In mathematics it happens at times that one and the same concept is given two different names to indicate a specific perspective, a certain attitude as to what to do with such objects.
A presheaf is just a contravariant functor. (More specifically, an “$S$-valued presheaf” is a contravariant functor with codomain $S$; in modern category theory the “default” value of $S$ for a presheaf is usually Set.) But one says presheaf instead of contravariant functor when one is interested in studying its sheafification, or even if one is just intersted in regarding the category of functors with its structure of a topos: the presheaf topos.
A quiver is just a directed graph (pseudograph, to be explicit). But one says quiver instead of directed graph when one is interested in studying quiver representations: functors from the free category on that graph to the category of finite-dimensional vector spaces.
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Last revised on October 1, 2018 at 13:11:42. See the history of this page for a list of all contributions to it.