Abstract Nonsense

Crushing one theorem at a time

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.

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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. \mathbf{Top,Grp,Ring,Set}, 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.

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Type of Morphisms

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Let us fix, for the duration of this post, some category \mathcal{C}. Suppose that we have two objects A,B\in\text{obj}(\mathcal{C}). We would like to lay out the different shapes a morphism A\xrightarrow{f}B could take. We lay out the big list of definitions as follows:

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The morphism f is called a monomorphism or is said to be mono/monic if whenever f\circ g_1=f\circ g_2 for X\underset{g_1}{\overset{g_2}{\longrightarrow}}A we may conclude that g_1=g_2. This property is perhaps more aptly dubbed left cancellable.

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The morphism f is called an epimorphism or is said to be epi/monic if whenenver g_1\circ f=g_2\circ f for B\underset{g_1}{\overset{g_2}{\longrightarrow}}X we may conclude that g_1=g_2. This property is, once again, perhaps more aptly named right cancellable.

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

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The morphism f is called a section if it has a left inverse, e.g. there exists  g:B\to A, such that g\circ f=1_A.

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The morphism f is called a retraction if it has a right inverse, e.g. there exists g:B\to A such that f\circ g=1_B.

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The morphism f is called an isomorphism if it has an inverse, e.g. there exists g:B\to A such that g\circ f=1_A and f\circ g=1_B. If there exists an isomorphism between A and B we call them isomorphic or say they are of the same isomorphism type. If A and B are of the same isomorphism type we are often prompted to denote this by A\cong B. It’s clear that if f is an isomorphism then its inverse is unique, and so we may unambiguously denote it by f^{-1}. Moreover, it’s clear that “is isomorphic to” is an equivalence relation on \text{obj}(\mathcal{C}).

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The morphism is called an endomorphism if A=B. Given an object A, the set of endomorphisms of A is denoted \text{End}(A), which of course is just shorthand for \text{Hom}(A,A).

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The morphism is called an automorphism if it is both an endomorphism and an isomorphism. The set of automorphisms for a given object A is denoted \text{Aut}(A).

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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 \mathbf{Grp,Ring,Rng,}R\text{-}\mathbf{Mod}, 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 \mathbf{Top} an ample place for counterexamples to such false beliefs.

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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 \mathbb{Z}\hookrightarrow\mathbb{Q}. Namely, if two maps agree on \mathbb{Z} then (being ring maps) they must agree on all of \mathbb{Q}. That said, there certainly does not exist a ring homomorphism \mathbb{Q}\to\mathbb{Z} which serves as a left inverse to the inclusion, since every ring homomorphism \mathbb{Q}\to\mathbb{Z} is zero (just think that f(n)=n and so f(nx)=nf(x) for every n\in\mathbb{N}, so n\mid f(x) for every n\in\mathbb{N})!

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We do have the following nice theorem though:

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Theorem: Let \mathcal{C} be a given category and A,B\in\text{obj}(\mathcal{C}). Then, for any A\xrightarrow{f}B the following are equivalent:

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\begin{aligned}&\mathbf{(1)}\quad f\textit{ is an isomorphism}\\ &\mathbf{(2)}\quad f\textit{ is a section and is epic}\\ &\mathbf{(3)}\quad f\textit{ is a retraction and is monic}\end{aligned}

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Proof: Clearly \mathbf{(1)}\implies\mathbf{(2)},\mathbf{(3)}. Now, to see that \mathbf{(2)}\implies\mathbf{(1)} let g:B\to A be a left inverse for f, we claim that g is also a right inverse. Indeed, f\circ g\circ f=f\circ(g\circ f)=f\circ 1_A=f and since f is right cancellable we may conclude that f\circ g=1_B. The proof for \mathbf{(3)}\implies\mathbf{(1)} is done similarly. \blacksquare

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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.

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December 27, 2011 - Posted by | Algebra, Category Theory | , , , , , , , , ,

9 Comments »

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