By Reid Barton

2019-03-12 17:09:56 8 Comments

Remark 1.30 of Adámek and Rosický, Locally Presentable and Accessible Categories claims that in any locally $\lambda$-presentable category, each $\mu$-presentable object (for $\mu\ge\lambda$) can be written as a $\mu$-small colimit of $\lambda$-presentable objects. I've also seen this stated in the literature without any reference given, suggesting it is considered "well-known to experts".

However, as Mike Shulman pointed out in a comment on the answer, it is unclear how the argument on pages 35 to 37 of Makkai and Paré cited in Remark 1.30 proves the claim. Not only is it unclear how to apply Lemma 2.5.2 of MP, but the category $\mathbf{K}$ constructed in its proof, which is the indexing category for the colimit produced by the lemma, has size which is not obviously bounded in terms of the sizes of the input diagrams, because it involves arbitrary morphisms between the given objects, not just ones that appear in the given diagrams.

Does anyone know how the claim of Remark 1.30 is to be proved? Alternatively, is there another, perhaps entirely different, proof in the literature?


@Jiří Rosický 2019-03-13 07:28:13

In our Remark 1.30, we should quote MP 2.3.11 (instead of MP, pp. 35-37). This Proposition shows how to replace a retract of a $\kappa$-filtered colimit by a $\kappa$-filtered colimit of a diagram obtained from the original by adding some morphisms, without changing the size of the diagram.

@Mike Shulman 2019-03-13 15:09:13

I thought of that; MP 2.3.11 is what you cite in Remark 2.15 for the analogous result in the case when $\mu\rhd \lambda$ and the colimits are $\lambda$-filtered. But doesn't the proof of MP 2.3.11 rely on the fact that the colimits are $\lambda$-filtered?

@Jiří Rosický 2019-03-13 16:07:59

Mike is right and I should take my answer back.

@Reid Barton 2019-03-12 20:21:58

Here's an argument which I currently believe. As Tim suggested, it does use the fat small object argument. References are to that paper.

Let $\mathcal{K}$ be a locally $\lambda$-presentable category and define $\mathcal{X}$ to be the class of all morphisms between $\lambda$-presentable objects. Assume $\mu \ge \lambda$ is uncountable; this is no loss of generality because the claim is trivial when $\mu = \lambda$.

Lemma 1: A map with the right lifting property with respect to $\mathcal{X}$ is an isomorphism.

Proof. Let $f : X \to Y$ be such a map. For each $\lambda$-presentable object $A$, the right lifting properties with respect to $\emptyset \to A$ and $A \amalg A \to A$ imply that $f_* : \mathrm{Hom}(A, X) \to \mathrm{Hom}(A, Y)$ is surjective and injective, respectively. Since the $\lambda$-presentable objects are dense in $\mathcal{K}$ it follows that $f$ is an isomorphism.

Lemma 2: Any map of $\mathcal{K}$ can be written as a transfinite composition of pushouts of morphisms of $\mathcal{X}$.

Proof. $\mathcal{X}$ is essentially small, so we may use the version of the small object which attaches one cell at a time to write any map $f : X \to Y$ as a transfinite composition of pushouts of maps of $\mathcal{X}$ followed by a map with the right lifting property with respect to $\mathcal{X}$. By Lemma 1, the latter map is an isomorphism so the original map is already a transfinite composition of the desired form.

Now, let $A$ be a $\mu$-presentable object of $\mathcal{K}$. By Lemma 2, we can write $\emptyset \to A$ as a transfinite composition of pushouts of morphisms of $\mathcal{X}$. However, the small object argument gives no control over the length of this composition (because there might be many lifting problems to solve at each stage). But we may apply Lemma 4.15 with $\kappa = \mu$ to replace this transfinite composition with one of length less than $\mu$ which is still a transfinite composition of pushouts of elements of $\mathcal{X}$.

This is not quite what we want, because objects after the first $\lambda$ need not be $\lambda$-presentable. But using Theorem 4.11 with $\kappa = \lambda$, we can convert this diagram to a $\lambda$-good $\lambda$-directed colimit whose links are pushouts of morphisms of $\mathcal{X}$. In particular, all the objects in this diagram are $\lambda$-presentable by Remark 4.14. Furthermore, inspection of the proof reveals that the links of this diagram are in one-to-one correspondence with the morphisms in the input transfinite composition, hence of cardinality less than $\mu$. The entire diagram may not be $\mu$-small, but we can throw away all the parts which are not below some isolated element. Because the diagram was $\lambda$-good, the resulting diagram has fewer than $\lambda \mu = \mu$ objects, and its colimit is still $A$ by Lemma 4.9. Thus, we have written $A$ as a $\mu$-small colimit of $\lambda$-presentable objects.

I'll leave this question open for a while in case someone can provide a reference to the literature.

@Kevin Carlson 2019-03-12 23:22:03

This looks to me like the kind of thing Garner's comonadic small object argument might also help with.

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