nLab
Isbell duality

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

---

Idea

A general abstract adjunction

$$(𝒪⊣\mathrm{Spec}):\mathrm{CoPresheaves}\stackrel{\stackrel{𝒪}{\leftarrow }}{\underset{\mathrm{Spec}}{\to }}\mathrm{Presheaves}$$(\mathcal{O} \dashv Spec) : CoPresheaves \stackrel{\overset{\mathcal{O}}{\leftarrow}}{\underset{Spec}{\to}} 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.

Definition

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

Let $𝒞$ be a small $𝒱$-enriched category.

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

Proposition

There is a $V$-adjunction

$$(𝒪⊣\mathrm{Spec}):[C,𝒱{]}^{\mathrm{op}}\stackrel{\stackrel{𝒪}{\leftarrow }}{\underset{\mathrm{Spec}}{\to }}[C^{\mathrm{op}},𝒱]$$(\mathcal{O} \dashv Spec) : [C, \mathcal{V}]^{op} \stackrel{\overset{\mathcal{O}}{\leftarrow}}{\underset{Spec}{\to}} [C^{op}, \mathcal{V}]

where

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

and

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

Remark

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

The proof is mostly a tautology after the notation is unwinded. 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

$$\begin{array}{rl}[C,𝒱{]}^{\mathrm{op}}(𝒪(X),A)& :={\int }_{c\in C}𝒱(A(c),𝒪(X)(c))\\ & :={\int }_{c\in C}𝒱(A(c),[C^{\mathrm{op}},𝒱](X,𝒱(-,c)))\\ & :={\int }_{c\in C}{\int }_{d\in C}𝒱(A(c),𝒱(X(d),𝒱(d,c)))\\ & \simeq {\int }_{d\in C}{\int }_{c\in C}𝒱(X(d),𝒱(A(c),𝒱(d,c)))\\ & =:{\int }_{d\in C}𝒱(X(d),[C,𝒱{]}^{\mathrm{op}}(𝒱(d,-),A))\\ & =:{\int }_{d\in C}𝒱(X(d),\mathrm{Spec}(A)(d))\\ & =:[C^{\mathrm{op}},𝒱](X,\mathrm{Spec}(A))\end{array}.$$\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, \mathcal{V}(-,c))) \\ & := \int_{c \in C} \int_{d \in C} \mathcal{V}(A(c), \mathcal{V}(X(d), \mathcal{V}(d,c))) \\ & \simeq \int_{d \in C} \int_{c \in C} \mathcal{V}(X(d), \mathcal{V}(A(c), \mathcal{V}(d,c))) \\ & =: \int_{d \in C} \mathcal{V}(X(d), [C,\mathcal{V}]^{op}(\mathcal{V}(d,-),A)) \\ & =: \int_{d \in C} \mathcal{V}(X(d), Spec(A)(d)) \\ & =: [C^{op}, \mathcal{V}](X, Spec(A)) \end{aligned} \,.

Notice that 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 $𝒱$ 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: $\mathrm{Spec}(𝒱(c,-))\simeq 𝒱(-,c)$

because we have a natural isomorphism

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

by the Yoneda lemma.

From this we get

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

by the sequence of natural isomorphisms

$$\begin{array}{rl}[C^{\mathrm{op}},𝒱](\mathrm{Spec}𝒱(c,-),\mathrm{Spec}A)& \simeq [C^{\mathrm{op}},𝒱](𝒱(-,c),\mathrm{Spec}A)\\ & \simeq (\mathrm{Spec}A)(c)\\ & :=[C,𝒱](A,𝒱(c,-))\end{array},$$\begin{aligned} [C^{op}, \mathcal{V}](Spec \mathcal{V}(c,-), Spec A) & \simeq [C^{op}, \mathcal{V}](\mathcal{V}(-,c), Spec A) \\ & \simeq (Spec A)(c) \\ & := [C, \mathcal{V}](A, \mathcal{V}(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 $X\in [C^{\mathrm{op}},𝒱]$ may be written as a colimit

$$X\simeq {\underrightarrow{\lim }}_i𝒱(-,c_i)$$X \simeq {\lim_\to}_i \mathcal{V}(-,c_i)

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

$$X\simeq {\underrightarrow{\lim }}_i\mathrm{Spec}(𝒱(c_i,-)).$$X \simeq {\lim_\to}_i Spec(\mathcal{V}(c_i,-)) \,.

