I am also looking forward to answers to your question. Meanwhile here is something pointing roughly into that direction:

One can study a category $C$ through its set-valued functor category $Set^C$. By the Yoneda lemma, $C$ sits as a full subcategory inside this functor category, and from it one can reconstruct something close to $C$ (I think the idempotent completion of $C$). But non-equivalent categories can give rise to equivalent functor categories, e.g. category $C$ in which not every idempotent splits and its idempotent completion, i.e. the category made from $C$ by adjoining objects such that each idempotent becomes a composition of projection to and inclusion of a subobject and thus splits. One calls such categories Morita-equivalent.

Now $Set^C$ is a Grothendieck topos (:=category of sheaves on a site, in this case with trivial topology) and there is the following theorem about those:

A locale is a distributive lattice closed under meets and finite joins, just like the lattice of open sets of a topological space, so it is a particular poset. The theorem of Joyal and Tierney, from their monograph "An extension of the Galois theory of Grothendieck", states that every Grothendieck topos is equivalent to the category of $G$-equivariant sheaves on a groupoid object in locales - see e.g. here.

Well at least it is a statement which separates a category into a groupoid and a poset part. So if you look from very far and take it with a boulder of salt you could read this as saying that every category is "Morita-equivalent" (not really!) to a groupoid internal to posets (it makes some intuitive sense to see this as an extension).

I am also looking forward to answers to your question. Meanwhile here is something pointing roughly into that direction:

One can study a category $C$ through its set-valued functor category $Set^{C^{op}}$. By the Yoneda lemma, $C$ sits as a full subcategory inside this functor category, and from it one can reconstruct something close to $C$ (namely the idempotent completion of $C$). But non-equivalent categories can give rise to equivalent functor categories, e.g. a category $C$ in which not every idempotent splits and its idempotent completion, i.e. the category made from $C$ by adjoining objects such that each idempotent becomes a composition of projection to and inclusion of a subobject and thus splits. One calls such categories Morita-equivalent.

Now $Set^{C^{op}}$ is a Grothendieck topos (:=category of sheaves on a site, in this case with trivial topology) and there is the following theorem about those:

A locale is a distributive lattice closed under meets and finite joins, just like the lattice of open sets of a topological space, so it is a particular poset. The theorem of Joyal and Tierney, from their monograph "An extension of the Galois theory of Grothendieck", states that every Grothendieck topos is equivalent to the category of $G$-equivariant sheaves on a groupoid object in locales - see e.g. here.

Well at least it is a statement which separates a category into a groupoid and a poset part. So if you look from very far and take it with a boulder of salt you could read this as saying that every category is "Morita-equivalent" (not really!) to a groupoid internal to posets (it makes some intuitive sense to see this as an extension).

I am also looking forward to answers to your question. Meanwhile here is something pointing roughly into that direction:

One can study a category $C$ through its set-valued functor category $Set^C$. By the Yoneda lemma, $C$ sits as a full subcategory inside this functor category, and from it one can reconstruct something close to $C$ (I think the idempotent completion of $C$). But non-equivalent categories can give rise to equivalent functor categories, e.g. category $C$ in which not every idempotent splits and its idempotent completion, i.e. the category made from $C$ by adjoining objects such that each idempotent becomes a composition of projection to and inclusion of a subobject and thus splits. One calls such categories Morita-equivalent.

Now $Set^C$ is a Grothendieck topos (:=category of sheaves on a site, in this case with trivial topology) and there is the following theorem about those:

A locale is a distributive lattice closed under meets and finite joins, just like the lattice of open sets of a topological space, so it is a particular poset. The theorem of Joyal and Tierney, from their monograph "An extension of the Galois theory of Grothendieck", states that every Grothendieck topos is equivalent to the category of $G$-equivariant sheaves on a groupoid object in locales - see e.g. here.

Well at least it is a statement which separates a category into a groupoid and a poset part. So if you look from very far and take it with a boulder of salt you could read this as saying that every category is "Morita-equivalent" (not really!) to a groupoid internal to posets (it makes some intuitive sense to see this as an extension).

I am also looking forward to answers to your question. Meanwhile here is something pointing roughly into that direction:

One can study a category $C$ through its set-valued functor category $Set^C$. By the Yoneda lemma, $C$ sits as a full subcategory inside this functor category, and from it one can reconstruct something close to $C$ (I think the idempotent completion of $C$). But non-equivalent categories can give rise to equivalent functor categories, e.g. category $C$ in which not every idempotent splits and its idempotent completion, i.e. the category made from $C$ by adjoining objects such that each idempotent becomes a composition of projection to and inclusion of a subobject and thus splits. One calls such categories Morita-equivalent.

Now $Set^C$ is a Grothendieck topos (:=category of sheaves on a site, in this case with trivial topology) and there is the following theorem about those:

A locale is a distributive lattice closed under meets and finite joins, just like the lattice of open sets of a topological space, so it is a particular poset. The theorem of Joyal and Tierney, from their monograph "An extension of the Galois theory of Grothendieck", states that every Grothendieck topos is equivalent to the category of $G$-equivariant sheaves on a groupoid object in locales - see e.g. here.

Well at least it is a statement which separates a category into a groupoid and a poset part. So if you look from very far and take it with a boulder of salt you could read this as saying that every category is "Morita-equivalent" (not really!) to a groupoid internal to posets (it makes some intuitive sense to see this as an extension).