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 actually have a very nice universal property that I'm very surprised I had never heard about before. So I'm really curious if the following is known, false, or if I should take some time to write the proof.

So the first thing is that, while it is not obviously the case when looking at the definition each simplicial Hom set $L^H \mathcal{C}(X,Y)$ of the Hammock localization actually is the nerve of a categories. So that the Hammock localization $L^H \mathcal{C}$ can identified with a strict $2$-category.

Moreover, that strict $2$-category has a very natural universal property:

Proposition: $L^H \mathcal{C}$ is the free $2$-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.

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!

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-cobar 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 actually have a very nice universal property that I'm very surprised I had never heard about before. So I'm really curious if the following is known, false, or if I should take some time to write the proof.

So the first thing is that, while it is not obviously the case when looking at the definition each simplicial Hom set $L^H \mathcal{C}(X,Y)$ of the Hammock localization actually is the nerve of a categories. So that the Hammock localization $L^H \mathcal{C}$ can identified with a strict $2$-category.

Moreover, that strict $2$-category has a very natural universal property:

Proposition: $L^H \mathcal{C}$ is the free $2$-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.

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 actually have a very nice universal property that I'm very surprised I had never heard about before. So I'm really curious if the following is known, false, or if I should take some time to write the proof.

So the first thing is that, while it is not obviously the case when looking at the definition each simplicial Hom set $L^H \mathcal{C}(X,Y)$ of the Hammock localization actually is the nerve of a categories. So that the Hammock localization $L^H \mathcal{C}$ can identified with a strict $2$-category.

Moreover, that strict $2$-category has a very natural universal property:

Proposition: $L^H \mathcal{C}$ is the free $2$-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.