ONT Re: Higher Order Categorical Logic
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Note 5
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| One we admit that functors $A$ -> $B$ are interesting objects to study,
| we should see in them the objects of yet another category. We shall
| study such functor categories in the next section. For the present,
| let us mention two other ways of forming new categories from old.
|
| Example C6. From any category (or graph) $A$ one forms
| a new category (respectively graph) $A$^op with the same
| objects but with arrows reversed, that is, with the two
| mappings "source" and "target" interchanged. $A$^op is
| called the 'opposite' or 'dual' of $A$. A functor from
| $A$^op to $B$ is often called a 'contravariant' functor
| from $A$ to $B$, but we shall avoid this terminology
| except for occasional emphasis.
|
| Example C7. Given two categories $A$ and $B$, one forms a new category
| $A$ x $B$ whose objects are pairs (A, B), A in $A$ and B in $B$, and whose
| arrows are pairs (f, g) : (A, B) -> (A', B'), where f : A -> A' in $A$ and
| g : B -> B' in $B$. Composition of arrows is defined componentwise.
|
| L&S, page 7.
|
| Lambek, J. & Scott, P.J.,
|'Introduction To Higher Order Categorical Logic',
| Cambridge University Press, Cambridge, UK, 1986.
|
| http://uk.cambridge.org/mathematics/catalogue/0521356539/
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