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

Lemma 3:

$$𝒪(X)\simeq {\underleftarrow{\lim }}_i𝒱(c_i,-)$$\mathcal{O}(X) \simeq {\lim_{\leftarrow}}_i \mathcal{V}(c_i,-)

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

$$\begin{array}{rl}𝒪(X)(d)& =[C^{\mathrm{op}},𝒱](X,𝒱(-,c))\\ & \simeq [C^{\mathrm{op}},𝒱]({\underrightarrow{\lim }}_i𝒱(-,c_i),𝒱(-,c))\\ & \simeq {\underleftarrow{\lim }}_i[C^{\mathrm{op}},𝒱](𝒱(-,c_i),𝒱(-,c))\\ & \simeq {\underleftarrow{\lim }}_i𝒱(c_i,c)\end{array}.$$\begin{aligned} \mathcal{O}(X)(d) &= [C^{op}, \mathcal{V}](X, \mathcal{V}(-,c)) \\ & \simeq [C^{op}, \mathcal{V}]({\lim_\to}_i \mathcal{V}(-,c_i), \mathcal{V}(-,c)) \\ & \simeq {\lim_\leftarrow}_i [C^{op}, \mathcal{V}](\mathcal{V}(-,c_i), \mathcal{V}(-,c)) \\ & \simeq {\lim_\leftarrow}_i \mathcal{V}(c_i,c) \end{aligned} \,.

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

$$\begin{array}{rl}[C,𝒱{]}^{\mathrm{op}}(𝒪(X),A)& \simeq [C,𝒱](A,{\underleftarrow{\lim }}_i𝒱(c_i,-),)\\ & \simeq {\underleftarrow{\lim }}_i[C,𝒱](A,𝒱(c_i,-))\\ & \simeq {\underleftarrow{\lim }}_i[C^{\mathrm{op}},𝒱](\mathrm{Spec}𝒱(c_i,-),\mathrm{Spec}A)\\ & \simeq [C^{\mathrm{op}},𝒱]({\underrightarrow{\lim }}_i\mathrm{Spec}𝒱(c_i,-),\mathrm{Spec}A)\\ & \simeq [C^{\mathrm{op}},𝒱](X,\mathrm{Spec}A)\end{array}.$$\begin{aligned} [C,\mathcal{V}]^{op}(\mathcal{O}(X), A) & \simeq [C,\mathcal{V}](A, {\lim_\leftarrow}_i \mathcal{V}(c_i,-), ) \\ & \simeq {\lim_{\leftarrow}}_i [C, \mathcal{V}](A, \mathcal{V}(c_i,-)) \\ & \simeq {\lim_{\leftarrow}}_i [C^{op}, \mathcal{V}](Spec \mathcal{V}(c_i,-), Spec A) \\ & \simeq [C^{op}, \mathcal{V}]({\lim_{\to}}_i Spec \mathcal{V}(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.

Definition

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

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

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

Remark

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

$$𝒪(F)(c)=[C^{\mathrm{op}},𝒱](F,C(-,c))\hookrightarrow {\int }_{s:S}𝒱(Fs,\mathrm{hom}(s,c))$$\mathcal{O}(F)(c) = [C^{op}, \mathcal{V}](F, C(-, c)) \hookrightarrow \int_{s: S} \mathcal{V}(F s, \hom(s, c))

$$\mathrm{Spec}(G)(c)=[C,𝒱](G,C(c,-))\hookrightarrow {\int }_{t:T}𝒱(Gt,\mathrm{hom}(c,t))$$Spec(G)(c) = [C, \mathcal{V}](G, C(c, -)) \hookrightarrow \int_{t: T} \mathcal{V}(G t, \hom(c, t))

Properties

Respect for limits

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

Proposition

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

$$(𝒪⊣\mathrm{Spec}):[C,𝒱{]}_{\times }^{\mathrm{op}}\stackrel{\stackrel{𝒪}{\leftarrow }}{\underset{\mathrm{Spec}}{\to }}[C^{\mathrm{op}},𝒱]$$(\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:

$$\begin{array}{rl}𝒪(X)({\underleftarrow{\lim }}_j{c}_j)& :=[C^{\mathrm{op}},𝒱](X,𝒱(-,{\underleftarrow{\lim }}_j{c}_j))\\ & \simeq [C^{\mathrm{op}},𝒱](X,{\underleftarrow{\lim }}_j𝒱(-,c_j))\\ & \simeq {\underleftarrow{\lim }}_j[C^{\mathrm{op}},𝒱](X,𝒱(-,c_j))\\ & =:{\underleftarrow{\lim }}_j𝒪(X)(c_j)\end{array}.$$\begin{aligned} \mathcal{O}(X)({\lim_{\leftarrow}}_j c_j) & := [C^{op}, \mathcal{V}](X,\mathcal{V}(-,{\lim_{\leftarrow}}_j c_j)) \\ & \simeq [C^{op}, \mathcal{V}](X,{\lim_{\leftarrow}}_j \mathcal{V}(-,c_j)) \\ & \simeq {\lim_{\leftarrow}}_j [C^{op}, \mathcal{V}](X,\mathcal{V}(-,c_j)) \\ & =: {\lim_{\leftarrow}}_j \mathcal{O}(X)(c_j) \end{aligned} \,.

