Abstract and concrete categories: the joy of cats by Jiri Adamek PDF

By Jiri Adamek

ISBN-10: 0471609226

ISBN-13: 9780471609223

This updated introductory remedy employs classification thought to discover the idea of buildings. Its special approach stresses concrete different types and provides a scientific view of factorization buildings, delivering a unifying viewpoint on prior paintings and summarizing fresh advancements. quite a few examples, starting from normal to precise, light up the textual content. 1990 version, up to date 2004.

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Pointed Categories (a) Show that there is a category whose objects are all pairs of the form (A, a), where A is a set and a ∈ A and hom((A, a), (B, b)) = { f | f : A → B and f (a) = b }. This is called the category of pointed sets (and base-point-preserving functions). It is denoted by pSet. (b) Show that there is a faithful functor U : pSet → Set with the property that U (f ) = f for each base-point-preserving function, f . Does U reflect isomorphisms? (c) Prove that Set and pSet are not equivalent.

Show, however, that BiComp = B1 ∩ B2 is not reflective in BiTop. , functions (X, α) −→ (Y, β) having the property that for each x ∈ α(F) it follows that f (x) ∈ β(f (F)), where f (F) = { M ⊆ Y | f −1 [M ] ∈ F}). , all spaces (X, α) with the property that each α(F) is a singleton set, is not reflective even though it is an intersection of a class of full reflective subcategories. ] 4G. Subcategories of Subcategories Let A be a subcategory of B and B be a subcategory of C. Prove that (a) A is a subcategory of C, (b) if A is reflective in B and B is reflective in C, then A is reflective in C, (c) if A is reflective in C and B is a full subcategory of C, then A is reflective in B, (d) if A is reflective in C, then A need not be reflective in B.

Then E2 is an embedding, G is an equivalence, and F = E1 ◦ G ◦ E2 . 6 DEFINITION A category A is said to be fully embeddable into B provided that there exists a full embedding A → B, or, equivalently, provided that A is isomorphic to a full subcategory of B. , the construct of unary algebras on two operations. Under an additional set-theoretical hypothesis (the non-existence of measurable cardinals), every construct is fully embeddable into Sgr (or Rel or Alg(1, 1)). 8 REMARK Because full subcategories are determined by their object classes, they are often regarded as “properties of objects”.

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Abstract and concrete categories: the joy of cats by Jiri Adamek


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