Handbook of Categorical Algebra: Volume 1, Basic Category Theory
The Handbook of Categorical Algebra is designed to give, in three volumes, a detailed account of what should be known by everybody working in, or using, category theory. As such it will be a unique reference. The volumes are written in sequence, with the first being essentially self-contained, and are accessible to graduate students with a good background in mathematics. In particular, Volume 1, which is devoted to general concepts, can be used for advanced undergraduate courses on category theory.
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2-category 2-cells abelian groups adjoint functor Applying arbitrary arrows axiom Banach spaces bidense bijection called canonical category Q choose closed cocone coequalizer colim colimit commutative composite cone Consider consider diagram constitute construction Conversely coproduct corresponding deduce deﬁned deﬁnition dense diagram elements equality example exists fact factorization faithful faithful functor ﬁltered Finally ﬁnite ﬁnite limits ﬁnitely complete category ﬁrst following conditions functor F given giving holds homomorphisms identity immediately implies indicates injective internal inverted isomorphism left adjoint limit linear means monomorphism Moreover morphism f natural transformation notation notion object observe obtained obvious particular phism precisely preserves projective Proof Proposition prove pullback quotient reﬂection regular result ring satisﬁes small category spaces square strong epimorphism subcategory subobject suppose theorem theory topological unique unique factorization unit universal volume write yields Yoneda