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Let $C$ be a category, and $\Gamma$ a graph in $C$. Under good conditions it makes sense to talk about the limit $\lim \Gamma$ of $\Gamma$ in $C$.

Suppose $\Gamma$ is the union of two subgraphs $\Gamma_1, \Gamma_2$. In general, what can one expect among $\lim \Gamma$, $\lim \Gamma_1$, $\lim \Gamma_2$, $\lim (\Gamma_1 \cap \Gamma_2)$? If any, does it admit any higher categorical generalization?

EDIT: A slicker way to formulate could be the following. Do categorical limits as functors preserve (co)limits?

As a concrete example, let $R$ be a commutative ring. Consider the category of schemes over $spec(R)$. Now let $spec(X), spec(Y), spec(S)$ be three affine schemes over $spec(R)$. There are six ways of pulling back, and they fit nicely together.

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  • $\begingroup$ If the question is "can I recover all of the data from some of the data?" then I think the answer is a sound "no": it strongly depends on the shape of the graphs $\Gamma, \Gamma_1,\Gamma_2$, on the codomain $C$ of the graph homomorphisms, etc. $\endgroup$
    – fosco
    Commented Dec 7, 2020 at 17:15
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    $\begingroup$ This, said, the functor that sends a diagram into its limit is (often) a right adjoint, so commutes with limits; it is rarely a left adjoint. $\endgroup$
    – fosco
    Commented Dec 7, 2020 at 17:16
  • $\begingroup$ memo: similar discussion can be found in Lurie's HTT, chapter 4.2.3. $\endgroup$
    – Student
    Commented Mar 10, 2021 at 18:13

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