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I'm working with Vistoli's definition of stacks [1]

Let $\Phi:C^{op}\to {Cat}$ be a psudeofunctor.

Is there always a minimal Grothendieck topology on $C$ such that it induces a stack? If not, are the conditions on $C$ or $\Phi$ that one can impose to derieve the existance of such a topology (maybe along with a description)?

1 - Notes on Grothendieck topologies, fibered categories and descent theory

Omer Rosler
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Well, every functor is certainly a stack with respect to the minimal Grothendieck topology, namely, that in which the only coverings are isomorphisms. So it is more reasonable to ask for a maximal topology with respect to which $\Phi$ is a stack. For this, you can just take the topology consisting of sieves $S$ such that $\Phi$ is a stack with respect to every pullback of $S$. There's nothing special about stacks, here-you tell the same story for strict sheaves.

Kevin Carlson
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  • Perhaps I was asking the wrong question, this seems right, though I still need to go through the details. I have another question, in a suitable category of sieves (maybe cover preserving morphisms that induce morphism of topoi on the categories of stacks?), if we restrict to the subcategory where $\Phi$ is a stack does this mximal sieve become a terminal object? – Omer Rosler Feb 18 '18 at 05:47
  • I didn't say anything about a maximal sieve for which $\Phi$ is a stack, which doesn't really make sense. We were talking about the maximal topology for which $\Phi$ is a stack, which is certainly a terminal object in the category of topologies whose morphisms are containments. But this kind of thing is true about the maximum of any partially ordered set, so it hasn't got anything to do with the problem at hand. – Kevin Carlson Feb 19 '18 at 00:56
  • Oh right, that was a typo, I meant to say site. Thank you for the answer, but I was thinking about a more geometric category of topologies, not the one induced by inclusion, where it is indeed an obvious fact, in these kinds of topologies it is not so obvious – Omer Rosler Feb 19 '18 at 05:43