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===Concrete Category===

===Concrete Category===

<pre>

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| Definition 1.1.  A 'concrete category' is a collection of two kinds

|

| of entities, called 'objects' and 'morphisms'.  The former are sets

<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>

| 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' 1_A : A -> A

| such that 1_A(a) = a for all a in A.  Moreover, morphisms

| f : A -> B and g : B -> C may be 'composed' to produce

| a morphism gf : A -> C such that (gf)(a) = g(f(a))

| for all a in A.

| We shall now progress from concrete categories

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

| to abstract ones, in three easy stages.

 

</pre>

<p>(Lambek & Scott, 4&ndash;5).</p>

===Graph===

===Graph===