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Let M be a model category presenting an ∞-category $\mathcal{M}$, and let $f : X \to Y$ and $g : Y \to Z$ be arrows of M. Consider the following propositions:

  1. The connected component of $f$ in $\mathcal{M}(X,Y)$ is contractible
  2. $\mathcal{M}(X, Y) \xrightarrow{g_*} \mathcal{M}(X, Z)$ restricts to a homotopy equivalence between the connected components of $f$ and $gf$

When can these propositions be expressed in an elementary way from the model structure on M?

I'm interested in the second proposition (it relates to homotopy uniqueness for properties expressed by arrows and extensions along arrows). But in the case $Z$ is fibrant, it can be reduced to the the first question for $f \in \mathbf{M}_{/Z}(gf, g)$. Conversely, the first proposition is the $Z=1$ case of the second.

Given a simplicial model category, when $X$ is cofibrant and $Y,Z$ are fibrant — or a general relative category by first computing a simplicial localization — one could answer these questions by appealing to the corresponding questions of simplicial sets.

However, I'm hoping there's a useful way to express these propositions somewhat more directly in terms of the the model structure on M rather than having to appeal to more elaborate consturctions.

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  • $\begingroup$ If you don't have simplicial enrichments, then you can use framings, Dwyer–Kan standard localization, or Dwyer–Kan simplicial localization to compute derived mapping simplicial sets, and provide a similar answer. But more context would be appropriate here, e.g., what kind of model category this is meant for, etc. $\endgroup$
    – Dmitri Pavlov
    Commented Apr 19, 2021 at 1:36
  • $\begingroup$ @DmitriPavlov The intent of my question about bypassing simplicial sets and the like; rather than churning these predicates out via simplicial machinery, I'm hoping there is some more elementary expression via weak equivalences, homotopies, basic categorical operations and the like, even if just for restricted cases (e.g. bifibrant source and target). The question is intended generally, but I expect to be interested in answers even for restricted classes of model categories. $\endgroup$
    – DiscretePointsInHomSpaces
    Commented Apr 19, 2021 at 2:33
  • $\begingroup$ In the simplicial case, if $f$ is a cofibration, $X$ (hence $Y$) is cofibrant, and $Y$ is fibrant, I think one can reduce (1) to lifting properties, along the lines indicated in this old comment. It's unclear to me from your comment whether something like that is of interest to you. $\endgroup$
    – Tim Campion
    Commented Apr 19, 2021 at 5:49
  • $\begingroup$ @TimCampion It is of interest to me, but also an example of what this question seeks to avoid. My original motivations are likely similar to yours: diagrammatic properties in $\infty$-categories via extension problems. I need to revisit how that works out along the lines of your old comment, but I decided it was more useful to take a predicate like in my question as a basic notion (originally I took an "exists homotopically unique" quantifier as basic), and I've found the basic idea appealing enough to want to apply it generally (thus the desire for a more direct interpretation). $\endgroup$
    – DiscretePointsInHomSpaces
    Commented Apr 19, 2021 at 8:08

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