Isbell duality


Higher algebra

Higher geometry


higher geometry \leftarrow Isbell duality \rightarrow higher algebra



A general abstract adjunction

(𝒪Spec):CoPresheavesSpec𝒪Presheaves (\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 CC: 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 CC (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 or the embedding of smooth manifolds into formal duals of R-algebras.)


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

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

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


There is a VV-adjunction

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


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


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

This is also called Isbell duality. Objects which are preserved by 𝒪Spec\mathcal{O} \circ Spec or Spec𝒪Spec \mathcal{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

[C,𝒱] op(𝒪(X),A) := cC𝒱(A(c),𝒪(X)(c)) := cC𝒱(A(c),[C op,𝒱](X,𝒱(,c))) := cC dC𝒱(A(c),𝒱(X(d),𝒱(d,c))) dC cC𝒱(X(d),𝒱(A(c),𝒱(d,c))) =: dC𝒱(X(d),[C,𝒱] op(𝒱(d,),A)) =: dC𝒱(X(d),Spec(A)(d)) =:[C op,𝒱](X,Spec(A)). \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} \,.

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

because we have a natural isomorphism

Spec(𝒱(c,))(d) :=[C,𝒱](𝒱(c,),𝒱(d,)) 𝒱(d,c) \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 op,𝒱](Spec𝒱(c,),SpecA)[C,𝒱](A,𝒱(c,))[C^{op}, \mathcal{V}](Spec \mathcal{V}(c,-), Spec A) \simeq [C,\mathcal{V}](A, \mathcal{V}(c,-))

by the sequence of natural isomorphisms

[C op,𝒱](Spec𝒱(c,),SpecA) [C op,𝒱](𝒱(,c),SpecA) (SpecA)(c) :=[C,𝒱](A,𝒱(c,)), \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[C op,𝒱]X \in [C^{op}, \mathcal{V}] may be written as a colimit

Xlim i𝒱(,c i) X \simeq {\lim_\to}_i \mathcal{V}(-,c_i)

over representables 𝒱(,c i)\mathcal{V}(-,c_i) we have

Xlim iSpec(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)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

𝒪(X)(d) =[C op,𝒱](X,𝒱(,c)) [C op,𝒱](lim i𝒱(,c i),𝒱(,c)) lim i[C op,𝒱](𝒱(,c i),𝒱(,c)) lim i𝒱(c i,c). \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

[C,𝒱] op(𝒪(X),A) [C,𝒱](A,lim i𝒱(c i,),) lim i[C,𝒱](A,𝒱(c i,)) lim i[C op,𝒱](Spec𝒱(c i,),SpecA) [C op,𝒱](lim iSpec𝒱(c i,),SpecA) [C op,𝒱](X,SpecA). \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.


An object XX or AA is Isbell-self-dual if

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

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


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

𝒪(F)(c)=[C op,𝒱](F,C(,c)) s:S𝒱(Fs,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))
Spec(G)(c)=[C,𝒱](G,C(c,)) t:T𝒱(Gt,hom(c,t))Spec(G)(c) = [C, \mathcal{V}](G, C(c, -)) \hookrightarrow \int_{t: T} \mathcal{V}(G t, \hom(c, t))


Respect for limits

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


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

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

Because the hom-functors preserves all limits:

𝒪(X)(lim jc j) :=[C op,𝒱](X,𝒱(,lim jc j)) [C op,𝒱](X,lim j𝒱(,c j)) lim j[C op,𝒱](X,𝒱(,c j)) =:lim j𝒪(X)(c j). \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


All representables are Isbell self-dual.


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

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

Therefore we have also the natural isomorphism

𝒪Spec𝒱(c,)(d) 𝒪𝒱(,c)(d) :=[C op,𝒱](𝒱(,c),𝒱(,d)) 𝒱(c,d), \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 and similar dualities

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

Function TT-Algebras on presheaves

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

Let C:=TC := T be the syntactic category of a 𝒱\mathcal{V}-enriched Lawvere theory, that is a 𝒱\mathcal{V}-category with finite products such that all objects are generated under products from a single object 11.

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

U T:TAlg𝒱:AA(1) U_T : T Alg \to \mathcal{V} : A \mapsto A(1)


F T:T opTAlg F_T : T^{op} \hookrightarrow T Alg

for the Yoneda embedding.



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

the TT-line object.


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

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

In particular

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

We have isomorphisms natural in kTk \in T

[C op,𝒱](X,Spec(F T(k))) TAlg(F T(k),𝒪(X)) 𝒪(X)(k) \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:=TC := T let more generally

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

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

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



SpecA:BTAlg(A,B) Spec A : B \mapsto T Alg(A,B)


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

Then we still have an adjunction

(𝒪Spec):TAlg opSpec𝒪[C op,𝒱]. (\mathcal{O} \dashv Spec) : T Alg^{op} \stackrel{\overset{\mathcal{O}}{\leftarrow}}{\underset{Spec}{\to}} [C^{op}, \mathcal{V}] \,.
TAlg op(𝒪(X),A) := kT𝒱(A(k),𝒪(X)(k)) := kT𝒱(A(k),[C op,𝒱](X,Spec(F T(k)))) := kT BC𝒱(A(k),𝒱(X(B),TAlg(F T(k),B))) kT BC𝒱(A(k),𝒱(X(B),B(k))) kT BC𝒱(X(B),𝒱(A(k),B(k))) =: BC𝒱(X(B),TAlg(A,B)) =: BC𝒱(X(B),Spec(A)(B)) =:[C op,Set](X,Spec(A)). \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\mathcal{V}.

Function kk-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 TT-algebras on \infty-stacks

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

Function 2-algebras on algebraic stacks

see Tannaka duality for geometric stacks

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.

duality between algebra and geometry in physics:

Poisson algebraPoisson manifold
deformation quantizationgeometric quantization
algebra of observablesspace of states
Heisenberg pictureSchrödinger picture
higher algebrahigher geometry
Poisson n-algebran-plectic manifold
En-algebrashigher symplectic geometry
BD-BV quantizationhigher geometric quantization
factorization algebra of observablesextended quantum field theory
factorization homologycobordism representation


The original articles on Isbell duality and the Isbell envelope are

  • John Isbell, Structure of categories, Bulletin of the American Mathematical Society 72 (1966), 619– 655. (project euclid)

  • John Isbell, Normal completions of categories, Reports of the Midwest Category Seminar, vol. 47, Springer, 1967, 110–155.

More recent discussion is in

  • William Lawvere, p. 17 of Taking categories seriously, Revista Colombiana de Matematicas, XX (1986) 147-178, reprinted as: Reprints in Theory and Applications of Categories, No. 8 (2005) pp. 1-24 (web)

  • Michael Barr, John Kennison, R. Raphael, Isbell Duality, Theory and Applications of Categories, Vol. 20, 2008, No. 15, pp 504-542. (web)

  • Richard Garner, The Isbell monad , Advances in Mathematics 274 (2015) pp.516-537. (draft)

  • Vaughan Pratt, Communes via Yoneda, from an elementary perspective, Fundamenta Informaticae 103 (2010), 203–218.

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


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)

See also

Last revised on June 13, 2018 at 04:48:47. See the history of this page for a list of all contributions to it.