Abstract Nonsense

Comma Categories (Pt. I)

Point of Post: In this post we introduce the notion of comma categories, give some examples, and discuss their relationship to some previously defined objects.

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Motivation

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Sometimes in mathematics very complicated ideas can be made simple by introducing the right notation, the right prerequisite constructions.  In this post we discuss such a construction: the comma category. Roughly the comma category takes prexisting categories and some functors between them, and constructs a new category whose objects are arrows in one of the categories. Having a predefined category whose objects are arrows of some set of categories we are interested in will be appealing to us, because we can apply object-oriented notions (e.g. initial object) to these arrows. We shall see that being able to make such object references to arrows will serve to be a universally (hint hint) clarifying idea in the category theory to come (we’ll even see that it can make already defined notions simpler!).

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Comma Categories

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To set up this construction we need to start with categories $\mathcal{C,D,E}$ and functors $\mathcal{D}\overset{F}{\longrightarrow}\mathcal{C}\overset{G}{\longleftarrow}\mathcal{E}$. Given objects $x\in\mathcal{D}$ and $y\in\mathcal{E}$ we can “connect” them by carrying them to their mutually related category $\mathcal{C}$. In other words, given two such objects we can consider arrows $F(x)\xrightarrow{f}G(y$ in $\mathcal{C}$. Such arrows are precisely the objects of the category we wish to construct. Of course, to be precise what we really need to say is that the objects of the coming-soon category are going to be triples $(x,y,f)$ where $x$ is an object in $\mathcal{D}$, $y$ an object in $\mathcal{E}$, and $f$ an arrow $F(x)\to G(y)$. Despite these formalities we shall often (except when it’s visually unappealing to do so) refer to these triples as just an arrow (with specified domain and codomain) i.e. $(x,y,f)\Leftrightarrow F(x)\xrightarrow{f}G(y)$.

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How exactly should we define the arrows of this category? Well, it seems a pretty standard idea that whatever the arrows of this category are, they should “respect” the structure of objects in the category. So, given two objects $F(x)\xrightarrow{f}G(y)$ and $F(x')\xrightarrow{f'}G(y')$ what should an arrow between them be? It seems intuitive that to respect these three-component objects (the two $\mathcal{D,E}$-objects, and the $\mathcal{C}$-arrow) we should respect each component separately. For the components $x,x'$ and $y,y'$ it’s clear what this should mean, we should have some arrows $x\xrightarrow{j}x'$ and $y\xrightarrow{h}y'$, but what should it mean to respect the arrows $f$ and $f'$? Well, considering that composition is king, in category theory at least, it seems as though respecting the arrows means that we should respect any pursuant compositions involving them–meaning concretely that we should get some type of commutative diagram. Luckily, we have just, by selecting arrows between the two pairs of $\mathcal{D,E}$-objects, created a diagram, namely:

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$\begin{matrix}F(x) & \overset{F(j)}{\longrightarrow} & F(x')\\ ^{f}\big\downarrow & & \big\downarrow ^{f'}\\ G(y) & \underset{G(h)}{\longrightarrow} & G(y')\end{matrix}\quad\mathbf{(1)}$

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thus it seems reasonable to define an arrow from $F(x)\xrightarrow{f}G(y)$ to $F(x')\xrightarrow{g}G(y')$ to be a pair of arrows $(j,h)$, with $x\xrightarrow{j}x'$ and $y\xrightarrow{h}y'$, which makes the pursuant diagram $\mathbf{(1)}$ commute.

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Given the above set-up we call the resulting category with composition done component-wise, the comma category of the diagram $\mathcal{D}\overset{F}{\longrightarrow}\mathcal{C}\overset{G}{\longleftarrow}\mathcal{E}$, and denote it by $(F\downarrow G)$.

