## Types of Morphisms

**Point of Post: **This post is mainly concerned with laying out the definitions of the various types of morphisms in a given category.

*Motivation*

As of now, with just the bare definition of categories, we have little to work with. From our current standpoint, all morphisms are created equal. But, keeping in mind that categories are modeled off the common categories we usually work with (e.g. , etc.) we know that this is certainly not true. Namely, the ideas of group epimorphisms, topological embeddings, and the like quickly show us that morphisms come in many different varieties. This post shall lay out the definitions to make these differences clear.

*Type of Morphisms*

Let us fix, for the duration of this post, some category . Suppose that we have two objects . We would like to lay out the different shapes a morphism could take. We lay out the big list of definitions as follows:

The morphism is called a *monomorphism *or is said to be *mono/monic *if whenever for we may conclude that . This property is perhaps more aptly dubbed *left cancellable*.

The morphism is called an *epimorphism *or is said to be *epi/monic* if whenenver for we may conclude that . This property is, once again, perhaps more aptly named *right cancellable.*

The morphism is called a *bimorphism *if it is both epi and mono.

The morphism is called a *section *if it has a left inverse, e.g. there exists , such that .

The morphism is called a *retraction *if it has a right inverse, e.g. there exists such that .

The morphism is called an *isomorphism *if it has an inverse, e.g. there exists such that and . If there exists an isomorphism between and we call them *isomorphic* or say they are of the same *isomorphism type. *If and are of the same isomorphism type we are often prompted to denote this by . It’s clear that if is an isomorphism then its inverse is unique, and so we may unambiguously denote it by . Moreover, it’s clear that “is isomorphic to” is an equivalence relation on .

The morphism is called an *endomorphism* if . Given an object , the set of endomorphisms of is denoted , which of course is just shorthand for .

The morphism is called an *automorphism* if it is both an endomorphism and an isomorphism. The set of automorphisms for a given object is denoted .

One must be careful not to use too much specific knowledge about any given category to try and deduce connections between these definitions. For example, in many “algebraic” categories (such as , etc.) being an isomorphism is equivalent to being a bimorphism, or it is also equivalent to being both a section and a retraction, but in general this isn’t true. The reason for the majority of these niceities enjoyed by algebraic categories, is due to the fact that an isomorphism is nothing more than a bijective morphism, but this is not true in general categories. For example, there are certainly examples of bijective continuous maps which aren’t homeomorphisms, making an ample place for counterexamples to such false beliefs.

All these warnings heeded, there are certainly some connections between the above definitions. For example every section is certainly monic and every retraction is certainly epic. That said, the converse to this is not true. The classical example of a non-retraction epic is the ring embedding . Namely, if two maps agree on then (being ring maps) they must agree on all of . That said, there certainly does not exist a ring homomorphism which serves as a left inverse to the inclusion, since every ring homomorphism is zero (just think that and so for every , so for every )!

We do have the following nice theorem though:

**Theorem: ***Let be a given category and . Then, for any the following are equivalent:*

**Proof: **Clearly . Now, to see that let be a left inverse for , we claim that is also a right inverse. Indeed, and since is right cancellable we may conclude that . The proof for is done similarly.

**References:**

[1] Mac, Lane Saunders. *Categories for the Working Mathematician*. New York: Springer-Verlag, 1994. Print.

[2] Adámek, Jirí, Horst Herrlich, and George E. Strecker. *Abstract and Concrete Categories: the Joy of Cats*. New York: John Wiley & Sons, 1990. Print.

[3] Berrick, A. J., and M. E. Keating. *Categories and Modules with K-theory in View*. Cambridge, UK: Cambridge UP, 2000. Print.

[4] Freyd, Peter J. *Abelian Categories.* New York: Harper & Row, 1964. Print.

[5] Rotman, Joseph J. *Introduction to Homological Algebra*. Springer-Verlag. Print.

[…] call a subcategory of isomorphism closed if whenenver is such that there exists of the same isomorphism type, then […]

Pingback by Subcategories and Skeleta « Abstract Nonsense | December 27, 2011 |

[…] to the case of different types of morphisms there are different types of functors. We fix two categories and and a functor . Then, we make […]

Pingback by Functors (Pt. II) « Abstract Nonsense | December 27, 2011 |

[…] transformation from to by . A natural transformation is called a natural equivalence if is an isomorphism for each . If is a natural equivalence, we are apt to notate this as […]

Pingback by Natural Transformations (Pt. I) « Abstract Nonsense | December 28, 2011 |

[…] have already discussed how there are different types of morphisms in a category, and so a natural question to ask, is what are the different types of objects? […]

Pingback by Initial, Terminal, and Zero Objects « Abstract Nonsense | December 28, 2011 |

[…] if and only if is an isomorphism. We have obvious extensions of these theorems for all the types of morphisms, but I’d like to mention one in […]

Pingback by The Opposite Category « Abstract Nonsense | December 30, 2011 |

[…] see that, just as functor categories “preserve” special types of morphsims they also preserve special types of […]

Pingback by Functor Categories (Pt. II) « Abstract Nonsense | January 7, 2012 |

[…] obvious things are obvious. For example, since (as we have previously proven) a morphism is mono if and only if it has zero kernel, and is an epi if and only if it has zero cokernel it’s […]

Pingback by Exact Sequences and Homology (Pt. II) « Abstract Nonsense | April 10, 2012 |

[…] is a simple example of why in general categories aren’t balanced. Indeed, the monos and epis in are the injective and surjective continuous maps. Thus, we see that is a bimorphism. […]

Pingback by The Exponential and Trigonometric Functions (Pt. II) « Abstract Nonsense | May 5, 2012 |

[…] create a fairly tame structure. In particular, call a category a groupoid if every arrow is an isomorphism. We claim then that if , called the fundamental groupoid of , is the category whose object set , […]

Pingback by The Fundamental Groupoid and Group (Pt. II) « Abstract Nonsense | August 30, 2012 |