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December 4, 2008

Question on Infinity-Yoneda

Posted by Urs Schreiber

What is known, maybe partially, about generalizations of the Yoneda lemma to any one of the existing \infty-categorical models?

For HStrucHStruc some category of “higher structures” (be it simplicial sets, Kan complexes, quasicategories, globular sets, nn-categories, ω\omega-categories, etc.) which I assume to

- come equipped with a faithful functor inj:SetsHStruc; inj : Sets \hookrightarrow HStruc \,;

- and to carry some closed structure which extends to an enrichment of the category of HStrucHStruc-valued (pseudo)presheaves [S op,HStruc][S^{op}, HStruc] over HStrucHStruc.

Then, given any HStrucHStruc-valued (pseudo)presheaf

X:S opHStruc X : S^{op} \to HStruc

I’d like to know what we can say about the HStrucHStruc-valued presheaf [S op,HStruc](injoHom S( 2, 1),X):S opHStruc, [S^{op}, HStruc](inj o Hom_S(-_2, -_1), X) : S^op \to HStruc \,, i.e. the presheaf which sends each USU \in S to U[S op,HStruc](injoHom S(,U),X)HStruc. U \mapsto [S^{op}, HStruc](inj o Hom_S(-, U), X) \in HStruc . or U[S op,HStruc](U,X) U \mapsto [S^{op}, HStruc](U, X) for short, with UU understood as the corresponding representable HStrucHStruc-valued (pseudo)presheaf.

In particular, how does it compare to XX itself?

What is known?

Posted at December 4, 2008 11:57 AM UTC

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Re: Question on Infinity-Yoneda

Okay, there is Lurie’s (,1)(\infty,1)-categorical Yoneda lemma, HTT, p. 260.

Still, is anything known in situations more general than (,1)(\infty,1)-categories?

Posted by: Urs Schreiber on December 4, 2008 12:38 PM | Permalink | Reply to this

Re: Question on Infinity-Yoneda

For my pet context of HStruc=ωCat(Spaces)HStruc = \omega Cat(Spaces) I suppose I can proceed entirely using the strong enriched Yoneda lemma from BCECT, p. page 39.

Using the notation there I’d have

V=ωCat(Spaces) V = \omega Cat(Spaces)

A=Spaces op A = Spaces^{op}

and for USpacesU \in Spaces and X:Spaces opωCat(Spaces)X : Spaces^{op} \to \omega Cat(Spaces) an omega-category valued presheaf the strong internal Yoneda theorem gives an isomorphism

X(U)Hom(U,X). X(U) \simeq Hom(U,X) \,.

I may need to figure out if an analogous statement holds for situations where XX itself is just a pseudo functor of sorts…

Posted by: Urs Schreiber on December 4, 2008 1:00 PM | Permalink | Reply to this

Re: Question on Infinity-Yoneda

Urs wrote: pseudo functor of sorts

so definition to be determined? what sort of pseudo do you have in mind? cf. lax?

(infty,1) - we need a better name

Posted by: jim stasheff on December 4, 2008 1:28 PM | Permalink | Reply to this

Re: Question on Infinity-Yoneda

what sort of pseudo do you have in mind?

Just the usual idea that in an nn-stack the restriction maps may usually be taken to respect composition only up to the relevant higher cells.

One point of the question in the above entry is:

given any notion of nn-stack, shouldn’t it be true that all nn-stacks can always be rectified, i.e. that every nn-stack is equivalent to one whose restriction maps respect composition strictly.

Posted by: Urs Schreiber on December 4, 2008 11:56 PM | Permalink | Reply to this

Re: Question on Infinity-Yoneda

Just the usual idea that in an n-stack the restriction maps may usually be taken to respect composition only up to the relevant higher cells.

But should these higher cells be invertible, at least in a sufficiently weak sense? If so, then to answer Jim’s question: a strong, not lax, sort of pseudo.

Posted by: Toby Bartels on December 5, 2008 5:12 AM | Permalink | Reply to this

Re: Question on Infinity-Yoneda

But should these higher cells be invertible, at least in a sufficiently weak sense?

