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

By Jiri Adamek

This up to date introductory remedy employs classification conception to discover the speculation of constructions. Its new angle stresses concrete different types and provides a scientific view of factorization buildings, delivering a unifying point of view on past paintings and summarizing contemporary advancements. a variety of examples, starting from common to express, remove darkness from the textual content. 1990 version, up-to-date 2004.

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Let X be a topological space, and let ≈ be the equivalence relation on X, given by: x ≈ y ⇔ the closure of {x} = the closure of {y}. Then the canonical map X → X/ ≈ is an A-reflection for X. (6) Making an abelian group torsion-free: B = Ab, A = TfAb, the full subcategory of torsion-free abelian groups. Let G be an abelian group and let T G be the torsion subgroup of G. Then the canonical map G → G/T G is an A-reflection for G. , those reachable acceptors with the property that no two different states are observably equivalent.

Here we list a few examples of full coreflective subcategories A of categories B. A. Modifications of the Structure 18th January 2005 Sec. 4] Subcategories 57 id X (1) Making a relation symmetric: B = Rel, A = Sym. (X, ρ ∩ ρ−1 ) −−− → (X, ρ) is an A-coreflection for (X, ρ). This example shows one of the rare instances where a subcategory is simultaneously reflective and coreflective [cf. 17(1)]. , spaces in which every sequentially closed set is closed). If (X, τ ) is a topological space, then τ = {A ⊆ X | X \ A is sequentially closed in (X, τ )} id X is a topology on X and (X, τ ) −−− → (X, τ ) is an A-coreflection for (X, τ ).

Show that every subcategory A of B can be embedded into a smallest isomorphism-closed subcategory A of B that contains A. The inclusion functor A → A is an equivalence iff all Bisomorphisms between A-objects belong to A. 4C. Full Subcategories (a) Show that a category is discrete if and only if each of its subcategories is full. (b) Show that in a poset, considered as a category, • every subcategory is isomorphism-closed, • every (co)reflective subcategory is full. 4D. Reflective Subcategories of Special Categories * (a) Show that Set has • precisely three full, isomorphism-closed, reflective subcategories, • precisely two full, isomorphism-closed, coreflective subcategories, • infinitely many reflective subcategories, • infinitely many coreflective subcategories.

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