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For ordinary category theory, we have the following fact.

A weighted colimit of a functor can always be equivalently expressed as a colimit of a different functor.

Specifically, the weighted colimit of $F\colon \mathcal{C} \to \mathcal{D}$ with weight $W\colon \mathcal{C}^{\mathop{op}} \to Set$ is equivalent to the colimit of $\int W \to \mathcal{C} \xrightarrow{F} \mathcal{D}$, where $\int W$ is the category of elements (or Grothendieck construction) of $W$: $$ \mathrm{colim}^W F \cong \mathrm{colim} \left( \int W \to \mathcal{C} \xrightarrow{F} \mathcal{D} \right). $$

This is presented e.g. in Riehl's book as (7.2.4).

Enriching in $\mathbf{Pos}$, the cartesian monoidal category of sets and monotone maps, the analogous statement is the following.

A weighted $\mathbf{Pos}$-colimit of a $\mathbf{Pos}$-functor can always be equivalently expressed as a conical $\mathbf{Pos}$-colimit of a different $\mathbf{Pos}$-functor.

Is this statement true? More than that, I am looking also for an intuitive reason for why this should be the case (or not).

Some observations

  1. This statement is famously not true for $\mathbf{Cat}$-enriched categories (strict 2-categories). The counterexample presented as 3.54 in Kelly's book (for the dual statement about limits) is based on the observation that the free 2-category generated by a single 2-cell $\alpha\colon f \Rightarrow f$ admits an underlying (1-)category which looks like $0 \xrightarrow{f} 1$, and that category admits $1 \times 1$ as a product, which should lift to a conical 2-limit but doesn't. This counterexample does not apply to $\mathbf{Pos}$-categories because the 2-cell structure is thin: an endo 2-cell must be the trivial identity 2-cell. I actually think I cannot build any counterexamples this way, because the functor $V = \mathbf{Pos}(\{*\}, -)\colon \mathbf{Pos} \to \mathbf{Set}$ is conservative (?) and hence any limit in the underlying category of a $\mathbf{Pos}$-category lifts to a conical $\mathbf{Pos}$-limit.

  2. Riehl says in her book (I believe) that this is true for $\mathbf{Set}$ in the dual case precisely because a functor is representable if and only if its category of elements has an initial object (7.1.12). I haven't thought too hard about whether or not this is true for $\mathbf{Pos}$, but even if it were I don't quite follow her argument in the $\mathbf{Set}$ case; she seems to be saying that a weighted limit is given by an end expression, which in $\mathbf{Set}$ can be rewritten as a particular limit which looks like it could be the result of writing down a different limit (the one in the category of elements) as products and equalisers, but I'm not sure how this generalises to $\mathcal{V}$-enriched or $\mathbf{Pos}$-enriched.

  3. Every $\mathcal{V}$-enriched presheaf $\mathcal{C}^{\mathop{op}} \xrightarrow{F} \mathcal{V}$ is a weighted colimit of $\mathcal{V}$-representable functors. Section 3.9 of Kelly's book (I believe) presents an argument that if one were to try to write down an expression for $F$ using just conical $\mathcal{V}$-colimits, this map would only go one way in general (this is the comparison map of 3.56) - it just so happens that for the case of $\mathcal{V} = \mathbf{Set}$ this is an isomorphism. This argument feels almost a bit circular, so at least I cannot extract an intuitive explanation from it.

  4. All $\mathcal{V}$-limits are built from conical $\mathcal{V}$-limits and cotensors. So the statement about ordinary categories can be thought of as $\mathbf{Set}$ having trivial cotensors. Indeed, in the $\mathbf{Cat}$-enriched case you only need a few different cotensors to get everything - I believe this is what PIE limits are about, the central idea of which is that all weighted $\mathbf{Cat}$-limits are given by products, inserters, and equifiers (with some caveats about strictness). I don't know what cotensors look like in $\mathbf{Pos}$ though, or how that would constitute a (dis)proof for my question.

Riehl, Emily, Categorical homotopy theory, New Mathematical Monographs 24. Cambridge: Cambridge University Press (ISBN 978-1-107-04845-4/hbk; 978-1-107-26145-7/ebook). xviii, 352 p. (2014). ZBL1317.18001.

Kelly, G. M., Basic concepts of enriched category theory, Repr. Theory Appl. Categ. 2005, No. 10, 1-136 (2005). ZBL1086.18001.

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    $\begingroup$ Pos-cotensors in Pos itself are just exponentials in the cartesian closed sense. I wouldn't even know how to express a cotensor with the nontrivial 2-element poset as a conical limit. But maybe that's possible since Kelly goes a more complicated route for his counterexample in Cat? Does anyone know how? $\endgroup$
    – Jonas Frey
    Commented Jan 13 at 22:52
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    $\begingroup$ The example in your Observation 1 is not relevant to the problem of expressing weighted limits as conical limits. It is an example of a diagram in an enriched category which admits a limit in the underlying category but not in the enriched category. $\endgroup$
    – Alexander Campbell
    Commented Jan 14 at 6:36
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    $\begingroup$ I might be nitpicking, but I think the question is too vague: Every weighted colimit $X= colim_W D(i)$ in $C$ is the colimit of the constant functor $* \to C$, $F(*) = X$. This is clearly not the answer you expect, but this answers the question which is asked. Given that the actual question probably has a negative answer, I feel the details of how one gets ride of this sort of trivial answer is actually important. For e.g. Giacomo's answer is making the additional assumption that the replacement needs to take value in the same full subcategory as the original weighted diagram. $\endgroup$
    – Simon Henry
    Commented Jan 22 at 20:09
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    $\begingroup$ Note that this is important as for example you can always express a Weigthed colimit as a conical colimit of tensor. $\endgroup$
    – Simon Henry
    Commented Jan 22 at 20:11
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    $\begingroup$ Simon makes a good point. In my head the question is asking whether the class of all weighted colimits is the "saturation" of the class of conical colimits (in the sense of Kelly-Schmitt). That is, whether every $\mathcal V$-category with conical colimits also has all weighted colimits, and every $\mathcal V$-functor preserving conical colimits, also preserves all weighted ones. The answer is negative for $\mathcal V=\mathbf{Pos}$ since discrete posets form a $\mathbf{Pos}$-category that has all conical colimits but lacks the weighted ones. $\endgroup$
    – Giacomo
    Commented Jan 23 at 12:25

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Consider the terminal object $1\in\mathbf{Pos}$.

Then the closure of $1$ in $\mathbf{Pos}$ under all weighted colimits is $\mathbf{Pos}$ itself: If $X\in\mathbf{Pos}$, then $X\cong X\cdot 1$ is the copower (= tensor) of $1$ by $X$.

On the other hand, the closure of $1$ in $\mathbf{Pos}$ undel all conical colimits is the full subcategory spanned by the dicrete posets (which is equivalent to $\mathbf{Set}$); this is because every discrete poset is a coproduct of copies of $1$ and any conical colimit of discrete posets is still discrete.

Thus, it is not possible to express any $\mathbf{Pos}$-weighted colimit as a conical one (otherwise the two $\mathbf{Pos}$-enriched categories above would be the same).

This counterexample works more generally whenever the unit $I$ of the base of enrichment $\mathcal V_0$ is not colimit-dense (that is, the closure of $I$ under ordinary colimits in $\mathcal V_0$ is strictly contained in $\mathcal V_0$).

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