added links for these references:

{#Isbell66} John Isbell,

*Structure of categories*, Bulletin of the American Mathematical Society**72**(1966), 619-655. [project euclid, ams:1966-72-04/S0002-9904-1966-11541-0]{#Isbell67} John Isbell,

*Normal completions of categories*, Reports of the Midwest Category Seminar,**47**, Springer (1967) 110-155 [doi:10.1007/BFb0074302]

Cite Kock’s result about the codensity monad of the Yoneda embedding.

]]>I have added (here) the doi-link doi.org/10.1090/noti2602, since that should eventually be the more stable one. On the other hand, it doesn’t work at the moment, but I guess they will fix this eventually.

]]>Added link to my short intro to Isbell duality in the AMS Notices.

]]>moved definition of self-dual to a better place

]]>Changed V(c,d) to C(c,d) per discussion; hopefully I got them all and didn’t introduce any mistakes.

]]>That might have been me, back then. Please fix as seems appropriate.

]]>I am having trouble understanding some of the notation here. We have a $V$-enriched category $C$ but then the expression $V(-,c)$ is used in many places; I assume this is meant to refer to the V-object $C(-,c)$? Or is this some common notational convention I was unaware of? If people are ok with $C(-,c)$ I’d be happy to change it, just wanted to make sure I’m not stepping on anyone’s toes.

]]>Added a reference to

- Tom Avery, Tom Leinster,
*Isbell conjugacy and the reflexive completion*, arXiv:2102.08290 (2021). (abstract)

This is also true for enriched categories, right?

]]>Sure, that’s exactly the way I think of it!

]]>I added another way to think about Isbell duality as an “example”; this is supposed to be good for people who only know ordinary categories up to and including the Yoneda embedding.

]]>replaced ’unwinded’ by ’unwound’.

]]>Just to check, after Proof B where it says

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,

can I add

]]>for instance, employing the 2-Yoneda lemma, there is an Isbell duality between presheaves and copresheaves of categories on a small 2-category?

Thanks. So perhaps Gabriel-Ulmer duality and the ultracategory one I’m talking about here can all be construed in Isbellian terms.

]]>I expect all of this would go through without much change for higher categories of all sorts.

]]>Added:

The codensity monad of the Yoneda embedding is isomorphic to the monad induced by the Isbell adjunction, $Spec \mathcal{O}$ (Di Liberti 19, Thrm 2.7).

Is there anything to stop the Yoneda embedding for bicategories similarly generating an Isbell adjunction? The results on the page are written for enriched categories, so I guess no problem for strict 2-categories.

Kan extensions for $Cat$ are as in Enriched categories as a free cocompletion, sec. 10?

]]>I was wondering about some kind of cohesive modality-algebra link over here based on an observation mentioned in #21, #44 there of a 7-term chain of adjoints.

I’m an interested observer of such general patterns. There were discussions on coalgebra over at the n-Café once, e.g., here. Not sure I have anything much to add at present, but I’m happy to see you think out such matters.

]]>Also, cohesion seems to exist for each type- there is no reason to prefer the spacial type as far as I can see. For instance, there is a cohesion between modules and topological modules. The functors are “discrete module”, “underlying module”, “indiscrete module”, and “$M$ maps to $M/N$ where $N$ is the smallest submodule containing the connected component of $0$”. Or for another example, take commutative rings and the full subcategory of functors $[I, \text{Ring}]$ from the interval category to ring consisting of surjective maps of nilpotent ideals. I think that the broader view here is that functors in cohesion preserve type, which follows for the three examples of type I gave.

David- from some of your posts you found it seemed like you share my fascination for making mathematics symmetrical like this, so let me know if you want to work on it with me.

]]>We were talking about cotoposes above…

More broadly, we could regard categories of space as distributive or extensive or lextensive (that is, let us momentarily take a step back from toposes). What is “codistributive”? Of course, the answer is “opposite category of distributive”, but how do we think of such categories? I would hesitantly contend that “codistributive” is “algebra-like”. For instance, coproducts distribute over products in $R \text{-alg}$. Also, $R \text{-alg}$ is coextensive (opposite category of an extensive category) - precisely the reverse from spacial categories. This “algebra-like” codistributivity is of course quite different than “linear” - it reminds more of rings/algebras than modules.

This observation is more or less expected if “algebra” categories stand in a sort of duality with “spacial” categories. But why do categories of algebras only seldom have “cosubobject classifiers”. I don’t know, nor do I know what it means to be “co-cartesian closed” (this seems like it could actually be more subtle than “opposite category of a cartesian closed category”).

Nevertheless, to continue, we may further observe that Isbell duals tend to be between “distributive” and “codistributive”. They are also often between linear (coproducts canonically isomorphic to products) and colinear, but the opposite of a linear category is of course the same as a linear category. For an example, take $R$-mod for a commutative ring $R$ and $\text{Hom}(-, M)$.

Contravariant Isbell duality is $[C, V] \leftrightarrow [C^{op}, V]$ when $C$ is enriched over $V$, where the brackets denote enriched functors. There is no mention of distributive, extensive, codistributive, topos, etc., which are yet to be supplied by specific context. Accordingly, we might notice that

1) If $V$ is distributive, then $[C, V]$ is distributive.

2) If $V$ is codistributive, then $[C, V]$ is codistributive.

3) If $V$ is linear, then $[C, V]$ is linear.

So it is the nature of the target category that tells the nature of the (co)presheaves. Of course, taking $C = D^{op}$ for another category $D$ will make no difference.

Now I am curious about putting conditions on $[C, V]$ such as limit preservation. Perhaps requiring limit preservation inverts type and colimit preservation preserves type? By type I mean, “spacial” (products distribute over coproducts), “linear” (products and coproducts canonically isomorphic), or “algebraic” (coproducts distribute over products).

Also, you guys will probably know the answer to this: if (the contravariant) Isbell duality pertains to $\text{Hom}(-, X)$ with $X$ playing two roles, then what is an analogously general duality between $\text{Hom}(X, -)$ and $X \otimes -$ with $X$ playing two roles?

P.S. I can’t seem to get Markdown+Itex to work for me. Has anyone else encountered this on Safari?

]]>https://mathoverflow.net/questions/313487/a-serendipitous-connection-between-isbell-duality-and-yoneda-structures ]]>

Looks good to me now, thanks!

]]>I have fixed it now, hopefully.

The symbol `\esh`

was one of the LaTeX symbols I added support for a while ago, along with a few others. The way it was done was not quite robust enough, I have tried to tighten it now. They are not present in the original Itex2MML, and are currently added by post-processing.

It appears to be the fault of the new renderer. The diff view, which uses the old renderer, works correctly.

https://ncatlab.org/nlab/show/diff/Isbell+duality

( arrgh I can’t use https://ncatlab.org/nlab/show/diff/Isbell+duality/25 to link the current diff in case somebody updates the file )

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