Abstract
The goal of this appendix is to make the reader familiar with the language of categories. We will explain the most important notions and give many examples. There will be almost no proofs in this chapter.
We start by introducing categories and functors. Important concepts are the notion of an equivalence of categories and, more generally, the notion of adjoint functors. We then give a short reminder on several notions of ordered sets and prove the principles of transfinite induction and transfinite recursion. In the last section we introduce limits and colimits in categories. This allows us to unify several important constructions such as kernels, products, fiber products, and final objects (which are all special cases of limits) or cokernels, coproducts, pushouts, and initial objects (which are all special cases of colimits).
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Notes
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Here we ignore all set-theoretical issues. To avoid set-theoretical difficulties one should work with a fixed universe in the sense of KS2 Definition 1.1.1 and assume that the sets of morphisms between two objects always lies in the given universe. Then a category \({\mathcal{C}}\) is called small if \({\rm Ob}({\mathcal{C}})\) is in that universe. Moreover it then will be sometimes necessary to pass to a bigger universe, for instance when considering the category of functors between two categories (see Definition 13.13). Finally one adds also the axiom that every set is an element of some universe to the axioms of Zermelo–Fraenkel set theory. We refer to SGA4 Exp. I for details. Alternatively one can also work with classes as explained in Sch.
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© 2016 Springer Fachmedien Wiesbaden
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Wedhorn, T. (2016). Appendix B: The Language of Categories. In: Manifolds, Sheaves, and Cohomology. Springer Studium Mathematik - Master. Springer Spektrum, Wiesbaden. https://doi.org/10.1007/978-3-658-10633-1_13
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DOI: https://doi.org/10.1007/978-3-658-10633-1_13
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