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Given a locally finitely presentable category $\mathscr C$ and an object $X \in \mathscr C$, it is not so hard to show that a morphism $(X \to Y)$ is compact in $\mathscr C_{X/}$ if it can be obtained from a pushout square $$\begin{array}{ccc}X_0 & \to & Y_0 \\ \downarrow & & \downarrow \\ X & \to & Y\end{array}$$ with $X_0, Y_0 \in \mathscr C^\omega$. In algebraic situations (e.g. commutative rings), this is the only thing that can happen: descend all the generators and relations down to a finitely generated subobject.

Question. Is the same true in any locally finitely presentable category? Are there any natural 'smallness' restrictions on $\mathscr C^\omega$ that give a positive result?

(In general, I am only able to prove that a compact object is a retract of an object of the type above.)

I am particularly looking for references that treat this type of question, as I'm sure the answer to this is well-known (I'm trying not to reinvent the wheel for once...).

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This is true. It is for example easy to see that the full subcategory of objects of this form is closed under all finite colimits*, dense and that these objects are all finitely presentable. I unfortunately don't know a reference for this fact.

*: this is probably the hardest part. You need to write X as a filtered colimits of finitely presentable object to see that any finite collection of such objects and arrows between them can all be considered to come from the same $X_0/C \to X/C$ for $X_0$ finitely presentable.

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  • $\begingroup$ Ah right, the first part is indeed the part I didn't check, but I now see how it goes. But how do these three facts imply the statement? $\endgroup$
    – R. van Dobben de Bruyn
    Commented Jan 25 at 11:30
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    $\begingroup$ Oh yes. This is a classical argument in locally presentable categories, see for example lemma 5.2.6 in Borceux's Handbook of categorical algebra vol 2. The general idea is that given a class P of objects satisfying these conditions in a category C then for every X in C, X is the colimit of P/X, but as P has colimit this is filtered, so if X is locally presentable it will be a retract of an object of P, and hence will be in P as P is closed under retract (which are special colimits) $\endgroup$
    – Simon Henry
    Commented Jan 25 at 13:30

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