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<p>'''Definition 1.1.'''  A ''concrete category'' is a collection of two kinds of entities, called ''objects'' and ''morphisms''.  The former are sets which are endowed with some kind of structure, and the latter are mappings, that is, functions from one object to another, in some sense preserving that structure.  Among the morphisms, there is attached to each object A the ''identity mapping'' <math>1_A : A \to A</math> such that <math>1_A(a) = a\!</math> for all <math>a \in A.\!</math>  Moreover, morphisms <math>f : A \to B</math> and <math>g : B \to C</math> may be ''composed'' to produce a morphism <math>gf : A \to C</math> such that <math>(gf)(a) = g(f(a))\!</math> for all <math>a \in A.\!</math></p>

<p>'''Definition 1.1.'''  A ''concrete category'' is a collection of two kinds of entities, called ''objects'' and ''morphisms''.  The former are sets which are endowed with some kind of structure, and the latter are mappings, that is, functions from one object to another, in some sense preserving that structure.  Among the morphisms, there is attached to each object <math>A\!</math> the ''identity mapping'' <math>1_A : A \to A</math> such that <math>1_A(a) = a\!</math> for all <math>a \in A.\!</math>  Moreover, morphisms <math>f : A \to B</math> and <math>g : B \to C</math> may be ''composed'' to produce a morphism <math>gf : A \to C</math> such that <math>(gf)(a) = g(f(a))\!</math> for all <math>a \in A.\!</math></p>

<p>We shall now progress from concrete categories to abstract ones, in three easy stages.</p>

<p>We shall now progress from concrete categories to abstract ones, in three easy stages.</p>