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When I work with various presheaf categories, and I need some lemma, I often am able to prove the lemma by proving the analogous lemma for sets. As a simple example, let $f_i :X_i\hookrightarrow Y$ be two monomorphisms for $i=1,2$. If $g:X_1 \rightarrow X_2$ be a map such that $f_2\circ g=f_1$ Then $g$ is also a monomorphism. Let us say that this is taking place in simplicial sets (or $\mathcal{A}$-sets). Well, to prove this for simplicial sets, we prove the lemma in sets, and apply pointwise.

What we were able to do is to prove a fact about simplicial sets by reducing to the case of sets. The question is, when can we do this in full generality.

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    $\begingroup$ If the statement can be formulated in terms of predicates of the form "diagram $C$ is a limit cone", "diagram $D$ is a colimit cocone", then it will be true in presheaf categories iff it is true in $Set$. This applies for instance to your monomorphism statement. $\endgroup$
    – Todd Trimble
    Commented Mar 7, 2012 at 2:34
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    $\begingroup$ Another thing you can say is that every presheaf category is a topos, so anything you can prove in the internal language of topoi will be valid there. This leads into a long story about categorical logic - the Wikipedia page is a reasonable entry point. $\endgroup$
    – Neil Strickland
    Commented Mar 7, 2012 at 8:47
  • $\begingroup$ Todd and Neil: Thank You for your comments. $\endgroup$
    – Spice the Bird
    Commented Mar 7, 2012 at 16:55

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