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In "Sheaves in Geometry and Logic", M&M define a sieve of an object $C$ as a downward-closed set of arrows $S$ with codomain $C$. They go on to say that for a locally small category, a sieve of $C$ is the same thing as a subfunctor of $\hom(-,C)$, under the rule

\[S = \{\, f : \exists A . f \in Q(A) \}\]

Is this always a set for a locally small category? It doesn't seem to be, taking for instance $Q = \hom(-,1)$. Then $S \cong \textrm{ob}(\mathcal C)$, which would imply that $\mathcal C$ is small. Is there something subtle going on here?

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  • $\begingroup$ I think your observation about a category with a terminal object is correct. You could even that $C = \mathcal Set$ itself as an example. Do you have some objection to a sieve being a class of arrows closed under precomposition? $\endgroup$
    – Eric Finster
    Commented Sep 7, 2010 at 21:03
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    $\begingroup$ In most applications, where one speaks of sieves and covering sieves (with respect to a topology), the underlying category $C$ of a site is assumed to be small. Otherwise, the sieve $S$ won't actually be a set, but a class (as you observe). Which isn't necessarily problematic; it depends what you want to do. $\endgroup$
    – Todd Trimble
    Commented Sep 7, 2010 at 23:13
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    $\begingroup$ Perhaps the objection was that M&M specifically say (on p37) that a sieve consists of a downward-closed set of arrows. However, note that on p36 the category C was introduced as "an arbitrary small category", so in the context they were working in, a sieve actually is a set. Furthermore, according to their "preliminaries" section, M&M are using the "one universe" foundation according to which even large categories are "sets" (just "large sets") rather than proper classes. $\endgroup$
    – Mike Shulman
    Commented Sep 8, 2010 at 4:19
  • $\begingroup$ Great! That clears it up. $\endgroup$
    – Aleks Kissinger
    Commented Sep 9, 2010 at 19:37

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