I have been studying weak factorization systems for their use in model categories. I am trying to use these to abstract away from a common phenomenon underlying fibrations. In brief, it seems as though, whenever we have a class of fibrations defined by a lifting property, there is some object which classifies them.

First, I will be using the notion when refering to the lifting property that occurs in factorization systems:

We say a morphism $f$ in a category $C$ satisfies the lifting property against $g$, and write $f \perp g$, if for each diagram as below:

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there is a unique $h$ such that this commutes:

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If $h$ is not required to be unique, we say that $f$ satisfies the weak lifting property against $g$, and write $f * g$. For $A \subset \text{Mor}(C)$, we write $A^\perp = \{ g \in \text{Mor}(C) : f \perp g \forall f \in A \}$ and $A^* = \{ g \in \text{Mor}(C) : f * g \forall f \in A \}$.

Like I said, it seems that morphisms which arise from the lifting property or the unique lifting property are often classified by some object, category, higher category, etc.. Here are some examples:

Example 1: Write $\text{Top}$ for the category of topological spaces.

(a) Let $\{ 0 \} \rightarrow I$ be the inclusion, where $I $ is the interval. A covering space map is an element of $\{ \{ 0 \} \rightarrow I \}^\perp$.

(b) $\text{Set}$ classifies covering spaces, in the sense that $\text{Cat}(\Pi_1 (X) , \text{Set})$ is naturally isomorphic to the category of covering spaces of $X$, for each topological space $X$. This is a reworking of the famous fundamental theorem of covering space theory.

Example 2: Now write $\text{Top}$ for the $(\infty, 1)$-category of topological spaces.

(a) For each topological space $X$, let $X \times \{ 0 \} \rightarrow X \times I$ be the canonical inclusion for each topological space $X$. $\{ X \times \{0 \} \rightarrow X \times I : X \in \text{Obj}(\text{Top}) \}^{*}$ is the class of fibrations in $\text{Top}$.

(b) $\text{Top}$ classifies fibrations, in the sense that $(\infty, 1 ) \text{-Cat}(\Pi_{\infty} (X), \text{Top})$ is naturally isomorphic to the category of fibrations over $X$.

Example 3: Write $\text{Cat}$ for the 2-category of small categories.

(a) Let $ 1 \rightarrow 2$ be the inclusion of the categories $1$ and $2$, sending the unique object of $1$ to the target of the unique map in $2$. A fibered category is (almost!) the same thing as an element of $\{ \delta \}^{*}$, but we need to tweak the lifting property so that we always have a universal lift.

(b) $\text{Cat}$ classifies fibered categories, in the sense that $2 \text{Cat}(C, \text{Cat})$ is naturally isomorphic to the $2$-category of fibered categories over $X$.

This next one might take some tweaking, but something like this should be true:

Example 4: Write $\text{Ring}$ for the category of simplicial rings.

(a) Let $N$ be the set of maps $\phi : A \rightarrow B$ which are surjective with kernel containing only nilpotents. Then $N^\perp$ is the class of formally étale maps.

(b) $\text{Set}$ classifies formally étale maps (or something close), in the sense that $[\pi_1(A), \text{Set}]$ is naturally isomorphic to the category of étale $A$-algebras (or something close).

In each case, it seems like the theory is very similar. In each example, we can view these classes of maps as arising from a lifting property or weak lifting property along some basic class of maps. And in each one, we can find some kind of classifying object. So my question is, is there some theorem which expresses all or most of these examples in one fell swoop? Is there some common explanation that they have? What is the explicit connection between these lifting properties and classifying objects?

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    $\begingroup$ There are two different phenomena happening here. If your class of fibrations is $\mathcal{J}^*$ for some set $\mathcal{J}$ of maps, then the latter is a class of generating trivial cofibrations (at least using the model categorical language.) So you're observing some examples of interesting weak factorization systems generated by a single trivial cofibration. This is not the same thing as having fibrations classified by an object, which means that every fibration is the pullback of some "universal" fibration. The latter is what's happening in your (b) examples: note that the (b) examples... $\endgroup$ – Kevin Carlson Jul 25 at 1:04
  • $\begingroup$ ...Are not defined in your presentation by a lifting property against a given class of maps (although they are all defined by a lifting property.) $\endgroup$ – Kevin Carlson Jul 25 at 1:04
  • $\begingroup$ In your example $2.b)$, you should replace $\mathbf{Set}$ by the category of non-empty sets (unless you're thinking of a different isomorphism than me). $\endgroup$ – Arnaud D. Jul 25 at 12:21
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    $\begingroup$ I do know of some fairly general occurence of this kind of phenomenom, but I don't think there is anything deep going on here. I think there is nothing more than the fact that many class of maps can be characterized by lifting property and many class of maps have some sort of classifier, so it is not so surprising that one can find many examples where both happen simultaneously. Especially that some of the example you give are somehow far fetch: ... $\endgroup$ – Simon Henry Jul 25 at 17:25
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    $\begingroup$ ...for Ex 2 the identification only works in a "up to homotopy sense" and up to homotopy every map is a fibration, so fibrations actually play no role in the story. Ex 3 is not quite a lifting property and Ex 4 is classified by something leaving in a very different category and using an object $\pi_1(A)$ artificially defined to fill that purpose. $\endgroup$ – Simon Henry Jul 25 at 17:27

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