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In category theory, a branch of mathematic … In category theory, a branch of mathematics, a limit or a colimit of presheaves on a category C is a limit or colimit in the functor category . The category admits small limits and small colimits. Explicitly, if is a functor from a small category I and U is an object in C, then is computed pointwise: The same is true for small limits. Concretely this means that, for example, a fiber product exists and is computed pointwise. When C is small, by the Yoneda lemma, one can view C as the full subcategory of . If is a functor, if is a functor from a small category I and if the colimit in is representable; i.e., isomorphic to an object in C, then, in D, (in particular the colimit on the right exists in D.) The density theorem states that every presheaf is a colimit of representable presheaves. is a colimit of representable presheaves.
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| rdfs:comment |
In category theory, a branch of mathematic … In category theory, a branch of mathematics, a limit or a colimit of presheaves on a category C is a limit or colimit in the functor category . The category admits small limits and small colimits. Explicitly, if is a functor from a small category I and U is an object in C, then is computed pointwise: The same is true for small limits. Concretely this means that, for example, a fiber product exists and is computed pointwise. (in particular the colimit on the right exists in D.) The density theorem states that every presheaf is a colimit of representable presheaves. is a colimit of representable presheaves.
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| rdfs:label |
Limit and colimit of presheaves
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