5
$\begingroup$

Suppose one has a model category $C$ with its class of weak equivalences $W$. It is possible to form a separate homotopical category $(C_{\operatorname{span}},W_{\operatorname{span}})$ which has objects given by the objects of $C$ and morphisms from $x$ to $z$ given by spans $x \overset{\simeq} \twoheadleftarrow y \twoheadrightarrow z$ where the first arrow is an acyclic fibration and the second arrow is a fibration. The class of weak equivalences $W_{\operatorname{span}}$ is given by the spans consisting of two acyclic fibrations. The composition of two spans is defined by pullback.

In case it is not obvious, the purpose of the fibration requirement is to ensure that the composite of two such spans is still a span with the backwards arrow an equivalence. It is well known that we may take a Hammock localization of a relative category in order to obtain a simplicially enriched category. The process involves considering commutative diagrams of zig-zags. Is it the case that the Hammock localization of $(C,W)$ is equivalent to the Hammock localization of $(C_{\operatorname{span}},W_{\operatorname{span}})$?

Improve this question
$\endgroup$
6
  • 2
    $\begingroup$ I considered a more restrictive notion here where the induced morphism $y \to z \times x$ is also required to be a fibration. I forget the precise reasons, but they had better technical properties. Amusingly, I used them to get a simple description of the hammock localisation... $\endgroup$
    – Zhen Lin
    Commented Dec 2, 2021 at 22:35
  • $\begingroup$ @ZhenLin Thanks for the relevant paper; the reason I am after such a result is because I want to be able to use zig-zags of weak equivalences for the vertical maps in the Hammock localizations $\endgroup$
    – Connor Malin
    Commented Dec 2, 2021 at 23:10
  • 4
    $\begingroup$ Of course, the "category" of spans is actually a bicategory. Are you thinking of a sort of Hammock localization that applies to a "relative bicategory"? $\endgroup$
    – Mike Shulman
    Commented Dec 3, 2021 at 1:50
  • 2
    $\begingroup$ If you invert the $2$-cells in $C_{span}$, you will get the Dwyer-Kan localization of $C$ by $W$ (at least if you restrict to fibrant objects). This is not presented exactly in these terms, but this is somehow discussed in this paper of Michael Weiss: core.ac.uk/download/pdf/82257797.pdf which is quite relevant, but this is more recent paper of Joost Nuiten is more likely to be useful: arxiv.org/abs/1612.03800 $\endgroup$
    – D.-C. Cisinski
    Commented Dec 3, 2021 at 23:38
  • 2
    $\begingroup$ A version for which $C$ is allowed to be an $\infty$-category is discussed in §7.2 on calculus of fractions, in my book on higher categories mathematik.uni-regensburg.de/cisinski/CatLR.pdf $\endgroup$
    – D.-C. Cisinski
    Commented Dec 3, 2021 at 23:38

1 Answer 1

Reset to default
2
$\begingroup$

Here's half an answer.

See Prop 5.2 of Dwyer and Kan's Function Complexes in Homotopical Algebra for a proof that the hammock localization of $C$ is the same as the hammock localization of the non-full subcategory of fibrations in $C$.

Then the remaining question is whether introducing the backwards weak equivalences affects the hammock localization. I think in order to see this you will have to spell out exactly what bicategory of spans you're using (what its universal property is), but I think the fact that this doesn't affect the hammock localization should follow from considering the universal properties involved.

Improve this answer
$\endgroup$

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .