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there is a span of concepts
higher geometry $\leftarrow$ Isbell duality $\to$ higher algebra
which is a pretty fundamental thing about math, I think (well, this observation is at least to Lawvere, of course).
I put this span of links at the top of these three entries. I am enjoying that, but let me know if it is once again a silly idea of mine.
(maybe it should also be higher Isbell duality )
hm, I forget, why are Isbell duality and Isbell conjugation separate? I think I will merge them…
I have merged Isbell conjugation into Isbell duality
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.
If
A sheaf is a presheaf that satisfies descent.
how do you fill in
An algebra is a copresheaf that satisfies ???
And sheafification is dual to what?
Oh, it looks like you told me something about this before. So codescent.
There’s a question on cosheaves at MO. Something to look at tomorrow.
how do you fill in
An algebra is a copresheaf that satisfies ???
A low-brow but nevertheless good answer is:
An algebra is a copresheaf that preserves finite products / maybe finite limits. As in algebra over a Lawvere theory.
I have to admit that I had not heard or thought of “Lawvere distributions” as are mentioned on MO. Let me try to have a look at that…
From the Preface of the reference mentioned at MO,
Distributions
“Following up on a 1966 Oberwolfach talk where I had proposed a theory of distributions (not only in but) on presheaf toposes, in 1983 at Aarhus I posed several questions concerning distributions on $\mathcal{S}$-toposes [where $\mathcal{S}$ is an elementary topos, thought of to be Set, the category of sets and functions]. The base for the definition and questions is a pair of analogies with known theories (commutative algebra and measure theory) for variable quantities, coupled with the fact that there are many important examples of variable $\mathcal{S}$-’quantities’ where the domains of variation are $\mathcal{S}$-toposes. The intensively variable quantities are taken to be the sheaves on the topos, i.e., simply the objects in the category. Of course, the term ’topos’ means ’place’ or ’situation’, but Grothendieck treats the general situation by dealing instead with the category of Set-valued quantities which vary continuously over it, as an affine k-scheme is described by dealing with the k-algebra of functions on it. (… ) Then we follow the lead of analysis and define a distribution or extensively variable quantity on an $\mathcal{S}$-topos to be a continuous linear functional, or generalized point, i.e., a functor to $\mathcal{S}$ which preserves $\mathcal{S}$-colimits, but not necessarily the finite limits.”
F. W. Lawvere, Comments on the Development of Topos Theory, in: Jean-Paul Pier (editor), Development of Mathematics 1950-2000, Birkhauser Verlag, Basel Boston Berlin, 2000, 715-734.
Thanks. I realize that I did hear of these distributions: Lawvere of course tallks about them also in his article on cohesive toposes.
I need to figure out what the precise relation to algebras over a Lawvere theory is.
I am still at home where I cannot access the book you pointed to. Will go to the institute now to check.
David,
I looked at the book now (excerpted Lawvere distribution and cosheaf).
I am not sure yet how one can use this to give a much better description of how Isbell duality relates geometry and algebra.
So one way to look at Isbell duality is as the adjunciton
$(\mathcal{O} \dashv Spec) : [C,Set]^{op} \stackrel{\leftarrow}{\to} [C^{op}, Set]$and observe conditions under which this restricts to an adjunction between sheaves on the right, and product-preserving functors on the left.
To get co-sheaves in the game, we’d probably want to use $[C,Set]^{op} \simeq [C^{op},Set^{op}]$. Then we could regard a product-preserving functor $C \to Set$ as a $Set^{op}$-valued cosheaf with respect to a coverage on $C^{op}$ where the covering families are the op-s of product projections in $C$. (Unless op-ing got me mixed up somewhere).
But now where does this help with seeing if the presheaf/copresheaf duality descends? The topologies used for the cosheaves is independent and in general different from that used for the sheaves.
Hm. I believe I need to think about this more.
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