User:Michiexile/MATH198/Lecture 2

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Morphisms and objects

Some morphisms and some objects are special enough to garner special names that we will use regularly.

• Isomorphisms and existence of inverses.
• Epi- and mono-morphisms and cancellability.
  • Examples in concrete categories.
  • Monomorphisms and subobjects:
    • Factoring through. Equivalence relation by mutual factoring.
    • Subobjects as equivalence classes of monomorphisms.
  • Splitting and the existence of inverses.
• Terminal and initial objects.
  • Constants. Pointless sets.

Morphisms

The arrows of a category are called morphisms. This is derived from homomorphisms.

Some arrows have special properties that make them extra helpful; and we'll name them:

Endomorphism

A morphism with the same object as source and target.

Monomorphism

A morphism that is left-cancellable. Corresponds to injective functions. We say that f is a monomorphism if for any $g_{1},g_{2}$, the equation $fg_{1}=fg_{2}$ implies $g_{1}=g_{2}$. In other words, with a concrete perspective, f doesn't introduce additional relations when applied.

Epimorphism

A morphism that is right-cancellable. Corresponds to surjective functions. We say that f is an epimorphism if for any $g_{1},g_{2}$, the equation $g_{1}f=g_{2}f$ implies $g_{1}=g_{2}$.

Note, by the way, that cancellability does not imply the existence of an inverse. Epi's and mono's that have inverses realizing their cancellability are called split.

Isomorphism

A morphism is an isomorphism if it has an inverse. Split epi and split mono imply isomorphism. Specifically, $f:v\to w$ is an isomorphism if there is a $g:w\to v$ such that $fg=1_{w}$ and $g=1_{v}$.

Automorphism

An automorphism is an endomorphism that is an isomorphism.

Objects

In a category, we use a different name for the vertices: objects. This comes from the roots in describing concrete categories - thus while objects may be actual mathematical objects, but they may just as well be completely different.

Just as with the morphisms, there are objects special enough to be named. An object v is

Initial

if [v,w] has exactly one element for all other objects w.

Terminal

if [w,v] has exactly one element for all other objects w.

A Zero object

if it is both initial and terminal.

All initial objects are isomorphic. If $i_{1},i_{2}$ are both initial, then there is exactly one map $i_{1}\to i_{2}$ and exactly one map $i_{2}\to i_{1}$. The two possible compositions are maps $i_{1}\to i_{1}$ and $i_{2}\to i_{2}$. However, the initiality condition holds even for the morphism set [v,v], so in these, the only existing morphism is $1_{i_{1}}$ and $1_{i_{2}}$ respectively. Hence, the compositions have to be this morphism, which proves the statement.

Dual category

The same proof carries over, word by word, to the terminal case. This is an illustration of a very commonly occurring phenomenon - dualization.