Monday, August 3, 2009

Hierarchy of Functors

We basically observed that functors are morphisms of categories. We've also just set up a hierarchy of morphisms. We'd like to set up a hierarchy of functors. First some light reading:

  • Jiri Ad├ímek, Horst Herrlich, George E. Strecker Abstract and Concrete Categories: The Joy of Cats freely available online (2004)
  • Saunders Mac Lane, Categories for the Working Mathematician Graduate Texts in Mathematics (vol 5) Springer-Verlag, Second Edition (1998);

We will use the term "isomorphism" a lot in category theory, the general intuition behind its use is clear from context. A functor that's an isomorphism tells us that the domain and codomain are "the same". The requirements for this to be true also are the same, that we need an inverse of the functor.

We'll use the term "isomorphism" in the future when discussing "higher morphisms", like a morphism between functors (aka "natural transformations").

Now we first would like to generalize the notion of injectivity, but with functors we have some extra freedom to specify "what" is injective: the function of $\hom(-,-)$ sets, or the function of objects. To let you in on a secret, all the important information in a category is contained in the morphisms. So we should be concerned about the function mapping $\hom(-,-)$ sets to $\hom(F(-),F(-))$ sets.

Such a functor is an important type of functor, which allows us to define our first property:

Definition 1. A "Faithful Functor" consists of
  • a functor $F:C\to{D}$
such that
  • the function $F_{X,Y}:\hom_{C}(X,Y)\to\hom_{D}(F(X),F(Y))$ is injective.

Now "dually" we can ask the function $F_{X,Y}$ to be surjective. It's a straightforward specification, if we have something that's injective, we should ask for surjectivity as well.

Definition 2. A "Full Functor" consists of
  • a functor $F:C\to{D}$
such that
  • the function $F_{X,Y}:\hom_{C}(X,Y)\to\hom_{D}(F(X),F(Y))$ is surjective.

Now we have specified how functors behave on the hom-sets, but we should probably specify how they behave on objects too.

The only really interesting behavior to worry about is, for $F:C\to{D}$, having $F(c)\cong{d}$ for some object $c\in{C}$ and some other object $d\in{D}$. Here the $\cong$ symbol indicates an isomorphism of the domain and codomain.

We now make our last specification (for the time being):

Definition 3. An "Essentially Surjective Functor" consists of
  • a functor $F:C\to{D}$
such that
  • for each $c\in{C}$, $F(c)\cong{d}$ for some corresponding $d\in{D}$.

(A note on notation here, I'm not certain if anyone mentions it explicitly, but $c\in{C}$ is the same as saying $c\in Ob(C)$ and later on we may see the notation $C(c,c^{\prime})$ which is really just $\hom_{C}(c,c^{\prime})$ the hom-set.)

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