Algebras and modules
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Geometry on formal duals of algebras
higher geometry Isbell duality higher algebra
A general abstract adjunction
relates (higher) presheaves with (higher) copresheaves on a given (higher) category : 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 be a good enriching category (a cosmos, i.e. a complete and cocomplete closed symmetric monoidal category).
Let be a small -enriched category.
Write and for the enriched functor categories.
There is a -adjunction
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
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.
because we have a natural isomorphism
by the Yoneda lemma.
From this we get
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 may be written as a colimit
over representables we have
In terms of the same diagram of representables it then follows that
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 or is Isbell-self-dual if
is an isomorphism in ;
is an isomorphism in , respectively.
Respect for limits
Choose any class of limits in and write for the full subcategory consisting of those functors preserving these limits.
The -adjunction does descend to this inclusion, in that we have an adjunction
Because the hom-functors preserves all limits:
Isbell self-dual objects
By Proof B , lemma 1 we have a natural isomorphisms in
Therefore we have also the natural isomorphism
where the second step is the Yoneda lemma. Similarly the other way round.
See Isbell envelope.
Examples and similar dualities
Isbell duality is a template for many other space/algebra-dualities in mathematics.
Function -Algebras on presheaves
Let be any cartesian closed category.
Let be the syntactic category of a -enriched Lawvere theory, that is a -category with finite products such that all objects are generated under products from a single object .
Then write for category of product-preserving functors: the category of -algebras. This comes with the canonical forgetful functor
for the Yoneda embedding.
the -line object.
For all we have
We have isomorphisms natural in
by the above adjunction and then by the Yoneda lemma.
All this generalizes to the following case:
instead of setting let more generally
be a small full subcategory of -algebras, containing all the free -algebras.
Then the original construction of no longer makes sense, but that in terms of the line object still does
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 .
Function -algebras on derived -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 -algebras on -stacks
for the moment see at function algebras on ∞-stacks.
Function 2-algebras on algebraic 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:
- Michael Barr, John Kennison, R. Raphael, Isbell Duality, PDF
Isbell conjugation is reviewed on page 17 of
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 -stacks over duals of algebras over arbitrary abelian Lawvere theories is the content of
- Herman Stel, -Stacks and their function algebras – with applications to -Lie theory , master thesis (2010) (web)