Relative Concreteness
In some parts of category theory, most notably topos theory, it is common to replace the category Set with a different category X, often called a base category. For this reason, it makes sense to call a pair (C,U) where C is a category and U a faithful functor C → X a concrete category over X. For example, it may be useful to think of the models of a theory with N sorts as forming a concrete category over SetN.
In this context, a concrete category over Set is sometimes called a construct.
Read more about this topic: Concrete Category
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