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I suspect there is a strong connection between these two results, but I'm not able to work out the details; in particular I sense that there is a strong analogy between the SOA, stated in the following way

Let $\Sigma$ be a set of arrows in a cocomplete category $\bf E$; if the domains of the maps in $\Sigma$ are $\alpha$-compact[1] for a regular cardinal $\alpha$, then there exists a functor $$ R\colon \bf E\to E $$ together with a natural transformation $\varrho\colon 1\to R$ such that:

  1. The object $R(X)$ is $\Sigma$-fibrant[2] for every object $X$
  2. the map $\varrho_X$ belongs to $\text{Cell}_\alpha(\Sigma)$[3] for every $X\in \bf E$.

and the solution to the OSP, as it is stated in the famous work by Freyd and Kelly, Categories of continuous functors 1, which says, more or less,

under suitable "compactness" conditions on a complete and cocomplete category $\cal A$ and on a class of objects $\Delta$, the class of arrows orthogonal to (all the objects of) $\Delta$ form a reflective subcategory.

To my eye the first statement is a direct consequence of the second: the functor $R$ is no more than the idempotent monad $iR$ induced by the reflection $R\colon \mathbf E \to \Phi(\Sigma)\colon i$, and $\Sigma$-fibrant objects are defined precisely via an orthogonality condition.

Skimming through Garner's "Understanding the SOA" paper I'm not able to see a precise statement in this sense ([FK] paper is cited only at the very beginning). Maybe I'm wrong, or this interpretation of the SOA is implicitly understood? If I'm wrong in some subtle detail, can you tell me which one? If I'm not wrong, I'm interested in making this analogy precise! Can you give me references where this is stated properly?

Thank you for your attention.

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Let me just fix notations:

1] It means that for each $S\in\Sigma$ the functor $\mathbf E(S,-)$ commutes with $\alpha$-filtered colimits.

2] An object $X$ is "$\Sigma$-fibrant" iff its terminal arrow $X\to 1$ belongs to the class $\text{rlp}(\Sigma)$ of all morphisms having the right lifting property with respect to each arrow in $\Sigma$.

3] A class of arrows in a category is said to be cellular if it contains all isomorphisms, is closed under finite and $\alpha$-transfinite composition, and under cobase change. $\text{Cell}_\alpha$, the cellularization operator, is the closure operator generated by the property "being cellular".

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    $\begingroup$ I'd recommend checking out the paper by Casacuberta and Chorny on the orthogonal subcategory problem. I'm swamped this week but can try to answer in a few days if no one else does in the meanwhile. If you look into papers which investigate the connection between Vopenka's Principle and homotopy theory I think it'll help. $\endgroup$ Jan 6, 2014 at 23:20
  • $\begingroup$ You can use the small object argument to show that small orthogonality classes are reflective: this is because small orthogonality classes are also small injective classes. $\endgroup$
    – Zhen Lin
    Jan 7, 2014 at 2:06
  • $\begingroup$ Thank you both! @Zhen: As you can see, I was interested in going the other way round $\endgroup$
    – fosco
    Jan 7, 2014 at 9:00
  • $\begingroup$ @DavidWhite: Waiting for you I found my answer :) iti.cs.tu-bs.de/~adamek/orthogonal.AHS.pdf were you aware of it? I'll be glad to hear you if you want to add something more $\endgroup$
    – fosco
    Jan 16, 2014 at 14:33
  • $\begingroup$ Hi. Sorry for never getting around to writing an answer. Job interviews have been eating my life. I was not aware of the paper you linked, and it seems to do better than what I had in mind. I was going to phrase my answer in terms of locally presentable categories, but the AHS paper does better by only requiring quasi-presentable. Zhen Lin's comment is also nice. It's rare to see "C admits the small object argument" as a hypothesis, but I for one would like to know exactly how much that hypothesis buys. Often we make stronger assumptions like locally presentable. $\endgroup$ Jan 20, 2014 at 15:13

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