Another FinVect to FinDimVect replacement

]]>Am occupied elsewhere, but just to note that this thread started out, 11 years ago, with chat on just this kind of size issue (comments #1, #2, #7 etc.). But maybe this was never recorded properly in the entry. If you feel you could add it, please do.

]]>Come to think of it again, we may not need such a restriction in the first place.

Here is an argument that should prove the strong Yoneda lemma for arbitrary large enriched categories.

We have to show that the map

V^C(hom_C(c,-), F) → F(c)

is an isomorphism. In particular, we must show that the left side exists.

If the left side exists, it satisfies the universal property for ends. Pick an arbitrary object v∈V and apply the functor hom(v,-). Here hom(v,v’) denotes the set of maps from v to v’.

We get V^C(hom_C(c,-), F) = ∫_x V(hom_C(c,x), F(X)), so applying the functor hom(v,-) yields

∫_x hom(hom_C(c,x), F^v(x)),

where F^v(x) = hom_V(v,F(x)).

Now, as long as this functor is well-defined (meaning the end exists in the category of sets) and representable (as a functor of v), then the original end exists.

To show that it is well-defined and representable, we invoke the weak enriched Yoneda lemma for locally small categories. We get

∫_x hom(hom_C(c,x), F^v(x)) ≅ F^v(c).

This isomorphism is natural v. Thus, we proved that the above functor is well-defined and isomorphic to a representable functor. Thus, the strong Yoneda lemma holds for large enriched categories.

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]]>Re #57: The limit is not small, but it exists for locally λ-presentable V. The easiest way may be to argue that the end may be computed on the subcategory of λ-presentable objects, which is essentially small.

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]]>Fixing a small typo is all.

Olivia Borghi

]]>In the section “For permutation representations” I replaced “permutation representation” with “G-set” throughout, starting with the title of the section.

]]>I added

- J.P.Pridham,
*Tannaka duality for enhanced triangulated categories*, arxiv/1309.0637

to the references at Tannaka duality. It is about Tannaka duality for dg-categories.

]]>Are there some internal Tannaka theorems and even internal reconstruction theorems in general (internal in the sense of internal category theory), e.g. things like Barr embedding theorem, Giraud’s theorem, Freyd-Mitchell etc in internal setup ? For Tannaka, I mean, consider some class of Grothendieck 1-topoi, and internal monoidal categories there and try to realize them as internal representations of something in such a topos. I am pondering how to proceed in going toward certain statement in model theory which is in fact about internal 2-categories in certain presheaf category; but I see that I am not clear about how to internalize reconstruction reasoning (I can kind of formulate the expected statements but lack the logic of the proofs).

]]>More about the general theme (of non-Tannakian) reconstruction: new entry Barr embedding theorem. Several person entries quoted at reconstruction, including Grigory Garkusha, Paul Balmer etc.

]]>I added stub fiber functor for easy reference. Please improve.

]]>Urs, the entry is already now scrolling over 8-9 pages in my window and it has still only formal content, what is small part of the story. Even if one restrains to algebraic case, as the original idea stated, the proofs have only group part so far and none on gerbes, coalgebras, Hopf algebroids and so on. It is a huge subject and I am thinking of further splittings; that is why I created a separate stub for Tannaka-Krein theorem in the narrow (classical) sense, that is for compact topological groups. One can not simultaneously discuss all the levels in reasonable length. Maybe you suggest the intermediate level, but I do not see it clearly. I mean Tannakian level is about representation categories which are monoidal and usually also rigid; one can consider some more general rep. categories without monoidal structure but then the reconstruction is very different, finally there are reconstruction theorems for categories of qcoh modules and so on and of sites, embedding theorems like that of Barr, Giraud etc. which are properly in reconstruction theorem of general type.

I have made numerous new references and links to bialgebroid, Hopf monad and Hopf algebroid.

]]>If you change it, please move the relevant bits to “reconstruction theorem”.

But maybe it would be easier to have it all discussed in one entry?

]]>I think I should further change the idea section of Tannaka duality. I mean the way it is stated it applies to more general reconstruction theorems (for which we have the entry reconstruction theorem): we should add that we mean exclusively for categories of representations which are monoidal! Now, there is an interesting search for better understanding in recent work of Kornél Szlachányi (see arXiv/0907.1578 and the category list question here; he also has some new results I was told which are yet not on the arXiv). He points out the role of flat functors and the Grothendieck topology interpretation; everything would be trivial monadic nonsense if there were not the problem of the monoidal structure.

]]>I also created a separate entry for Tannaka-Krein theorem, that is the original Tannaka-type duality for compact topological groups; edits to Tadao Tannaka.

]]>I have added numerous references and links to Tannaka duality. Many other microscopic changes.

]]>Is his description explicit enough for you?

Okay, I spent some time with the article now. I see it now.

More on this should be put in the nLab entry, but I am afraid I won’t have the time to do it.

]]>I got that same insight myself when I first read Daniel’s paper, although he phrases the first point a bit differently. Is his description explicit enough for you?

]]>I know no coherent treatment in this generality.

Okay, if you come across anything, let me know.

Maybe it is common knowledge, but to me the two simple statements we have worked out here give me a much clearer picture of what is going on with Tannaka duality than I got from looking at the literature:

for unrestricted representations, Tannaka duality $End(F : A Mod \to V) = \int_{N \in A Mod} V(F(N), F(N))\simeq A$ is a tautology , the enriched Yoneda lemma in slight disguise, ($V$ any enriching category/cosmos)

so the crucial point is that some ends

$\int_{N \in A Mod} V(F(N), F(N))$over the category of all modules are already computed when restricting to just dualizable modules

$\cdots \simeq \int_{N \in A Mod_{dual}} V(F(N), F(N)) \,.$

To me, personally, this is a useful insight that helps me put Tannaka duality in perspective. If this is mentioned explicitly in the literature anywhere, I’d be interested.

]]>Where is this discussed explicitly?

I know no coherent treatment in this generality. It is being remarked here and there and some people who discovered some new generalizations like Rosenberg know it in various depth levels.

]]>Is any interest being shown in the idea that alongside comodules people should look at contramodules?

]]>all reconstruction theorems like Barr embedding theorem, Tannaka theorem, Krein theorem, Ulrich’s theorem, Giraud’s theorem boil down to Yoneda arguments, and this is well known.

Where is this discussed explicitly?

]]>the original statement by Tannaka (and Krein, I gather) were about the simple case of a plain group and its linear representation category

No. Tannaka’s reconstruction in general is all kinds of things, while Tannaka’s theorem and Krein’s theorem are two different particular theorems in functional analysis setup. So Krein theorem is not the same as Tannaka theorem. Both are special cases of many generalization. In modern generalizations one should also distinguish neutral and mixed Tannakian reconstruction, in first one looks at one fiber functor while in another one looks at a more stacky picture where one has fiber functors to a family of categories which correspond to something like algebraically closed extensions of the ground field. This leads to gerbes.

Edit: 1. slight correction: classical Tannaka and Krein theorems for compact topological groups are equivalent though differently formulated. Now we have a stub Tannaka-Krein theorem waiting for context. 2. It would be nice to eventually have the mixed Tannaka duality treated in $n$Lab.

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