [C^{op}, \mathcal{V}](X, Spec(A)) \end{aligned} \,.
###### Remark

Here apart from writing out or hiding the ends, the only step that is not a definition is precisely the middle one, where we used that $\mathcal{V}$ is a symmetric closed monoidal category.

The following proof does not use ends and needs instead slightly more preparation, but has then the advantage that its structure goes through also in great generality in higher category theory.

###### Proof B

Notice that

Lemma 1: $Spec(C(c,-)) \simeq C(-,c)$

because we have a natural isomorphism

\begin{aligned} Spec(C(c,-))(d) & := [C,\mathcal{V}](C(c,-), C(d,-)) \\ & \simeq C(d,c) \end{aligned}

by the Yoneda lemma.

From this we get

Lemma 2: $[C^{op}, \mathcal{V}](Spec C(c,-), Spec A) \simeq [C,\mathcal{V}](A, C(c,-))$

by the sequence of natural isomorphisms

\begin{aligned} [C^{op}, \mathcal{V}](Spec C(c,-), Spec A) & \simeq [C^{op}, \mathcal{V}](C(-,c), Spec A) \\ & \simeq (Spec A)(c) \\ & := [C, \mathcal{V}](A, C(c,-)) \end{aligned} \,,

where the first is Lemma 1 and the second the Yoneda lemma.

Since (by what is sometimes called the co-Yoneda lemma) every object $X \in [C^{op}, \mathcal{V}]$ may be written as a colimit

$X \simeq {\lim_\to}_i C(-,c_i)$

over representables $C(-,c_i)$ we have

$X \simeq {\lim_\to}_i Spec(C(c_i,-)) \,.$

In terms of the same diagram of representables it then follows that

Lemma 3:

$\mathcal{O}(X) \simeq {\lim_{\leftarrow}}_i C(c_i,-)$

because using the above colimit representation and the Yoneda lemma we have natural isomorphisms

\begin{aligned} \mathcal{O}(X)(d) &= [C^{op}, \mathcal{V}](X, C(-,c)) \\ & \simeq [C^{op}, \mathcal{V}]({\lim_\to}_i C(-,c_i), C(-,c)) \\ & \simeq {\lim_\leftarrow}_i [C^{op}, \mathcal{V}](C(-,c_i), C(-,c)) \\ & \simeq {\lim_\leftarrow}_i C(c_i,c) \end{aligned} \,.

Using all this we can finally obtain the adjunction in question by the following sequence of natural isomorphisms

\begin{aligned} [C,\mathcal{V}]^{op}(\mathcal{O}(X), A) & \simeq [C,\mathcal{V}](A, {\lim_\leftarrow}_i C(c_i,-)) \\ & \simeq {\lim_{\leftarrow}}_i [C, \mathcal{V}](A, C(c_i,-)) \\ & \simeq {\lim_{\leftarrow}}_i [C^{op}, \mathcal{V}](Spec C(c_i,-), Spec A) \\ & \simeq [C^{op}, \mathcal{V}]({\lim_{\to}}_i Spec C(c_i,-), Spec A) \\ & \simeq [C^{op}, \mathcal{V}](X, Spec A) \end{aligned} \,.

The pattern of this proof has the advantage that it goes through in great generality also on higher category theory without reference to a higher notion of enriched category theory.

###### Remark

Under certain circumstances, Isbell duality can be extended to large $\mathcal{V}$-enriched categories $C$. For example, if $C$ has a small generating subcategory $S$ and a small cogenerating subcategory $T$, then for each $F: C^{op} \to \mathcal{V}$ and $G: C \to \mathcal{V}$, one may construct $\mathcal{O}(F)$ and $Spec(G)$ objectwise as appropriate subobjects in $\mathcal{V}$:

$\mathcal{O}(F)(c) = [C^{op}, \mathcal{V}](F, C(-, c)) \hookrightarrow \int_{s: S} \mathcal{V}(F s, \hom(s, c))$
$Spec(G)(c) = [C, \mathcal{V}](G, C(c, -)) \hookrightarrow \int_{t: T} \mathcal{V}(G t, \hom(c, t))$

## Example

In the simplest case, namely for an ordinary category $\mathcal{C}$, the adjunction between presheaves and copresheaves arises as follows.

