Colimits & Nonfaithful Functors: Real-Life Examples

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Colimits are defined through functors from indexing categories to other categories, typically involving faithful functors. The discussion raises the question of whether nonfaithful functors appear in real-life examples of colimits. Participants explore specific cases, including the coproduct of an object with itself, to illustrate potential nonfaithful scenarios. The conversation emphasizes the importance of understanding how nonfaithful functors can impact the structure of colimits. Overall, the exploration of nonfaithful functors in colimits reveals their significance in category theory applications.
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The colimit is defined in terms of a functor from an indexing category to another category(a diagram).

In the examples I've seen of the colimit, the functor is always faithful, so my question is, are there any nonfaithful functors from index categories that come up in real life?
 
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What about particular cases of the diagrams you already know?

e.g. the coproduct of an object with itself.
 
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