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Let us verify that this is, indeed, a category. The only things that really need checked is that the composition of two arrows in $(F\downarrow G)$ is an arrow in $(F\downarrow G)$ and that $(1_x,1_y)$ serves as an identity arrow for $F(x)\xrightarrow{f}G(y)$. Of course, the former of these two is tantamount to showing that the commutativity holds. To this end, suppose that we have arrows $(x,y,f)\xrightarrow{(j,h)}(x',y',f')\xrightarrow{(j',h')}(x'',y'',f'')$ in $(F\downarrow G)$. We wish to show that $f''\circ F(j'\circ j)=G(h'\circ ')\circ f$. But, of course since $F$ and $G$ are functors we must merely expand to see that

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\begin{aligned}f''\circ F(f'\circ f) &=f''\circ F(f')\circ F(f)\\ &=G(h')\circ f'\circ F(f)\\ &=G(h')\circ G(h)\circ f\\ &=G(h'\circ h)\circ f\end{aligned}

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The fact that $(1_x,1_y)$ is an identity for $F(x)\xrightarrow{f}G(y)$ is trivial.

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If we have a diagram of the form $\mathcal{D}\overset{F}{\longrightarrow}\mathcal{C}\overset{1_\mathcal{C}}{\longleftarrow}\mathcal{C}$ it is common to denote $(F\downarrow 1_\mathcal{C})$ as $(F\downarrow \mathcal{C})$ and similarly for $(\mathcal{C}\downarrow G)$. More generally, if $\mathcal{D},\mathcal{E}$ are subcategories of $\mathcal{C}$ and we have the inclusion diagram $\mathcal{D}\overset{I}{\hookrightarrow}\mathcal{C}\overset{I}{\hookleftarrow}\mathcal{E}$ we denote the resulting comma category as $(\mathcal{D}\downarrow\mathcal{E})$. If $\mathcal{D}$ or $\mathcal{E}$ is a subcategory with one object with only the identity arrow (i.e. the discrete subcategory with that one object), we are apt to just forego writing $\mathcal{D}$ or $\mathcal{E}$ in place of the object (e.g. $(x\downarrow\mathcal{E}$) and $(\mathcal{D}\downarrow y)$).

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Ok, fine, so now that we know how to define comma categories, let’s see if we can’t get a better grasp of them by looking at some particular examples.

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Let $\bullet$ represent some singleton in $\mathbf{Set}$ and conflate the object $\bullet$ with the inclusion functor $I:\bullet\hookrightarrow\mathbf{Set}$. What does $(\bullet\downarrow\mathbf{Set})$ look like? The objects of this comma category look like arrows of the form $\bullet\xrightarrow{f}X$ where $X$ is some set. So, what exactly does this really mean, what information is really contained in this arrow? Clearly all the function $f$ does is “select” a point $f(\bullet)$ of $X$. Thus, we can really think of an object in $(\bullet\downarrow\mathbf{Set})$ as being an ordered pair $(X,x_0)$ where $x_0\in X$. Ok, fine, so what do the arrows in $(\bullet\downarrow\mathbf{Set})$ look like? Well, by definition an arrow between $\bullet\xrightarrow{f}X$ and $\bullet\xrightarrow{g}Y$ is going to be a pair of arrows $(j:\bullet\to\bullet,h:X\to Y)$ such that $g\circ F(j)=G(h)\circ f$ but note that since $j$  is necessarily the identity arrow in $latex\ bullet$ and $G$ is the identity functor this reduces to $g=h\circ f$. Or, if we think of $\bullet\xrightarrow{f}X$ and $\bullet\xrightarrow{g}Y$ as $(X,x_0)$ and $(Y,y_0)$ as already described then the equation $g=h\circ f$ becomes $h(x_0)=y_0$. Thus, we can really interpret $(\bullet\downarrow\mathbf{Set})$ as having objects $(X,x_0)$ with $x_0\in X$ and arrows $f:(X,x_0)\to (Y,y_0)$ as just being set functions $f:X\to Y$ which respect the distinguished points, i.e. $f(x_0)=y_0$. Thus, we see that $(\bullet\downarrow\mathbf{Set})$ is nothing more than the commonly used category $\mathbf{Set}_\ast$ of pointed sets. Similarly, one can check that $(\bullet\downarrow\mathbf{Top})$ is the category of pointed topological spaces.

