In a model category, I have tools to show that mapping spaces are *contractible*. But if I want to show a mapping space is *empty or contractible*, is there anything I can do on general grounds?

The idea is this. Suppose I have a type of structure I'm interested in, and a collection of objects. To each object I assign a space parameterizing the choices required to equip that object with the structure in question. If all these parameter spaces are *empty-or-contractible*, that means the structure is "property-like": an object either has the structure or it doesn't; it can't carry different versions of the same structure. If the parameter spaces are all *contractible*, this means that in fact every object carries a unique version of the structure. Let me illustrate with an example of each.

**A contractible space of choices: property-like structures that always exist.** For example, in Joyal's model structure for quasicategories, suppose I want to show that the space of composites of two composable morphisms in a quasicategory $X$ is always contractible. This reduces to showing that the horn inclusion $\Lambda^1[2] \to \Delta[2]$ is an acyclic cofibration (since this implies that the fibers of $X^{\Delta[2]} \to X^{\Lambda^1[2]}$ are contractible).

Of course, this is immediate from typical descriptions of the model structure, but more generally I should consider myself lucky if all I have to do to answer a question is show that one explicit map is an acyclic cofibration, since I can work with explicit generators to make this a matter of combinatorics. (There's an asterisk in this case because in this model structure really I only know how to understand anodyne extensions combinatorially and not acyclic cofibrations in general, but in practice this understanding usually suffices.)

**An empty-or-contractible space of choices: property-like structures that don't always exist.** But now, again working in the Joyal model structure, suppose I want to show that the space of retracts of a given idempotent in a quasicategory is always either empty or contractible. This doesn't reduce to showing that the inclusion of the free idempotent into the free retract is an acyclic cofibration, because that would be too strong -- it would imply that every idempotent has a retract, which is not the case.

The closest thing I can think of is to perform a Bousfield localization to force every idempotent to have a retract, and then show that this map is an acyclic cofibration in the new model structure. But this only shows that if a quasicategory has *all* split idempotents (and I have to check that these are exactly the fibrant objects in the localized model structure), then the space of retracts for a given idempotent is contractible -- it doesn't tell me anything about quasicategories where some but not all idempotents split.

In this particular case, Lurie shows this fact using the theory of cofinality. But this is something specific to the example, not a general approach to the understanding property-like structures, even if we restrict our attention to quasicategories.

**Question:** Do model categories afford some general method for understanding property-like structures on their objects/morphisms, analogous to the method of showing that certain maps are acyclic cofibrations? Or can these things only be understood on a case-by-case basis?