Abstract
Here we study systematically the category-theoretic concepts corresponding to universal constructions: initial and terminal objects, representable functors, adjoint functors and limits and colimits; how one of these constructions can often be expressed in terms of another, and when such constructions exist. We prove such results as that limits always “respect” other limits, and colimits other colimits, and also examine special but important situations where limits and colimits respect one another. (For instance, in the category of sets, though not in a general category, direct limits respect finite products. This is why, given a directed system of finitary algebras, one gets an induced algebra structure on the direct limit of their underlying sets.)
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Bergman, G.M. (2015). Universal Constructions in Category-Theoretic Terms. In: An Invitation to General Algebra and Universal Constructions. Universitext. Springer, Cham. https://doi.org/10.1007/978-3-319-11478-1_8
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DOI: https://doi.org/10.1007/978-3-319-11478-1_8
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-11477-4
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