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In a similar vein, let’s discuss the algebraic variant of the above. Namely, let’s see if we can get a handle on the category $(\mathbb{Z}\downarrow\mathbf{Ab})$. Once again, let’s start by looking at what the objects of this category look like. Such an object looks like an arrow $\mathbb{Z}\xrightarrow{f}A$ in $\mathbf{Ab}$. So how precisely does this generalize the pointed categories in the previous paragraph? The key is that any such arrow is completely determined by where it sends $1\in\mathbb{Z}$ and moreover any such element of $A$ can be the image of $1$. Thus, we can think that arrows $\mathbb{Z}\xrightarrow{f}A$ in $\mathbf{Ab}$ can be uniquely identified with ordered pairs $(A,a_0)$ where $a_0\in A$ (i.e. by $a_0\leftrightarrow f(1)$). Ok, so what do the arrows in $(\mathbb{Z}\downarrow\mathbf{Ab})$ look like? So, suppose we had two objects in the comma category, say $\mathbb{Z}\xrightarrow{f}A$ and $\mathbb{Z}\xrightarrow{g}B$. We know then a morphism between these two objects is a pair of arrows $\mathbb{Z}\xrightarrow{j}\mathbb{Z}$ and $A\xrightarrow{h}B$ (both in $\mathbf{Ab}$) such that (considering we are dealing with inclusion functors) $g\circ j=h\circ f$. Now, note that since these are both arrows out of $\mathbb{Z}$ their equivalence as maps is equivalent to their equivalence on $1$. In other words, $g\circ j=h\circ f$ is equivalent to $g(j(1))=h(f(1))$. But, note that since the only arrow we are considering $\mathbb{Z}\to\mathbb{Z}$ is the indentity arrow (since we are thinking of $\mathbb{Z}$ as a single object discrete category) we must have that $j=1_\mathbb{Z}$ and so $g(j(1))=h(f(1))$ reduces to $g(1)=h(f(1))$. Thus, we see that $(\mathbb{Z}\downarrow\mathbf{Ab})$ is nothing more than the category whose objects are $(A,a_0)$ where $A$ is an abelian group and $a_0\in A$ and a morphism $(A,a_0)\to (B,b_0)$ is nothing more than a group homomorphism $f:A\to B$ such that $f(a_0)=b_0$–we are apt to call this $\mathbf{Ab}_\ast$, the category of pointed abelian groups. Of course, the only operative thing in the above analysis was that $\mathbb{Z}$ is a rank one free abelian group. More generally it’s easy to see that if we are given a unital ring $R$ then $(R\downarrow R\text{-}\mathbf{Mod})$ is the category of pointed (unital, as always) left $R$-modules.

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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.

[6] Herrlich, Horst, and George E. Strecker. Category Theory: An Introduction. Lemgo: Heldermann, 2007. Print.

January 10, 2012 -

1. […] Comma Categories (Pt. II) Point of Post: This is a continuation of this post. […]

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2. […] Theorem: Let and be two categories and a functor . Then, if is an object of an ordered pair where is an object in and an arrow , is a universal arrow from to , if and only if is an initial object in the comma category . […]

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3. […] and . Since any two products of and are going to be unique up to a unique isomorphism in the comma category we often do not distinguish between different products, and (possibly to some ambiguity) denote […]

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4. […] co-slice category $textbf{C}/y$.  (note: I owe these above cool examples to this fantastic post: https://drexel28.wordpress.com/2012/01/10/comma-categories-pt-i/.  You should really visit this guy’s blog.) […]

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