Isbell self-dual objects

Proposition

All representables are Isbell self-dual.

Proof

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

$$\mathrm{Spec}(𝒱(c,-))\simeq 𝒱(-,c).$$Spec(\mathcal{V}(c,-)) \simeq \mathcal{V}(-,c) \,.

Therefore we have also the natural isomorphism

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

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

Isbell envelope

See Isbell envelope.

Examples

Gelfand duality

Gelfand duality is the equivalence of categories between (nonunital) commutative C-star 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-star algebra are equivalently modules of sections of vector bundles over the Gelfand-dual topological space.

Function $T$-Algebras on presheaves

Let $𝒱$ be any cartesian closed category.

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

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

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

Write

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

for the Yoneda embedding.

Definition

Call

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

the $T$-line object.

Observation

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

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

In particular

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

Proof

We have isomorphisms natural in $k\in T$

$$\begin{array}{rl}[C^{\mathrm{op}},𝒱](X,\mathrm{Spec}(F_T(k)))& \simeq T\mathrm{Alg}(F_T(k),𝒪(X))\\ & \simeq 𝒪(X)(k)\end{array}$$\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

$$T\subset C\subset T{\mathrm{Alg}}^{\mathrm{op}}$$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 $𝒪⊣\mathrm{Spec}$ no longer makes sense, but that in terms of the line object still does

Proposition

Set

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

and

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

Then we still have an adjunction

$$(𝒪⊣\mathrm{Spec}):T{\mathrm{Alg}}^{\mathrm{op}}\stackrel{\stackrel{𝒪}{\leftarrow }}{\underset{\mathrm{Spec}}{\to }}[C^{\mathrm{op}},𝒱].$$(\mathcal{O} \dashv Spec) : T Alg^{op} \stackrel{\overset{\mathcal{O}}{\leftarrow}}{\underset{Spec}{\to}} [C^{op}, \mathcal{V}] \,.

Proof

$$\begin{array}{rl}T{\mathrm{Alg}}^{\mathrm{op}}(𝒪(X),A)& :={\int }_{k\in T}𝒱(A(k),𝒪(X)(k))\\ & :={\int }_{k\in T}𝒱(A(k),[C^{\mathrm{op}},𝒱](X,\mathrm{Spec}(F_T(k))))\\ & :={\int }_{k\in T}{\int }_{B\in C}𝒱(A(k),𝒱(X(B),T\mathrm{Alg}(F_T(k),B)))\\ & \simeq {\int }_{k\in T}{\int }_{B\in C}𝒱(A(k),𝒱(X(B),B(k)))\\ & \simeq {\int }_{k\in T}{\int }_{B\in C}𝒱(X(B),𝒱(A(k),B(k)))\\ & =:{\int }_{B\in C}𝒱(X(B),T\mathrm{Alg}(A,B))\\ & =:{\int }_{B\in C}𝒱(X(B),\mathrm{Spec}(A)(B))\\ & =:[C^{\mathrm{op}},\mathrm{Set}](X,\mathrm{Spec}(A))\end{array}.$$\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 $𝒱$.

Function $k$-algebras on derived $\mathrm{\infty }$-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 $\mathrm{\infty }$-stacks

for the moment see function algebras on ∞-stacks

Function 2-algebras on algebraic stacks

see Tannaka duality for geometric stacks

Related concepts

duality between algebra and geometry in physics:

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

References

Isbell conjugation is reviewed on page 17 of

• Bill Lawvere, Taking categories seriously (pdf)

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

• David Ben-Zvi, David Nadler, Loop spaces and connections (arXiv:1002.3636)

in

• Bertrand Toën, Champs affine (arXiv:math/0012219)

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 $\mathrm{\infty }$-stacks over duals of algebras over arbitrary abelian Lawvere theories is the content of

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

Some discussion at MathOverflow

• theme-of-isbell-duality