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I'm interested in locally small, cocomplete categories $\mathbf{C}$ such that every limit preserving functor

$$\mathbf{C}^\mathrm{op}\to\mathbf{Set}$$

is representable. Is there a name for such categories?


I thought they might be equivalent to some existing notion such as locally presentable categories or total categories, but I can't prove the equivalence in either case.

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    $\begingroup$ Call Cunnigham category your notion. Any loc. pres category is Cunnigham, because of the AFT. The converse is not true, in fact, Top is Cunnigham but not locally presentable. Any cocomplete well-copowered category with a generator is Cunnigham. $\endgroup$ – Ivan Di Liberti Oct 9 '18 at 16:47
  • $\begingroup$ @IvanDiLiberti Good example! Is there also an easy example of a cocomplete category that is well-copowered but not total? $\endgroup$ – Oscar Cunningham Oct 9 '18 at 16:56
  • $\begingroup$ I am not sure to know such an example. But your question is related to this one: mathoverflow.net/questions/312051/virtual-generators $\endgroup$ – Ivan Di Liberti Oct 9 '18 at 17:00
  • $\begingroup$ @IvanDiLiberti I found some references, so I gave my own answer. Maybe it will help your question? $\endgroup$ – Oscar Cunningham Oct 9 '18 at 21:26
  • $\begingroup$ Thanks Oscar :). $\endgroup$ – Ivan Di Liberti Oct 9 '18 at 21:27
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Without the cocompleteness condition, these were studied in G. M. Kelly's paper "A survey of totality for enriched and ordinary categories" under the name compact categories, and in this mathoverflow question under the much better name saft categories (after the Special Adjoint Functor Theorem).

In that question, Theo Johnson-Freyd gives the following equivalent characterisations:

Theorem. Let $\mathbf{C}$ be a locally small category. Then the following are equivalent:

  1. Every continuous functor $\mathbf{C}^\mathrm{op}\to\mathbf{Set}$ is representable.

  2. For all locally small $\mathbf{D}$, every cocontinuous functor $\mathbf{C}\to\mathbf{D}$ has a right adjoint.

Proof. (1) $\Rightarrow$ (2) Let $F:\mathbf{C}\to\mathbf{D}$ be cocontinuous. For $d\in\mathbf{D}$ the functor $\mathrm{Hom}(F(-),d):\mathbf{C}^\mathrm{op}\to\mathbf{Set}$ is continuous, and hence representable by an object $G(d)\in\mathbf{C}$. Then given any $c\in\mathbf{C}$ we have $\mathrm{Hom}(F(c),d)\simeq\mathrm{Hom}(c,G(d))$, which [by Categories for the Working Mathematician IV.2.ii] is enough to establish that $F$ has a right adjoint agreeing with $G$ on objects.

(2) $\Rightarrow$ (1) Let $F:\mathbf{C}^\mathrm{op}\to\mathbf{Set}$ be continuous. Then $F^\mathrm{op}:\mathbf{C}\to\mathbf{Set}^\mathrm{op}$ is cocontinuous and $\mathbf{Set}^\mathrm{op}$ is locally small, so $F^\mathrm{op}$ has a right adjoint $G$. Then $F$ is represented by $G(1)$ since $F(c)\simeq\mathrm{Hom}_\mathbf{Set}(1,F(c))\simeq\mathrm{Hom}_{\mathbf{Set}^\mathrm{op}}(F(c),1)\simeq\mathrm{Hom}_\mathbf{C}(c,G(1))$. $\square$


They also point out that every saft category must be complete (since for any small diagram $F:\mathbf{K}\to\mathbf{C}$ the functor $\lim_{k\in K}\mathrm{Hom}(-,F(k)):\mathbf{C}^\mathrm{op}\to\mathbf{Set}$ is continuous, and hence representable by an object of $\mathbf{C}$ which is therefore the limit of $F$). Interestingly they don't have to be cocomplete, a property normally needed for the SAFT.

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    $\begingroup$ I might suggest that the failure of cocompleteness is an indication that total categories are a more natural class than saft ones. This is further indicated by the weirdness of the Adamek example of a saft, non-total category. $\endgroup$ – Kevin Carlson Oct 10 '18 at 5:26

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