The category of presheaves $[\mathcal{C}^{op}, \mathrm{Set}]$ is the free cocompletion of $\mathcal{C}$. This means that any functor

$f \colon \mathcal{C} \to \mathcal{D}$

to a cocomplete category $\mathcal{D}$ extends along the Yoneda embedding $y \colon \mathcal{C} \to [\mathcal{C}^{op}, \mathrm{Set}]$ to a cocontinuous functor

$F \colon [\mathcal{C}^{op}, \mathrm{Set}] \to \mathcal{D}$

in a manner unique up to natural isomorphism.

Dually, the category of copresheaves $[\mathcal{C}, \mathrm{Set}]^{op}$ is the free completion of $\mathcal{C}$. This means that any functor

$g \colon \mathcal{C} \to \mathcal{D}$

to a complete category $\mathcal{D}$ extends along the co-Yoneda embedding $z \colon \mathcal{C} \to [\mathcal{C}, \mathrm{Set}]^{op}$ to a continuous functor

$G \colon [\mathcal{C}, \mathrm{Set}]^{op} \to \mathcal{D}$

in a manner unique up to natural isomorphism.

We can apply these ideas to get the functors involved in Isbell duality. The presheaf category $[\mathcal{C}^{op}, \mathrm{Set}]$ has all limits, so we can extend the Yoneda embedding to a continuous functor

$Y \colon [\mathcal{C}, \mathrm{Set}]^{op} \to [\mathcal{C}^{op}, \mathrm{Set}]$

from copresheaves to presheaves. Dually, the copresheaf category $[\mathcal{C}, \mathrm{Set}]^{op}$ has all colimits, so we can extend the co-Yoneda embedding to a cocontinuous functor

$Z \colon [\mathcal{C}^{op}, \mathrm{Set}] \to [\mathcal{C}, \mathrm{Set}]^{op}$

from presheaves to copresheaves.

Isbell duality says that these are adjoint functors: $Y$ is right adjoint to $Z$.

## Properties

### Relation to Yoneda embedding

$Spec$ is the left Kan extension of the Yoneda embedding along the contravariant Yoneda embedding, while $\mathcal{O}$ is the left Kan extension of the contravariant Yoneda embedding along the Yoneda embedding.

The codensity monad of the Yoneda embedding is isomorphic to the monad induced by the Isbell adjunction, $Spec \mathcal{O}$ (Kock 66, Theorem 4.1 and Di Liberti 19, Theorem 2.7).

### Respect for limits

Choose any class $L$ of limits in $C$ and write $[C,\mathcal{V}]_\times \subset [C,\mathcal{V}]$ for the full subcategory consisting of those functors preserving these limits.

###### Proposition

The $(\mathcal{O} \dashv Spec)$-adjunction does descend to this inclusion, in that we have an adjunction

$(\mathcal{O} \dashv Spec) : [C, \mathcal{V}]_{\times}^{op} \stackrel{\overset{\mathcal{O}}{\leftarrow}}{\underset{Spec}{\to}} [C^{op}, \mathcal{V}]$
###### Proof

Because the hom-functors preserves all limits:

\begin{aligned} \mathcal{O}(X)({\lim_{\leftarrow}}_j c_j) & := [C^{op}, \mathcal{V}](X,C(-,{\lim_{\leftarrow}}_j c_j)) \\ & \simeq [C^{op}, \mathcal{V}](X,{\lim_{\leftarrow}}_j C(-,c_j)) \\ & \simeq {\lim_{\leftarrow}}_j [C^{op}, \mathcal{V}](X,C(-,c_j)) \\ & =: {\lim_{\leftarrow}}_j \mathcal{O}(X)(c_j) \end{aligned} \,.

