symmetric monoidal (∞,1)-category of spectra
higher geometry / derived geometry
geometric little (∞,1)-toposes
geometric big (∞,1)-toposes
derived smooth geometry
abstract duality: opposite category,
concrete duality: dual object, dualizable object, fully dualizable object, dualizing object
between higher geometry/higher algebra
Langlands duality, geometric Langlands duality, quantum geometric Langlands duality
higher geometry $\leftarrow$ Isbell duality $\rightarrow$ higher algebra
A general abstract adjunction
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.
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.
There is a $V$-adjunction
where
and
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 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.
Use the end-expression for the hom-objects of the enriched functor categories to compute
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.
Notice that
Lemma 1: $Spec(\mathcal{V}(c,-)) \simeq \mathcal{V}(-,c)$
because we have a natural isomorphism
by the Yoneda lemma.
From this we get
Lemma 2: $[C^{op}, \mathcal{V}](Spec \mathcal{V}(c,-), Spec A) \simeq [C,\mathcal{V}](A, \mathcal{V}(c,-))$
by the sequence of natural isomorphisms
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
over representables $\mathcal{V}(-,c_i)$ we have
In terms of the same diagram of representables it then follows that
Lemma 3:
because using the above colimit representation and the Yoneda lemma we have natural isomorphisms
Using all this we can finally obtain the adjunction in question by the following sequence of natural isomorphisms
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 $X$ 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.
Under certain circumstances, Isbell duality can be extended to large $\mathcal{V}$-enriched categories $C$. For example, if $C$ has a small generating ssubcategory $S$ and a small cogenerating subcategory $T$, then for each $F: C^{op} \to \mathcal{V}$ and $G: C \to \mathcal{V}$, one may construct $\mathcal{O}(F)$ and $Spec(G)$ objectwise as appropriate subobjects in $\mathcal{V}$:
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.
The $(\mathcal{O} \dashv Spec)$-adjunction does descend to this inclusion, in that we have an adjunction
Because the hom-functors preserves all limits:
All representables are Isbell self-dual.
By Proof B , lemma 1 we have a natural isomorphisms in $c \in C$
Therefore we have also the natural isomorphism
where the second step is the Yoneda lemma. Similarly the other way round.
See Isbell envelope.
Isbell duality is a template for many other space/algebra-dualities in mathematics.
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
Write
for the Yoneda embedding.
Call
the $T$-line object.
For all $X \in [C^{op}, \mathcal{V}]$ we have
In particular
We have isomorphisms natural in $k \in T$
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
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
Set
and
Then we still have an adjunction
The first step that is not a definition is the Yoneda lemma. The step after that is the symmetric-closed-monoidal structure of $\mathcal{V}$.
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).
for the moment see at function algebras on ∞-stacks.
see Tannaka duality for geometric stacks
Gelfand duality is the equivalence of categories between (nonunital) commutative C-star algebras and (locally) compact topological spaces. See there for more details.
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:
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
Bill Lawvere, p. 17 of Taking categories seriously, 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)
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
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
See also