@EspenNielsen yeah, and Lurie's proof that Piotr cited basically goes by reducing to that case

Now we should figure out appropriate references for descriptions of slices over other presheaves. I think it should go like this: if $F : C^{op} \to \mathcal{S}$ is a presheaf of spaces on $C$, it classifies a right fibration $E \to C$ and $P(C)/F \cong P(E)$.

Maybe that follows easily from straightening-unstraightening: $P(C) \cong RFib(C)$ with $F$ corresponding to $E \to C$, so $P(C)/F \cong RFib(C)_{/E} \cong RFib(E)$. That last equivalence requires argument...

If V and W are monoidal categories, then the category Lax(V,W) of lax monoidal functors from V to W (with monoidal transformations) doesn't have a "natural" monoidal structure. If V and W are symmetric, then Lax(V,W) does have a natural (symmetric) monoidal structure. I feel like there's a 2-categorical "explanation" to be given here, but at the moment I can't figure out what it is.

This has a nice explanation in terms of Barwick's "external" Boardman-Vogt tensor product: Suppose you have two operator categories P and Q (e.g. finite sets, which gives symmetric operads and symmetric monoidal categories, or ordered sets, which gives non-symmetric operads/monoidal categories), and a P-operad O and a $P\times Q$-operad O'. Then the category of O-algebras in O' (viewed as a P-operad) has a natural structure of a Q-operad.

(This is right adjoint to a natural construction of a $P\times Q$-operad from a P-operad and a Q-operad.)

For monoidal categories, you've "used up" the monoidal structure in forming the category of algebras/lax mon. functors. But since F={finite sets} is the terminal operator category, a symmetric operad can also be viewed as an $F\times F$-operad, which explains the symmetric monoidal structure in the second case.

Similarly, if V is monoidal and W is braided monoidal, you get a natural monoidal structure (but not a braided one) on Lax(V,W).

@OmarAntolín-Camarena The equivalence of RFib(C)_/E and RFib(E) follows quite immediately from properties of the contravariant model structure: For p : E -> C you obviously have an equivalence of categories p_! : sSet_/E -> (sSet_/C)_/p, so you need to check that's a left Quillen equivalence between the contravariant model structure and the slice model structure.

It's actually an equivalence of model categories: cofibrations are monomorphisms in both, so it's enough to see they have the same fibrant objects. In sSet/E the fibrant objects are the right fibrations (2.2.3.12), and the same holds in (sSet/C)/p by 2.2.3.14 in HTT.

@MikeMiller Dear Mike, due to the fact that I'm quite a newbie in surgery theory, what are the differences between your surgery and the one described in Milnor's A procedure for killing homotopy groups of Differentiable manifolds"? I looked at that paper but it seems his notion of surgery is "stronger" than yours, due to the fact that according to him, surgery preserves pontrjagin numbers i.e. signature, which clearly isn't true in the procedure you described.

My opinion is that there are problem with the choice of framing and trivialisations of tangent bundle of the imbedded spheres, but again, my lack of expertise in this field is preventing me to see what's going on

@MarcHoyois Thanks! I was able to get what I needed from Rezk's paper.

@Riccardo You're totally right. My apologies.

@Riccardo Apparently you can employ similar methods as in Teichner's thesis to construct rational homology spheres with fundamental group Z/n to do so for groups whose cohomology is 4-periodic. This includes cyclic-by-cyclic groups like the the dihedral groups. See 6.3.

(But it's certainly much harder than I suggested. I don't know if there are smooth constructions.)

@RuneHaugseng Thanks! I knew I was missing something like 2.2.3.14

What are examples of abelian categories where infinite direct products commute with infinite direct sums? I can't think of any at the moment

zero category of course, but apart from that?