### Isbell self-dual objects

###### Definition

An object $X$ or $A$ is Isbell-self-dual if

• $A \stackrel{}{\to} \mathcal{O} Spec(A)$ is an isomorphism in $[C,\mathcal{V}]$;

• $X \to Spec \mathcal{O} X$ is an isomorphism in $[C^{op}, \mathcal{V}]$, respectively.

###### Proposition

All representables are Isbell self-dual.

###### Proof

By Proof B , lemma 1 we have a natural isomorphisms in $c \in C$

$Spec(C(c,-)) \simeq C(-,c) \,.$

Therefore we have also the natural isomorphism

\begin{aligned} \mathcal{O} Spec C(c,-)(d) & \simeq \mathcal{O} C(-,c) (d) \\ & := [C^{op}, \mathcal{V}](C(-,c), C(-,d)) \\ & \simeq C(c,d) \end{aligned} \,,

where the second step is the Yoneda lemma. Similarly the other way round.

### Isbell envelope

See Isbell envelope.

## Examples and similar dualities

Isbell duality is a template for many other space/algebra-dualities in mathematics.

### Function $T$-Algebras on presheaves

Let $\mathcal{V}$ be any cartesian closed category.

Let $C := T$ be the syntactic category of a $\mathcal{V}$-enriched Lawvere theory, that is a $\mathcal{V}$-category with finite products such that all objects are generated under products from a single object $1$.

Then write $T Alg := [C,\mathcal{V}]_\times$ for category of product-preserving functors: the category of $T$-algebras. This comes with the canonical forgetful functor

$U_T : T Alg \to \mathcal{V} : A \mapsto A(1)$

Write

$F_T : T^{op} \hookrightarrow T Alg$

for the Yoneda embedding.

###### Definition

Call

$\mathbb{A}_T := Spec(F_T(1)) \in [C^{op}, \mathcal{V}]$

the $T$-line object.

###### Observation

For all $X \in [C^{op}, \mathcal{V}]$ we have

$\mathcal{O}(X) \simeq [C^{op}, \mathcal{V}](X, Spec(F_T(-))) \,.$

In particular

$U_T(\mathcal{O}(X)) \simeq [C^{op}, \mathcal{V}](X,\mathbb{A}_T) \,.$
###### Proof

We have isomorphisms natural in $k \in T$

\begin{aligned} [C^{op}, \mathcal{V}](X, Spec(F_T(k))) & \simeq T Alg(F_T(k), \mathcal{O}(X)) \\ & \simeq \mathcal{O}(X)(k) \end{aligned}

by the above adjunction and then by the Yoneda lemma.

All this generalizes to the following case:

instead of setting $C := T$ let more generally

$T \subset C \subset T Alg^{op}$

be a small full subcategory of $T$-algebras, containing all the free $T$-algebras.

Then the original construction of $\mathcal{O} \dashv Spec$ no longer makes sense, but that in terms of the line object still does

