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I created locally bounded category with basic results from papers of Kelly and Lack. My motivation (unfortunately not reflected in the current stub) is to provide a reference for convergence conditions for the free monad construction.
On this topic, does anyone know whether there are reasonable conditions under which the dual “free comonad” construction would converge? I’m concerned by the result at locally presentable category (new to me; does anyone have a reference?) which says that the opposite of a locally presentable category is locally presentable only if the category is a poset.
Excellent, thanks! I added some links and a redirect for the plural form.
I think the cofree comonad construction (e.g. for coinductive types) is actually often easier, because the endofunctors we start with are usually defined by mapping out of things, and therefore preserve limits automatically. They don’t generally preserve colimits, which is why the free monad is harder to construct; we have to look at highly filtered colimits that they do preserve, or pass to factorization systems instead.
But if you start with an arbitrary endofunctor, then no, I don’t know how to ensure that the cofree comonad construction converges.
The theorem that the opposite of a locally presentable category is locally presentable only if the category is a poset is attributed in Makkai-Pare (p63) to Gabriel and Ulmer’s original “Lokal präsentierbare Kategorien”, which I don’t have a copy of.
It is Satz 7.13:
Sei $A$ eine Kategorie derart, dass $A$ und $A^\circ$ lokal präsentierbar sind. Dann ist $A$ eine “Hängematte”, d.h. $A$ ist äquivalent zu einer inf- und supvollständigen geordnete Menge (vgl. 7.2g). Gibt es in $A$ ein Nullobjekt, dann ist $A \cong \{0\}$.
Apparently eine Hängematte is a hammock. 7.2g is the example
Eine inf- und supvollständige partiell geordnete Menge $M$ ist lokal präsentierbar und lokal kopräsentierbar.
So 7.13 is really an if-and-only-if.
Apparently eine Hängematte is a hammock.
Yes. Verbatim “a hanging mat” :-)
Vielen Dank!
I have added the reference to locally presentable category.
(OT: according to de.wikipedia, Hängematte was originally Hamach, from the same word as the English hammock, but was ’retconned’ over the years into its current form. A fascinating insight into the German-speaking mind!)
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