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All representable functors are continuous. This makes it possible to associate additional natural operations with them, which are absent for arbitrary presheaves.

  1. What are the sufficient and what are the necessary conditions on the subcanonical site $I$ for all sheaves to be continuous?
  2. Is there a canonical way to introduce the structure of such a site on an arbitrary small category? (by canonical I mean functorial with respect to category equivalences)
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  • $\begingroup$ Can you be more precise about how you define “continuous”? Particularly in the case where the site does not have “all” colimits. $\endgroup$
    – Zhen Lin
    Commented Dec 15, 2022 at 3:11
  • $\begingroup$ @ZhenLin I mean: functor $F : I^{op} \to \mathrm{Set}$ is continuous iff $F$ preserves all the limits that $I^{op}$ has. $\endgroup$
    – Arshak Aivazian
    Commented Dec 15, 2022 at 3:19
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    $\begingroup$ @AivazianArshak That definition is wild (= badly behaved, uncontrolled). For example, if $I$ has a terminal object then $\textrm{id} : I^\textrm{op} \to I^\textrm{op}$ has a limit, so you would demand that $F : I^\textrm{op} \to \textbf{Set}$ preserves it. $\endgroup$
    – Zhen Lin
    Commented Dec 15, 2022 at 7:07
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    $\begingroup$ For any topos $T$ you can take $T$ with its canonical topology and then sheaves = continuous presheaf = representable presheaf. But If you want a small site $C$ where sheaf are all continuous this is is a strange question because a small category (which isn't a poset) never has all limits so this is about finding a generating subcategory of the topos that isn't going to have colimits that aren't colimits in the topos... $\endgroup$
    – Simon Henry
    Commented Dec 15, 2022 at 8:30
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    $\begingroup$ By "strange question" what I mean is: you are very probably not asking what you actually want to know, you should tell use more about what it is you are trying to do, because that specific question is hard for reason that are really not interesting. $\endgroup$
    – Simon Henry
    Commented Dec 15, 2022 at 8:33

1 Answer 1

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Given a category $C$, and a familly of co-cone in $C$ (you can take all colimit cocone in $C$ if you want - the family doesn't even have to be small) there is a smallest topology on $C$ so that sheaves for this topology sends these cocone to limit cones.

The detail of the construction below should also give some answer to your question 1.

For each of the special cocone $F:I^\triangleright \to C$, we consider the map $colim_I F \to F(*)$ in the category $P(C)$ of presheaves over $C$. A presheaf sends this cocone to a colimit if and only if it is right orthogonal to this map. A Grothendieck topology is the same as a left exact localization of $P(C)$, so we are looking for the smallest left exact localization that inverse these maps.

A Grothendieck topology invert a map $f:X \to Y$ if and only if it invert the two monomorphism $Im(f) \to Y$ and $X \to X \times_Y X$ and it inverts a monomorphism $A \hookrightarrow X$ if and only if for each representable $c$ and each map $c \to X$ (that each element of $X(c)$) the Sieve on $c$ obtain as the pullback of $A \to X$ is a covering sieve.

So, putting everything together, saying that some class of maps $colim_I F \to F(*)$ is inverted by the topology (that is that sheaves sends the corresponds cocone to a limit cone) can be written as a fact that a bunch of sieve are covering sieve. and hence there is a smallest topology which satisfies these conditions. (the Grothendieck topology generated by those sieve).

Of course this topology has no reason to be subcanonical in general. For example, assuming that the category $C$ has pullback, this topology being subcanonical implies in particular that colimits in $C$ are universal (and this is probably not sufficient).

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