###### Proposition

Set

$Spec A : B \mapsto T Alg(A,B)$

and

$\mathcal{O}(X) : k \mapsto [C^{op}, \mathcal{V}](X, Spec(F_T(k))) \,.$

Then we still have an adjunction

$(\mathcal{O} \dashv Spec) : T Alg^{op} \stackrel{\overset{\mathcal{O}}{\leftarrow}}{\underset{Spec}{\to}} [C^{op}, \mathcal{V}] \,.$
###### Proof
\begin{aligned} T Alg^{op}(\mathcal{O}(X), A) & := \int_{k \in T} \mathcal{V}( A(k), \mathcal{O}(X)(k) ) \\ & := \int_{k \in T} \mathcal{V}( A(k), [C^{op}, \mathcal{V}](X, Spec(F_T(k))) ) \\ & := \int_{k \in T} \int_{B \in C} \mathcal{V}(A(k), \mathcal{V}(X(B), T Alg(F_T(k), B) )) \\ & \simeq \int_{k \in T} \int_{B \in C} \mathcal{V}(A(k), \mathcal{V}(X(B), B(k) )) \\ & \simeq \int_{k \in T} \int_{B \in C} \mathcal{V}(X(B), \mathcal{V}(A(k), B(k) )) \\ & =: \int_{B \in C} \mathcal{V}(X(B), T Alg(A,B)) \\ & =: \int_{B \in C} \mathcal{V}(X(B), Spec(A)(B)) \\ & =: [C^{op}, Set](X,Spec(A)) \end{aligned} \,.

The first step that is not a definition is the Yoneda lemma. The step after that is the symmetric-closed-monoidal structure of $\mathcal{V}$.

### Function $k$-algebras on derived $\infty$-stacks

The structure of our Proof B above goes through in higher category theory.

Formulated in terms of derived stacks over the (∞,1)-category of dg-algebras, this is essentially the argument appearing on page 23 of (Ben-ZviNadler).

### Function $T$-algebras on $\infty$-stacks

for the moment see at function algebras on ∞-stacks.

### Gelfand duality

Gelfand duality is the equivalence of categories between (nonunital) commutative C*-algebras and (locally) compact topological spaces. See there for more details.

### Serre-Swan theorem

The Serre-Swan theorem says that suitable modules over an commutative C*-algebra are equivalently modules of sections of vector bundles over the Gelfand-dual topological space.

duality between $\;$algebra and geometry

$\phantom{A}$geometry$\phantom{A}$$\phantom{A}$category$\phantom{A}$$\phantom{A}$dual category$\phantom{A}$$\phantom{A}$algebra$\phantom{A}$
$\phantom{A}$topology$\phantom{A}$$\phantom{A}$$\phantom{NC}TopSpaces_{H,cpt}$$\phantom{A}$$\phantom{A}$$\overset{\text{Gelfand-Kolmogorov}}{\hookrightarrow} Alg^{op}_{\mathbb{R}}$$\phantom{A}$$\phantom{A}$commutative algebra$\phantom{A}$
$\phantom{A}$topology$\phantom{A}$$\phantom{A}$$\phantom{NC}TopSpaces_{H,cpt}$$\phantom{A}$$\phantom{A}$$\overset{\text{Gelfand duality}}{\simeq} TopAlg^{op}_{C^\ast, comm}$$\phantom{A}$$\phantom{A}$comm. C-star-algebra$\phantom{A}$
$\phantom{A}$noncomm. topology$\phantom{A}$$\phantom{A}$$NCTopSpaces_{H,cpt}$$\phantom{A}$$\phantom{A}$$\overset{\phantom{\text{Gelfand duality}}}{\coloneqq} TopAlg^{op}_{C^\ast}$$\phantom{A}$$\phantom{A}$general C-star-algebra$\phantom{A}$
$\phantom{A}$algebraic geometry$\phantom{A}$$\phantom{A}$$\phantom{NC}Schemes_{Aff}$$\phantom{A}$$\phantom{A}$$\overset{\text{almost by def.}}{\hookrightarrow} \phantom{Top}Alg^{op}_{fin}$$\phantom{A}$$\phantom{A}$fin. gen.$\phantom{A}$
$\phantom{A}$commutative algebra$\phantom{A}$
$\phantom{A}$noncomm. algebraic$\phantom{A}$
$\phantom{A}$geometry$\phantom{A}$
$\phantom{A}$$NCSchemes_{Aff}$$\phantom{A}$$\phantom{A}$$\overset{\phantom{\text{Gelfand duality}}}{\coloneqq} \phantom{Top}Alg^{op}_{fin, red}$$\phantom{A}$$\phantom{A}$fin. gen.
$\phantom{A}$associative algebra$\phantom{A}$$\phantom{A}$
$\phantom{A}$differential geometry$\phantom{A}$$\phantom{A}$$SmoothManifolds$$\phantom{A}$$\phantom{A}$$\overset{\text{Milnor's exercise}}{\hookrightarrow} \phantom{Top}Alg^{op}_{comm}$$\phantom{A}$$\phantom{A}$commutative algebra$\phantom{A}$
$\phantom{A}$supergeometry$\phantom{A}$$\phantom{A}$$\array{SuperSpaces_{Cart} \\ \\ \mathbb{R}^{n\vert q}}$$\phantom{A}$$\phantom{A}$$\array{ \overset{\phantom{\text{Milnor's exercise}}}{\hookrightarrow} & Alg^{op}_{\mathbb{Z}_2 \phantom{AAAA}} \\ \mapsto & C^\infty(\mathbb{R}^n) \otimes \wedge^\bullet \mathbb{R}^q }$$\phantom{A}$$\phantom{A}$supercommutative$\phantom{A}$
$\phantom{A}$superalgebra$\phantom{A}$
$\phantom{A}$formal higher$\phantom{A}$
$\phantom{A}$supergeometry$\phantom{A}$
$\phantom{A}$(super Lie theory)$\phantom{A}$
$\phantom{A}\array{ Super L_\infty Alg_{fin} \\ \mathfrak{g} }\phantom{A}$$\phantom{A}\array{ \overset{ \phantom{A}\text{Lada-Markl}\phantom{A} }{\hookrightarrow} & sdgcAlg^{op} \\ \mapsto & CE(\mathfrak{g}) }\phantom{A}$$\phantom{A}$differential graded-commutative$\phantom{A}$
$\phantom{A}$superalgebra
$\phantom{A}$ (“FDAs”)

