Category Theory/Functors
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The notion of category being established as that which gives precision to the concept of domain of mathematical discourse, the formalization of the precise notion corresponding to the intuitive idea of the interrelation or connection between different domains is now considered. Within any particular category not only the objects must be specified but also the morphisms. Furthermore, these morphisms are, in the principal examples, required in some sense to transport the structure characteristic of the objects of the category; e.g., group structure, topological structure. In the same way a functor Φ from the category \mathcal C to the category \mathcal D is defined as a rule that associates with every object A of \mathcal C an object Φ(A) of \mathcal D, and with every morphism f: A \to_{\mathcal C} B a morphism \Phi(f) : \Phi(A) \to_{\mathcal D} \Phi(B), subject to the conditions of transport of structure (see 347, 348):
\Phi(1_{\mathcal C, A}) = 1_{\mathcal D, \Phi(A)}
\Phi (g \circ_{\mathcal C} f) = \Phi(g) \circ_{\mathcal D} \Phi(f)
In expressing the relation, there is a tacit assumption that the composition fb is defined. This is standard practice, to avoid unnecessarily complicated statements.
A functor f always sends isomorphisms to isomorphisms, a very crucial property. Otherwise expressed, if Ff is not an isomorphism, then f cannot be an isomorphism. This yields a basic methodological procedure; to demonstrate that A and B, two objects of , really are different—i.e., non-isomorphic—it may be possible to find a functor f from to a category in which it is far easier to make the comparison. This procedure may be regarded as a massive generalization of the classical arithmetical test of casting out nines.
[edit] Examples of functors
- For any group \mathfrak G it is possible to render \mathfrak G commutative by adjoining the relations xy = yx for all x, y in its universe G. The resulting group may be written \mathfrak G_{\mathrm{ab}}. If f: \mathfrak G \to_{\mathbf{Grp}} \mathfrak H is a homomorphism (in the category of groups notated as Grp), it is easy to see that f induces a uniquely determined homomorphism f_{\mathrm{ab}}: \mathfrak G_{\mathrm{ab}} \to_{\mathbf{Ab}} \mathfrak H_{\mathrm{ab}} in the category of Abelian groups (notated as Ab). Thus there is a functor from the category of groups Grp to the category of Abelian groups Ab. This should be the first functor tried in an attempt to prove two groups nonisomorphic.
- For any pointed topological space (X, x)—that is, a topological space X, together with a distinguished point X in X—it is possible to construct the Poincaré fundamental group p(X, x). Then p is a functor from the category of pointed topological spaces and pointed continuous functions to the category of groups. Because, for example, the sphere and the torus (a surface in the shape of a doughnut) have non-isomorphic fundamental groups, it follows that they are not homeomorphic.
- If x, y are two pre-ordered sets regarded as categories, as above, then a functor from X to Y is merely an order-preserving function.
- There are many important functors, called forgetful or underlying functors, that simply forget part of the structure present in the objects and transported by the morphisms. Thus, the objects in the examples in The notion of a category all have underlying set structures and the morphisms are all functions; moreover, the law of composition is simply that of functions. Thus, in all these cases, there is an underlying functor to . It is also possible to take a ring, forget the multiplication in it, and thereby obtain an Abelian group; this yields an underlying functor. Underlying functors may seem trivial things; actually, they are fundamental to categorical language.
[edit] Properties of functors
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