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Let $\mathcal{A}$ be a category. The Yoneda embedding $Y : \mathcal{A} \hookrightarrow \mathrm{Hom}(\mathcal{A}^{\mathrm{op}},\mathbf{Set})$ corestricts to an embedding $$y : \mathcal{A} \hookrightarrow \mathrm{Hom}_{\mathrm{c}}(\mathcal{A}^{\mathrm{op}},\mathbf{Set}),$$ where the subscript $\mathrm{c}$ indicates the subcategory of all continuous functors (this is just a very complicated way of saying that every representable functor is continuous). You can view them as "generalized sheaves" on $\mathcal{A}$.

Question. Is there a classification of those categories $\mathcal{A}$ for which $y$ is an equivalence of categories? Is there perhaps an established name for those categories? Can you provide a reference to the literature?

Some observations:

  1. The notion of a total category is similar, but I think that it is not equivalent.
  2. Every locally presentable category has this property. This includes lots of examples, of course.
  3. More generally, if $\mathcal{A}$ has the property that every cocontinuous functor on $\mathcal{A}$ is a left adjoint (I call them SAFT-categories, but I don't know if there is a more established name), then it has this property. In fact, continuous functors $\mathcal{A}^{\mathrm{op}} \to \mathbf{Set}$ are then automatically right adjoint, thus dually correspond to left adjoints $\mathbf{Set} \to \mathcal{A}^{\mathrm{op}}$, i.e. to objects of $\mathcal{A}$.
  4. Probably this is worth a separate question, but I wonder if $\mathbf{Top}$ has this property (this is work in progress).
  5. Brown's representability theorem is the "homotopy version" of this property for the homotopy category of connected pointed CW-complexes.
Martin Brandenburg
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    This must be related to limit-sketchability, but size issues make the connection complicated. The category of models of a small limit sketch is locally presentable, and in the other direction a category with your property is the category of models of a large limit sketch. Therefore the natural question to ask is whether every category of models of a (possibly large) limit sketch has the property you want – but this question is not well formulated because the category of models of a large limit sketch may fail to be locally small. But maybe that is the only problem. – Zhen Lin Apr 11 '21 at 01:51
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    "SAFT categories" have been called "compact" by Isbell. I learned this from the introduction to this paper. I think I prefer the term "SAFT category". – tcamps Apr 11 '21 at 01:52
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    The paper I linked to also uses the term "SAFT category" with a different meaning, though. That paper also shows that Isbell-compactness (=Brandenburg-SAFTness) lifts along semitopological functors. Since the forgetful functor $Top \to Set$ is topological and in particular semitopological, it follows that $Top$ is Isbell-compact, and thus has the property of the question. – tcamps Apr 11 '21 at 02:02
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    The property of the question is equivalent to Isbell-compactness. One implication is given in (3) of the question. Conversely, if $A$ has the property of the question and $F: A \to B$ is cocontinuous, then for every $b \in B$ we have that $Hom(F-,b): A^{op} \to Set$ is continuous, and hence representable by hypothesis. Call the representing object $Gb$. Then $G$ gives a right adjoint to $F$. – tcamps Apr 11 '21 at 02:09
  • @tcamps I noticed that in the paper "compact and hypercomplete categories" you mentioned compactness is actually defined by the condition that every hypercontinuous functor on $\mathcal{A}^{op}$ is representable. Hypercontinuity means that even large limits are preserved. So this will probably be a weaker notion? – Martin Brandenburg May 21 '21 at 22:05
  • @tcamps Because with "continuous" in my question I meant that small limits are preserved. – Martin Brandenburg May 21 '21 at 22:18

1 Answers1

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I think my comments have turned into an answer. The property of the question is equivalent to being a "SAFT category" in the terminology of the question, a.k.a. to being "compact" in the sense of Isbell. A reference for this notion is Borger, Tholen, Wischnewsky, and Wolff, which also gives several theorems that imply that $Top$ is Isbell-compact and compares to some of the other notions listed in the question. They also show, for example, that some of these notions coincide under (co)generation hypotheses.

I believe there may be other papers of Borger which compare this notion to totality.

tcamps
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  • Great answer! Thank you. – Martin Brandenburg Apr 11 '21 at 09:46
  • Do you see a direct argument why $\mathbf{Top}$ is Isbell-compact? (I am afraid that I won't be able to understand the theory of these papers.) I have already some ideas for the proof, but maybe it's way too complicated. – Martin Brandenburg Apr 11 '21 at 10:42
  • Namely, for a sheaf $F$ on $\mathbf{Top}$ I can construct a topology on the set $F(1)$ using the net convergence characterization (see https://math.stackexchange.com/questions/4096812) and this space must represent $F$. – Martin Brandenburg Apr 11 '21 at 11:26
  • @MartinBrandenburg Actually Top has a generator and cogenerator and is well powered and co well powered. So the usual special adjoint functor theorem and its dual simply apply, straight up, to show that Top is both Isbell compact and co compact. Also the nlab lists a totality version of SAFT, so totality implies isbell compactness. – tcamps Apr 11 '21 at 12:58
  • Er it might be co totality which implies compactness. I always get turned around like this with SAFT! – tcamps Apr 11 '21 at 13:12
  • Ah, right. Thank you :) – Martin Brandenburg Apr 11 '21 at 14:21
  • Now I feel a bit stupid for asking these questions, since both proofs are just one line. This is what happens when you don't do math for months, I guess. :) – Martin Brandenburg Apr 11 '21 at 14:27
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    Haha! We've all been there. Thanks for asking the question -- it gave me a chance to look up and get straight on some of this terminology and some of the implications! – tcamps Apr 11 '21 at 14:29