in physics:

$\phantom{A}$algebra$\phantom{A}$$\phantom{A}$geometry$\phantom{A}$
$\phantom{A}$Poisson algebra$\phantom{A}$$\phantom{A}$Poisson manifold$\phantom{A}$
$\phantom{A}$deformation quantization$\phantom{A}$$\phantom{A}$geometric quantization$\phantom{A}$
$\phantom{A}$algebra of observables$\phantom{A}$space of states$\phantom{A}$
$\phantom{A}$Heisenberg picture$\phantom{A}$Schrödinger picture$\phantom{A}$
$\phantom{A}$AQFT$\phantom{A}$$\phantom{A}$FQFT$\phantom{A}$
$\phantom{A}$higher algebra$\phantom{A}$$\phantom{A}$higher geometry$\phantom{A}$
$\phantom{A}$Poisson n-algebra$\phantom{A}$$\phantom{A}$n-plectic manifold$\phantom{A}$
$\phantom{A}$En-algebras$\phantom{A}$$\phantom{A}$higher symplectic geometry$\phantom{A}$
$\phantom{A}$BD-BV quantization$\phantom{A}$$\phantom{A}$higher geometric quantization$\phantom{A}$
$\phantom{A}$factorization algebra of observables$\phantom{A}$$\phantom{A}$extended quantum field theory$\phantom{A}$
$\phantom{A}$factorization homology$\phantom{A}$$\phantom{A}$cobordism representation$\phantom{A}$

## References

The original articles on Isbell duality and the Isbell envelope are

More recent discussion:

Isbell conjugacy for (∞,1)-presheaves over the (∞,1)-category of duals of dg-algebras is discussed around page 32 of

in

Isbell self-dual ∞-stacks over duals of commutative associative algebras are called affine stacks. They are characterized as those objects that are small in a sense and local with respect to the cohomology with coefficients in the canonical line object.

A generalization of this latter to $\infty$-stacks over duals of algebras over arbitrary abelian Lawvere theories is the content of

• Herman Stel, $\infty$-Stacks and their function algebras – with applications to $\infty$-Lie theory, master thesis (2010) (web)