The category $\textbf{Set}$ can be given a Grothendieck topology where the covering families are jointly surjective families of set inclusions $\{X_i\stackrel{\phi_i}{\hookrightarrow} X\}\in\mathrm{Cov}(X)$, $X\in\mathrm{ob}(\textbf{Set})$.

Are there any other Grothendieck topologies on $\textbf{Set}$, not equivalent to the above one?

Two Grothendieck topologies on $\mathcal{C}$ are equivalent when they give rise to sheaf categories which are equivalent as reflective subcategories of the category of presheaves on $\mathcal{C}$.

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    $\begingroup$ The trivial topology? $\endgroup$ – skd Feb 17 '17 at 19:58
  • $\begingroup$ Ok. Maybe I should make some assumptions in order to make the question more meaningful. What if I require the topology to be subcanonical? (which the trivial topology is not, if I'm not mistaken?) $\endgroup$ – Qfwfq Feb 17 '17 at 20:04
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    $\begingroup$ The minimal topology is the one where the only covering sieve of an object X is the maximal sieve of X. For this topology, every presheaf is a sheaf. The maximal topology is the one where every sieve on every object is a covering sieve. For this topology the only way a presheaf F can be a sheaf is if F(X) is a singleton for every X. $\endgroup$ – Tom Goodwillie Feb 17 '17 at 20:21
  • $\begingroup$ (I'm not sure which of those is called the trivial topology. I would guess the former.) There is also the slightly less than maximal topology in which the covering sieves are the nonempty sieves. For this the sheaves are the constant presheaves. $\endgroup$ – Tom Goodwillie Feb 17 '17 at 20:23
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    $\begingroup$ @TomGoodwillie There must be lots of similar examples - presheaves on any full subcategory of sets give a reflective subcategory in all presheaves. The reflector is restriction, and the embedding is given by the right Kan extension along the inclusion of the subcategory. Your last example corresponds to the full subcategory with the singleton as the only object. $\endgroup$ – მამუკა ჯიბლაძე Feb 17 '17 at 20:27

(Much of this has basically been said by someone in the comments already.)

Here is a way of making examples of topologies on Set. Let $\mathcal C$ be a class of sets. Define a topology on Set by saying that a sieve $S$ on an object $X$ is a cover if and only if for every $Y\in\mathcal C$ every morphism $Y\to X$ belongs to $S$. An object is a point (in the sense that the only cover of that object is its maximal sieve) if it belongs to $\mathcal C$, and more generally if it is a retract of such an object. If we enlarge $\mathcal C$ to make it closed under retraction the topology is unchanged. Any presheaf that is a right Kan extension from (the full subcategory) $\mathcal C$ is a sheaf, and conversely any sheaf coincides with the Kan extension of its restriction.

(All of that is valid for any category, not just for Set.)

In the case when $\mathcal C$ consists of just a singleton, this topology on Set is the canonical one, where the sheaves are the representable funtors. If $\mathcal C$ has at least one nonempty set in it then the sheaf is subcanonical.

In general a class of sets closed under retraction must be either: all sets with cardinality less than a fixed cardinal, or all non-empty sets with cardinality less than a fixed cardinal. For example, $\mathcal C$ might be all nonempty sets having at most $n$ elements, or all nonempty finite sets, or all nonempty countable sets, or all nonempty sets (or any of these together with the empty set).

I suppose there are examples of topologies on Set not of this kind. That is, I suppose that there is a topology $T$ on Set such that $T$ does not coincide with the largest one that has the same points as $T$. Does somebody know?

EDIT: I realize now that of course there are perfectly everyday examples not of that kind. For example, the topology generated by finite covers in the ordinary sense. ($X$ is covered by ${Y_i\to X}$ if $X$ is the union of $Y_1,\dots ,Y_n$.)

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    $\begingroup$ A quick consideration, sorry I do not see right away if it works - one might try "very epic" covers, meaning maps $Y\to X$ with cardinalities of point preimages bounded from below by some fixed cardinality. I think this should be not of the kind described in the answer... $\endgroup$ – მამუკა ჯიბლაძე Feb 18 '17 at 6:43
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    $\begingroup$ I'm not sure what you mean, but on one interpretation this does not work: If every covering sieve of $X$ has a surjective map $Y\to X$ in it then every covering sieve of $X$ has the identity map $X\to X$ in it, so the maximal sieve is the only covering sieve. $\endgroup$ – Tom Goodwillie Feb 19 '17 at 14:21
  • $\begingroup$ Yes I understand, my version does not work. I wanted to somehow describe the topology which would admit as sheaves the functors preserving "weakly filtered" colimits, by which I mean colimits over diagrams with every subdiagram of cardinality $\geqslant\varkappa$ coneable, but made mistake. And now I realize that probably this would again be a particular case of what you describe anyway. $\endgroup$ – მამუკა ჯიბლაძე Feb 19 '17 at 14:31

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