Questions tagged [forgetful-functors]

For questions involving the forgetful functor, the functor that drops some of the properties of the input structure before mapping to the output.

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Do subobjects in concrete categories correspond to subsets?

A concrete category is a category $C$ endowed with a faithful functor $U:C\rightarrow Set$. And if $a$ is an object in $C$, then a subobject of $a$ is an isomorphism class of monomorphisms with ...
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Completeness of over category (slice)

Let $\mathcal J$ be a (small) category (denote $I:= \mathcal J_0$) and $\mathcal C$ a category that has all (small) limits (all limits of shape $\mathcal J$ for all $\mathcal J$). Prop 3.4 states then ...