Let $C$ be a small (1-)category. There is always a poset $D$ and a functor $p : D \to C$ such that:

• $p$ is surjective on objects, i.e. for every $c$ in $C$ there is a $d$ in $D$ such that $p (d) = c$, and

• $p$ is surjective on morphisms, i.e. for every $f : c_0 \to c_1$ in $C$ there is a $g : d_0 \to d_1$ such that $p (g) = f$.

Thus we could say that every category is the surjective image of some poset, but we can do much better than just "surjective image". In fact, we can get $p$ universally localising, hence simultaneously homotopy coinitial and homotopy cofinal: see here and here in the Kerodon. In particular, every category is equivalent to the localisation of a poset!

There are many variants of results like this, the most commonly used being perhaps the one that tells us we can find for every filtered category $C$ a homotopy cofinal functor $D \to C$ where $D$ is a directed poset. Another one says that, for a Grothendieck site $(C, J)$, there is a Grothendieck site $(D, K)$ and a functor $p : D \to C$ such that $p^*$ is the inverse image functor of a surjective open geometric morphism $\textbf{Sh} (D, K) \to \textbf{Sh} (C, J)$. The proofs of these facts have a similar flavour: we take for $D$ (resp. $D^\textrm{op}$) a set (partially ordered by inclusion) of certain subcategories of either $C \times \omega$ (resp. $C \times \omega^\textrm{op}$) which have (as a property) a certain kind of distinguished object, where $\omega = \{ 0 < 1 < 2 < \cdots \}$, and $p : D \to C$ is a sort of projection functor that sends such a subcategory to its distinguished object.

Question. Is the similarity superficial, or is there something deeper going on – perhaps a more general result?

Let me elaborate a bit more on the two proofs I am alluding to.

Say that a functor strictly reflects identity morphisms if it has the right lifting property with respect to $[1] \to [0]$, i.e. $f : A \to B$ such that if $f (a_0 \to a_1)$ is an identity morphism in $B$, then $a_0 = a_1$ and the morphism is the identity. Let $\mathbf{ON}$ be the partially ordered class of (small) ordinal numbers and let $A$ be a (small) category. Then there exists a functor $A \to \mathbf{ON}$ that strictly reflects identity morphisms if and only if $A$ is a direct category; dually, there exists a functor $A \to \mathbf{ON}^\textrm{op}$ that strictly reflects identity morphisms if and only if $A$ is an inverse category.

Say that a subcategory $A \subseteq C \times \omega$ is direct if the projection $A \to \omega$ strictly reflects identity morphisms. (This is a slight abuse of terminology since a subcategory that is direct as a category may not be direct in this narrower sense.) Say that a subcategory of $C \times \omega$ is a direct cocone if it is direct (as a subcategory) and (as a category) has a terminal object; note that a terminal object in a direct category is strictly unique if it exists. Given $A \subseteq B \subseteq C \times \omega$, if $A$ and $B$ are direct cocones, then (by definition of terminal object in $B$), we have a unique morphism in $B$ from the terminal object of $A$ to the terminal object of $B$. Thus, projecting down to $C$, we obtain a functor from the poset of direct cocones to $C$.

In the case where $C$ is filtered, we take $D$ to be the set of finite direct cocones, partially ordered by inclusion. The union of finitely many finite direct subcategories generates a finite direct subcategory, and any finite direct subcategory is contained in a (not necessarily unique) finite direct cocone, so $D$ is a directed poset. It is not hard to see that, for every object $c$ in $C$, the comma poset $(c \downarrow p)$ is directed, hence weakly contractible. The projection $p : D \to C$ is then homotopy cofinal, as desired.

In the case where we want a surjective open geometric morphism, we take $D^\textrm{op}$ to be the set (partially ordered by inclusion) of finite terminal segments of $C \times \omega^\textrm{op}$, by which we mean subcategories $A$ such that the projection $A \to \omega^\textrm{op}$ is strictly injective on objects and morphisms and has image $\{ n > \cdots > 0 \}$ for some $n \ge 0$. The projection $p : D \to C$ is defined by sending a finite terminal segment to its initial object. It has a lifting property for morphisms: given a morphism $c \to p (A)$ in $C$, there is an object $B$ in $D$ such that $A \subseteq B$ and the morphism $p (B) \to p (A)$ is the one we were given; in short, the projection $(c \downarrow p) \to {}^{c /} C$ is surjective on objects. This suffices to ensure that $\textbf{Psh} (D) \to \textbf{Psh} (C)$ is an open surjection, and pulling back the Grothendieck topology $J$ on $C$ yields the desired Grothendieck topology $K$ on $D$.

It seems to me that the second factor of $C \times \omega$ (or $C \times \omega^\textrm{op}$) serves several purposes at once:

• It allows us to define direct (or inverse) subcategories.

• It allows us to "unfurl" finite direct (or inverse) diagrams in $C$ that are not injective on objects or morphisms, enabling the (normally abusive) conflation of diagrams and subcategories.

• It provides a uniform system of "coordinates" so that there is no ambiguity about how one diagram is embedded in another.

Is this a mere trick, or does it hide something deeper? Could the construction of a universally localising functor $D \to C$ also be expressed this way? (The fact that it is a kind of double subdivision where the first step does not necessarily yield a poset suggests not...)