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  "body": "\n## Summary\n\nA concrete category is a [[category|A2DE66937D]] $\\mathcal{C}$ together with a [[faithful functor|GRG8WEBSH5]] $U$ from $\\mathcal{C}$ to the category of sets.\nThe functor $U$ assigns an underlying set to each object, and morphisms are determined by their underlying functions.\nGroups, rings, vector spaces and topological spaces with their usual forgetful functors are concrete categories.\nNot every category admits such a functor, so concreteness is a real condition on large categories.\n\n## Explanations\n\n### L1 human\n\nIn many categories the objects are sets with extra structure and the arrows are functions that respect the structure.\nA concrete category records this through a forgetful functor that remembers only the underlying set and the underlying function.\nBeing faithful means that two arrows with the same underlying function are the same arrow.\nFor example, a continuous map between two topological spaces is nothing more than a function with a property.\nSo topological spaces with the forgetful functor form a concrete category.\n",
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