Yes.

If so, then to answer Jim’s question: a strong, not lax, sort of pseudo.

Ah. I have become used to the convention

“lax”:= up to directed cells,

“psedo”:=up to weak isos.

But I intentionally did not want to be specific about this weakining of the respect for pullbacks in \infty-stacks here. The question I ask above is really: can’t we just restrict to \infty-stacks where everything pulls back strictly.

A kind of nnLab:semi-strictification.

We expect for any notion of \infty-category an \infty-Yoneda lemma. Using this as described above would seem to provide an explicit way to rectify any \infty-stack.

(I should mention that this goes back to discussion I am having with Thomas Nikolaus.)

More concretely, one can read one of the main results of my work with Konrad Waldorf as saying that for instance

- the 1-stack of bundles with connections has a natural rectification; ()

- the 2-stack of 2-bundles with connection has a natural rectification

These statement follow a pattern which one clearly expects to continue to \infty:

nn-bundles with connection are given by the nn-stack which is the \infty-stackification of the (naturally recitified) prestack of globally smooth parallel transport functors:

GTrivBund ():Uhom(P n(U),BG). G-TrivBund_\nabla(-) : U \mapsto hom(P_n(U), \mathbf{B}G) \,.

See for instance the first two pages of these notes.

Konrad and I naturally rectify (= make restriction maps be strictly preserved) the \infty-stackification of this by using “parallel transport nn-functors”, namely by taking

GBund () G-Bund_\nabla(-)

to send UU to the nn-category of “parallel transport” nn-functors Uhom nCat(Sets)(P n(U),BG){haslocalsmoothtrivialization}, U \mapsto hom_{nCat(Sets)}(P_n(U), \mathbf{B}G) \cap \{has local smooth trivialization\} \,, i.e. nn-functors as before, but now allowed to not respect the ambient topological/smooth structure globally, but required to have the property of admitting a “smooth local trivialization”.

We prove that such “(locally smoothly trivializable) =: (parallel transport)-nn-functors” are equivalent to differential nonabelian nn-cocycles, hence to nn-bundles with connection (as I said, done for n2n \leq 2, but the general construction is certainly expected to go through for all nn). This means that the nn-stack of parallel transport nn-functors is a natural rectification of the nn-stack pseudofunctor of nn-bundles with connection.

The above Yoneda question is supposed to support the expectation that this goes through for nn \to \infty from a different angle.

Posted by: Urs Schreiber on December 5, 2008 9:26 AM | Permalink | Reply to this

Re: Question on Infinity-Yoneda

In fact, there is a model for \infty-stacks which are fully rectified (i.e. are just plain contravariant functors on the underlying site) in the context of Segal categories, one model for (,1)(\infty,1)-categories:

this is recalled on the top of page 12 in Toën’s review:

it says there that, in this context, a model for \infty-stacks on a site CC is given by precisely those presheaves on CC with values in simplicial sets (“rectified pre-\infty-stacks”) which satisfy descent.

This is precisely the kind of statement I am referring to. Similarly we say, following Street, that an ω\omega-category valued (co)presheaf is an ω\omega-(co)stack if it satisfies ω\omega-(co)descent.

Posted by: Urs Schreiber on December 5, 2008 3:03 PM | Permalink | Reply to this

Re: Question on Infinity-Yoneda

Jim writes:

what sort of pseudo do you have in mind? cf. lax?

In Australian nn-category theory, ‘pseudo’ refers to replacing equations by isomorphisms, while ‘lax’ means replacing equations by morphisms, and ‘oplax’ means replacing equations by morphisms going the other way.

Of course homotopy theorists use ‘lax’ differently.

Posted by: John Baez on December 7, 2008 1:33 AM | Permalink | Reply to this

Re: Question on Infinity-Yoneda

Of course homotopy theorists use ‘lax’ differently.

I didn’t know that!

My own answer used ‘lax’ in the Australian sense.

Posted by: Toby Bartels on December 7, 2008 5:54 AM | Permalink | Reply to this

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