Forgetful functor

Forgetful functor

between categories that "forget" certain structure or properties of objects in one category and only retain their underlying sets or elements. This functor is called "forgetful" because it discards information that is not relevant to the new category it is mapping to. For example, the forgetful functor from the category of groups to the category of sets forgets the group operation and only retains the underlying set of elements.From: Invariance Theory, the Heat Equation, and the Atiyah-Singer Index Theorem [2018]

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Published in Peter B. Gilkey, Invariance Theory, the Heat Equation, and the Atiyah-Singer Index Theorem, 2018

Peter B. Gilkey

We say V ∈ Vect G (M) is Riemannian if G ∈ {U (k), O (k)}. We say V is orientable if we can choose a consistent orientation for the fibers; complex vector bundles are always orientable and have natural orientations; real vector bundles need not be orientable. For example, the Möbius vector bundle over the circle is not orientable. Let Vect SO (k) (M) be the set of vector bundles V ∈ Vect O (k) (M) which are orientable and for which a fixed orientation is chosen. The forgetful functor defines maps () ℱ:VectU(j)(M)→VectSO(2j)(M)ℱ:VectSO(j)(M)→VectO(j)(M)

Constructing condensed memories in functorial time

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Published in Journal of Experimental & Theoretical Artificial Intelligence, 2023

Shanna Dobson, Chris Fields

Sheafification (Theorem 1.2.9 of Alper (2021)) Letbe a site andandbe the categories of sheaves and presheaves, respectively, on. The forgetful functoradmits a left adjoint, called the sheafification.

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Constructing condensed memories in functorial time