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As stated in this question, colimits of $\mathsf{Set}$-valued diagrams $F:D\to \mathsf{Set}$ can be calculated as $\varinjlim F=\coprod_{x\in D}T(x)/\sim$, where $\sim$ is the minimal equivalence relation such that $(x,y)\sim(x',(Ff)(y))$ for all morphisms $f:x\to x'$ (this can be seen as "forcing the diagram to commute" in $\mathsf{Set}$).

Such constructions of colimits are used in many other cases - for example, the stalk of a presheaf of rings $\mathcal{F}$ on $X$ at a point $x\in X$ is the colimit $$\mathcal{F}_x=\underset{U\ni x}{\varinjlim\,} \mathcal{F}(U).$$ It can be more practically thought of as the ring of germs, which are indeed equivalence classes $[(U, s)]$, for $x\in U$ and $s\in \mathcal F (U)$, under the equivalence relation $(U, s)\sim (V,t)$ iff there exists $W\subseteq U \cap V$ s.t. $s\big|_W=t\big|_W$.

In what categories can one generaly calculate colimits as a disjoint union of the diagram modulu this kind of equivalence relation?

Robert
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    If a category has all coproducts and coequalizers, it is cocomplete and any colimit can be expressed as a coequalizer of maps into a coproduct. I'd argue this is morally the right generalization of the statement in Sets. – Thorgott Nov 26 '23 at 01:25
  • @Thorgott In the context of sheaves, I've mainly heard that the generalization is for any category in which filtered inductive limits exist. I don't quite understand if that's stronger or weaker, and what that gives you. – Robert Nov 26 '23 at 01:33
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    The comment by Thorgott is the answer to the question. – Martin Brandenburg Nov 26 '23 at 08:57
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    Depends on what you mean by "this kind of equivalence relation. In general, colimits are quotients of equivalence relations if the category has small coproducts and kernel pairs. I believe aspects of the colimit (e.g. being filtered) would correspond to aspects of the equivalence relation. Is it the latter correspondence you're asking about? – Vladimir Sotirov Nov 26 '23 at 18:58

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