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Enriched categories, topoi, abelian categories, monoidal categories, homological algebra.

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The category of Abelian groups with selected elements

Here is an easy example which illustrates Francesco Polizzi's answer: Consider the morphisms $\mathbb{Z} \rightarrow\mathbb{Z}/12$ and $\mathbb{Z} \rightarrow\mathbb{Z}/6$, sending 1 to the class of 3 …
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