202411121911

Tags : Category Theory

Examples of Functors

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A Functor consists of a mapping of objects and a mapping of morphisms that preserves all of the structure of categories. namely domain, codomain, composition and identities.

Covariant Functors

• There is an endofunctor $P:Set\to Set$ that sends a set $A$ to its power set and a function $f:A\to B$ to its direct-image function $f_{\ast }:PA\to PB$
• Most Algebraic (and other) object that are sets with other operations and restrictions on it have a forgetful functor from their category to the category of sets, where each element is mapped to its base set.
• The fundamental group is a functor from category of Topological spaces to the category of groups, that sends continuous functions to group homomorphisms.

Contravariant Functor

• The functor $(-{)}^{\text{op}}:\text{CAT}\to \text{CAT}$ defines a non-trivial automorphism of categories of categories.
• For any topological space $X$, there is a contravariant functor isomorphism of poset categories $O(X)\cong C(X{)}^{\text{op}}$ that associates an open subset of $X$ to its closeed complement.

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References

• Functors