Hammock localization and free adjoints

Question

The Hammock localization $L^H \mathcal{C}$ of a relative category $(\mathcal{C},\mathcal {W})$ is a simplicial category defined by Dwyer and Kan as a way to compute the $\infty$-categorical localization or ordinary category, simpler than using what they call the "standard localization" (which boils down to localizing the bar resolution of $\mathcal{C}$).

It looks like a fairly natural construction, yet a little artificial as well. But I was thinking about it today and it strike me that it really feel like there should be a nice universal property. I'm really curious if the following is known, false, or if something like this is true.

Proposition: For any relative category $\mathcal{C}$, the simplicial category $L^H \mathcal{C}$ is the hom-wise nerve of a strict $2$-category, and it is the free strict $2$-category equipped with a functor $\mathcal{C} \to L^H\mathcal{C}$, such that each arrow $f$ in $\mathcal{W}$ has a right adjoint $f_*$ in $L^H\mathcal{C}(X,Y)$, and the canonical isomorphism $(fg)_* \simeq g_* f_*$ is actually an identity.

To be clear, what I claim in the first part is that each Hom-simplicial set of $L^H \mathcal{C}$ is a nerve, so that it can be identified with a strict 2-category.

Note: I no longer believe this to be the case in general - it probably is only true for some nice categories, like categories free on graphs. But I still want to understand this better!

Answer 1

A connection between Dwyer–Kan hammock localizations and adjoints in 2-categories certainly exists. As already mentioned in the comments, as early as 2002, Dawson–Paré–Pronk in “Adjoining adjoints” made essentially the same type of an observation. In particular, the hammock localization of a category with two objects and a single nonidentity arrow is the simplicial category Adj given by the free adjunction. (Another written proof can be found in Riehl–Verity “Homotopy coherent adjunctions and the formal theory of monads”, Remark 3.2.5.)

That being said, one observation that makes establishing such a connection more difficult than it seems (but does not necessarily preclude it) is that simplicial sets of morphisms in a hammock localization need not be nerves of categories.

Counterexamples can be exhibited using free constructions. Consider a category with objects $\{0,1,2,3,A,B,F,G\}$, morphisms generated by $f:0→1$, $g:1→2$, $h:2→3$, $u:0→A$, $v:B→3$, weak equivalences $p:A→2$, $q:1→B$, $w:F→A$, $x:B→G$, with relations $pu=gf$, $vq=hg$.

Then the following two hammocks from~0 to~3, given by 1-simplices in the simplicial set of morphisms from~0 to~3, cannot be composed.

The first hammock is as follows.

• The top row is $u:0→A$, $w:A←F$, $hpw:F→3$.
• The middle vertical maps are $p:A→2$, $pw:F→2$.
• The bottom row is $gf:0→2$, $\def\id{{\rm id}}\id:2←2$, $h:2→3$.

The second hammock is as follows.

• The top row is $f:0→1$, $\id:1←1$, $hg:1→3$.
• The middle vertical maps are $xq:1→G$, $q:1→B$.
• The bottom row is $xqf:0→G$, $x:G←B$, $v:B→3$.

These two hammocks have matching source and target because of the following chain of equivalent rows:

• $(gf:0→2,\id:2←2,h:2→3)$;
• $(gf:0→2,h:2→3)$;
• $(hgf:0→3)$;
• $(f:0→1,hg:1→3)$;
• $(f:0→1,\id:1←1,hg:1→3)$.

These two hammocks cannot be composed: any potential compositon hammock would have to decompose $u$ as a composition of morphisms that pass through an object weakly equivalent to 1, which is impossible.