Abstract Nonsense

Initial, Terminal, and Zero Objects

Point of Post: In this post we discuss the notion of initial, terminal, and zero objects in general categories, and give examples of each.

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Motivation

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We 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? Namely, are there objects in a category that are distinguishable? Well, considering our mantra that objects are invisible (there direct reference in constructions should be avoided) whatever these distinguished objects are, they’re ‘special qualities’ should be formulated entirely in terms of morphisms. In this post we consider three particular kinds of objects: initial, terminal, and zero. Intuitively these are the objects that the simplest possible morphism structure in the category. Namely, initial, terminal, and zero objects are such that given any other object in the category one (or both) the Hom set between them (which Hom set [e.g. $\text{Hom}(\text{initial object},\text{other object})$ vs. $\text{Hom}(\text{other object},\text{initial object})$] depends upon which type of object we are discussing) are completely determined, and moreover their determination is simple. For example, the object $x$ in a category will be initial if $\text{Hom}(x,y)$ is a singleton for each object $y$. Similarly, an object $x$ is terminal if $\text{Hom}(y,x)$ is a singleton for each object $y$. An object is a zero object if it’s both initial and terminal. The naming for initial and terminal object should be fairly transparent, and if you’re wondering why a zero object was named as such, just think that in the category $\mathbf{Ab}$ the zero group $\{0\}$ serves as the zero object.

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Initial, Terminal, and Zero Objects

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Suppose we are given a fixed category $\mathcal{C}$. We call an element $x\in\text{obj}\left(\mathcal{C}\right)$ initial if for every other element $y\in\text{obj}(\mathcal{C})$ there exists a unique morphism $x\to y$. We call $x$ terminal if there exists a unique morphism $y\to x$ for every other object $y$. Finally, we call an element $z$ a zero object if it’s both terminal and initial.

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Let’s look at some examples:

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The empty set $\varnothing$ is a terminal object in $\mathbf{Set}$. Given any other set $X$ there is a unique morphism $\varnothing\to X$ (the empty mapping).

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The ring of integers $\mathbb{Z}$ is an initial object in the category $\mathbf{Ring}$ of unital rings and unital ring maps.

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A singleton $\{x\}$ is terminal in $\mathbf{Set}$. The only mapping $X\to\{x\}$ is the constant map.

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A singleton $\{x\}$ (given the only possible topology) is also a terminal object in $\mathbf{Top}$ since given any topological space $X$ the constant map $X\to\{x\}$ is continuous, and it’s the only such continuous map (in fact any map) $X\to\{x\}$.

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The zero group, zero ring, zero module are all zero objects in $\mathbf{Grp,Rng,}R\text{-}\mathbf{Mod}$ respectively.

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Now, let’s look at some non-examples:

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While $\varnothing$ is an initial object in $\mathbf{Set}$ it is certainly not a terminal object, since given $X\ne\varnothing$ there does not even exist a map $X\to\varnothing.$latex \text{ }\$

Singletons $\{x\}$ are not initial objects in either $\mathbf{Set}$ or $\mathbf{Top}$ since, for example, there are two set/continuous maps $\{x\}\to\{1,2\}$ (where in the case of $\mathbf{Top}$, $\{1,2\}$ is given the discrete topology).

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$\mathbb{Z}$ is not a terminal object in $\mathbf{Ring}$ since there are two distinct projections $\mathbb{Z}\times\mathbb{Z}\to\mathbb{Z}$.

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Now, let’s look at some theorems:

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Theorem: Let $\mathcal{C}$ be a category, then initial and terminal objects in $\mathcal{C}$ are unique up to unique isomorphism.

Proof: Suppose first that $x$ and $x'$ are two initial objects in $\mathcal{C}$. The fact that any isomorphism $x\to x'$ is unique is clear, and such an isomorphism exists because if $x\xrightarrow{f}x'$ and $x'\xrightarrow{g}x$ are guaranteed maps, then we know that $x\xrightarrow{g\circ f}x$ and $x'\xrightarrow{f\circ g}x'$ are morphisms, but since $x\xrightarrow{1_x}x$ and $x'\xrightarrow{1_{x'}}x'$ are also morphisms we may conclude by uniqueness that $g\circ f=1_x$ and $f\circ g=1_{x'}$. Thus, $f$ is an isomorphism.

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The case for terminal objects follows similarly. $\blacksquare$

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Note that if $z$ is a zero object for the category $\mathcal{C}$ and $a,b$ are any two other objects then there exists a unique map $a\to z\to b$, this is called the zero morphism $a\to b$. Note that the composition of zero morphisms is a zero morphism.

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The above contains an important kernel of information. Namely, while it’s often easy to forget that this is true, given objects $x,y$ in some category $\mathcal{C}$ there is no reason that $\text{Hom}(x,y)$ isn’t empty–i.e. that there are no arrows $x\to y$. For example, in the category $\mathbf{Ring}$ of untial rings with unital homomorphisms one has that $\text{Hom}_{\mathbf{Ring}}(\mathbb{Z}_5,\mathbb{Z}_7)=\varnothing$. Indeed, we can recall that the existence of a unital homomorphism $R\to S$ implies that $\text{char}(S)\mid\text{char}(R)$ (this is just a rephrasing of the simple fact of group theory that $|f(x)|\mid |x|$ applied to the fact that $1\mapsto 1$), and since $7\nmid 5$ we have the above equality (in fact, this also shows that $\text{Hom}_\mathbf{Ring}(\mathbb{Z}_7,\mathbb{Z}_5)=\varnothing$ as well). That said, it’s easy to see that given any two objects $x,y$ in $\mathbf{Rng}$, rings (not necessarily unital) with ring maps (not necessarily unital rings maps), one has that $\text{Hom}_{\mathbf{Rng}}(x,y)\ne\varnothing$. Indeed, this stems from the simple fact that given any two rings $R$ and $S$ the zero map $R\to S$ is a ring homomorphism (albeit not unital!).

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Basically the saving grace (assuming we wanted non-empty Hom sets) was that $\mathbf{Rng}$ has a zero object–the zero ring. The categorical zero map $R\to S$ factoring through the zero ring is nothing more than the described zero map $x\mapsto 0$. Thus, this shows us that if a category $\mathcal{C}$ has a zero object $z$ then $\text{Hom}_\mathcal{C}(x,y)\ne\varnothing$ for any two objects $x,y$ in $\mathcal{C}$ because we have the zero arrow $x\to z\to y$. The converse of this tells us that $\mathbf{Ring}$, and in fact any category $\mathcal{C}$ with an empty Hom set (e.g. the category $\mathbf{Field}$ and the category $\mathbf{Domain}$ of integral domains, thought of as full subcategories of $\mathbf{Ring}$) cannot have a zero object. Of course, the converse to this is unfortunately not true. For example, consider the category $\mathcal{C}$ of non-empty topological spaces thought of as a full subcategory of $\mathbf{Top}$. Note then that for any non-empty topological spaces $X,Y$ one has that $\text{Hom}_\mathcal{C}(X,Y)$ is non-empty as choosing $y\in Y$ the constant map $X\ni x\mapsto y$ is a continuous map $X\to Y$. That said, $\mathcal{C}$ does not have a zero object, for there are always uncountably many continuous maps $X\to\mathbb{R}$ for any non-empty topological space $\mathbb{R}$ corresponding to the constant maps associated to each element of $\mathbb{R}$.

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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.

December 28, 2011 -

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