Is there a "james periodicity" for stunted lens spaces? If so where can I read about it?

@SaalHardali Kambe-Matsunaga-Toda projecteuclid.org/euclid.kjm/1250524531

@DylanWilson Thanks! Relieved to see it's a rather elegant statement. Incredibly old reference though, are these results (periodicity type results for lens/projective spaces) collected in a single more modern reference that you know of? It would be nice to have a self contained reference for this.

Maybe its standard and in every textbook on k theory in which case sorry ^^

@AdrianClough This follows immediately from computation of limits and colimits in a product of categories

@HarryGindi Can you give some more details? The proof that I had in mind is to compute the left Kan extension of $A \times B \to E$ along $A \times B \to A$, and then observe that $A \times B \to E$ and the resulting functor $A \to E$ have the same colimit.

The above argument relies on the fact that a left Kan extension along a coCartesian fibration $E \to B$ can be computed taking colimits of the fibres $E_b$ rather than the slice categories $E_{b/}$.

That's right

that's the more general theorem.

I have a principal G bundle E\rightarrow B, G compact Lie. I don't know much about smoothness of B. Is there a notion of parallel transport in this setting? This page seems to say so: encyclopediaofmath.org/wiki/Parallel_transport but I can't find references.

I think such a thing should exist as the space of choices of lifts of a given curve in the base is contractible.

@ThomasRot I would define a parallel transport operator = connection in this context as a continuous functor $\mathcal PB \to \text{Iso}(E)$, where the latter category is the topological category with space of objects B and Hom(x, y) = Iso(E_x, E_y). This is basically equivalent to a continuous homomorphism $\Omega B \to G$.

This loops to a homomorphism $\Omega B \to \Omega BG$. The latter guy is equivalent as a group in some homotopy sense to G. So "up to homotopy" your map B -> BG loops to the desired parallel transport operator.

I guess there are serious subtleties in making this strict and I haven't tried to make a claim that the space of such homomorphisms of "given topological type" is contractible, so maybe this is the wrong notion.

The definition via finite simplicial sets is equivalent to being a finite colimit of Delta^n's, but that's a bit circular...

@RuneHaugseng Why is that not ok? It looks to me like the definition of finite spaces

The ∞-cat of finite ∞-cats is the smallest subcategory containing Δ^0 and Δ^1 and closed under finite colimits

It seems to me that it captures well the intuition of "having finitely many objects and finitely many arrows"

Maybe you can find something like the Wall finiteness obstruction of ∞-cats, but it's unclear to me that that would be a "more model independent" or "better" definition

It's sort of circular since a finite colimit is defined to be a colimit over a finite infinity-category

@RuneHaugseng Ok, ok generated by $\varnothing$, $\Delta^0$ and $\Delta^1$ under pushouts

I guess you could turn a proposition into a definition and say "containing the empty set, the point, and the arrow, and closed under pushouts"

Jinx!

haha- beat me to it

(I swear I sometimes forget that finite colimits are not defined by "generated by initial object and pushouts)

@DylanWilson Also this is not really turning a theorem into a definition, because you still need to prove that finite colimits are generated by initial object and pushouts in arbitrary ∞-cats. That's not too hard though if you set up everything model independently

(it's probably not going to be surprising to you that I recently spent a lot of time discussing with Jay the equivariant version of this)

That's certainly a reasonable definition, though it is curious that the notion of finiteness has to be put in "by hand" in a sense (though admittedly starting with some generators that are pretty obviously finite).

it's almost like it's more natural to say that, "whatever finite colimits are the collection of them should be closed under finite colimits" and then declare that something like 'finite posets' are an example and ask for the collection to be 'closed under diagrams indexed by things in the collection'... but having to write all that out feels less natural in the end

@RuneHaugseng Do you have a better description for finite spaces? I don't, so I kind of accept this as the best we can do

Is there something wrong with 'obtained via finitely many pushouts from the collection of finite (possibly empty) posets'?

@IanColey No, you are just adding a lot of superfluous generators

Okay sure, but I mean the general principle of the thing. I suppose Dylan's answer feels more infinity-ish

@DenisNardin Yes, I think you're right about that

I don't suppose there's anything better one can say for compact objects in infinity-categories? (I.e. retracts of finite ones.)

@RuneHaugseng is it enough to say mapping out commutes with sequential colimits? (like- if we said omega-filtered that'd be circular, but maybe you just need N-shaped?)

I think you can. Compact objects can be characterised as those whose hom-out-of functor commutes with filtered colimits. So we can look at $\text{Flt}/C$ of filtered categories mapping to some infty-category $C$. I think you can then get some kind of 2-simplex like $C\to \text{Flt}/C\to C$ where the first functor is the diagonal and the second is `compute the colimit', and the long edge $C\to C$ is the identity.

The natural transformation should give you the canonical comparison between $C(X,\text{colim}_D F_d)$ and $\text{colim}_D C(X,F_d)$. Then you can identify the compact objects are exactly ones that gives you an equivalence

Unless I've misunderstood the question

@IanColey I think one issue is whether the definition of "filtered" uses the notion of "finite" which Rune wasn't happy about? but maybe I'm the one misunderstanding?

Nuts

Yeah `every finite diagram admits a cocone'

Axiom: $\varnothing$, 1, and 2 are finite

@IanColey I think you also need the span