# Abstract Nonsense

## Categorical Products

Point of Post: In this post we discuss the notion of products in the sense of category theory.

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Motivation

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We have indicated in our description of universal arrows that one of their key features was the ability to take constructions found in various parts of mathematics that, while ostensibly unrelated, are really the same thing when viewed as the solution to certain universal mapping problems. In this post we give a typifying example of this to show how the notion of “product” found throughout mathematics (e.g. module theory, ring theory, group theory, topology, etc.) is really just a guised version of general, categorical, notion of “product” which is going to be the solution to a certain universal mapping property. This should be seen as a triumph since not only does it allow us to understand why the particular definitions of “product” mentioned before make sense, but it also allows us to be able to smartly define “product” in any new category we wish to explore. So, what exactly are categorical products? Roughly the product of two objects $x,y$ should be an object $p$ for which the arrows $z\to p$  into this third object are in one-to-one correspondence with pairs of arrows $z\to x,z\to y$ into the factor objects.

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Categorical Products

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Let $\mathcal{C}$ be some category. There is a functor $\Delta:\mathcal{C}\to\mathcal{C}\times\mathcal{C}$ called the diagional functor taking an object $x$ to the object $(x,x)$ and an arrow $x\xrightarrow{f}y$ to $(x\xrightarrow{f}y,x\xrightarrow{f}y)$. Given two objects $x$ and $y$ in $\mathcal{C}$ we define a product of $x$ by $y$ to be a universal arrow $(p,(\pi_1,\pi_2))$ from $(x,y)$ to $\Delta$. Let’s unravel what this means. In full transparency this tells us that $p$ is an object of $\mathcal{C}$ equipped with arrows $x\overset{\pi_1}{\longleftarrow}p\overset{\pi_2}{\longrightarrow}y$ with the property that given any two arrows $z\xrightarrow{f}x$ and $z\xrightarrow{g}y$ there exists a unique arrow $z\xrightarrow{j}p$ such that $\pi_1\circ j=f$ and $\pi_2\circ j=g$. Since any two products of $x$ and $y$ are going to be unique up to a unique isomorphism in the comma category $((x,y)\downarrow\Delta)$ we often do not distinguish between different products, and (possibly to some ambiguity) denote a/the object of a product of $x$ and $y$ by $x\times y$.

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Note that, like all categorical constructions, products need not exist in categories. Namely, it is possible to find a category $\mathcal{C}$ and objects $x$ and $y$ in $\mathcal{C}$ for which $x\times y$ does not exist, we shall discuss such an example below.

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Let’s now look at some examples:

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In $\mathbf{Set}$ there is a product, and it’s the usual Cartesian product of sets. Namely, given two sets $X$ and $Y$ we can form their Cartesian product $X\times Y$ and consider the usual projections $\pi_1:X\times Y\to X:(x,y)\mapsto x$ and $\pi_2:X\times Y\to X:(x,y)\mapsto y$. It’s a simple fact of set theory then that every pair of functions $j_1:Z\to X$ and $j_2:Z\to Y$ admits a unique function $j:Z\to X\times Y$ with $\pi_i\circ j=j_i$, namely the function $j(z)=(j_1(z),j_2(z))$. Thus, we see that $(X\times Y,(\pi_1,\pi_2))$ is a/the product of $X$ and $Y$ in $\mathbf{Set}$.

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In the category $\mathbf{Top}$ the category of two topological spaces $X$ and $Y$ is the topological space whose underlying set is $X\times Y$ with the product topology. Indeed, we know that if we consider the usual projections $\pi_1$ and $\pi_2$ then a basic fact of topology is that a map $j:Z\to X\times Y$ is continuous if and only if $\pi_1\circ j:Z\to X$ and $\pi_2\circ j:Z\to Y$ , known as the coordinate functions, are continuous. Thus, we see that if given two continuous maps $j_1:Z\to X$ and $j_2:Z\to Y$ the map $j:X\times Y\to Z$ defined by $j(z)=(j_1(x),j_2(z))$ is continuous (since each coordinate function is continuous) and satisfies $\pi_2\circ j=j_i$. Moreover, it’s clear that such a map is unique by uniqueness in $\mathbf{Set}$.

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In the category $\mathbf{Ring}$ of unital rings with unital ring maps there are products, as we have already proven that the usual product of rings with the canonical projections form a product.

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Given some ring $R$ the usual product was defined to be a product!

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Let’s now consider a category that does not have products. Namely, let $\mathbf{Field}$ be the full subcategory of $\mathbf{Ring}$ whose objects are just fields. We claim that $\mathbb{Z}_2$ and $\mathbb{Q}$ do not have a product in $\mathbf{Field}$. Indeed, suppose that $\mathbb{Z}_2\overset{\pi_1}{\longleftarrow}k\overset{\pi_2}{\longrightarrow}\mathbb{Q}$ was a product of $\mathbb{Z}_2$ by $\mathbb{Q}$ in $\mathbf{Field}$. Forget for a second that everything actually lives in $\mathbf{Field}$ and think about this diagram in $\mathbf{Ring}$. Since we have unital maps out of single object into the two objects $\mathbb{Z}_2$ and $\mathbb{Q}$ the fact that the usual product of rings $\mathbb{Z}_2\times\mathbb{Q}$ is a product in $\mathbf{Ring}$ tells us that we have some unital map $j:k\to\mathbb{Z}_2\times\mathbb{Q}$ such that $\pi_i\circ j=\pi_i$ (where, confusingly I am denoting the usual projection on the left and the ‘supposed’ projection on the right by $\pi_i$). The important thing to note though is that $j$ is, if anything, a unital map $k\to\mathbb{Z}_2\times\mathbb{Q}$. Now, since all unital maps out of fields are injective we have that $\text{im }j$ is a field contained in $\mathbb{Z}_2\times\mathbb{Q}$ containing $(1,1)$. But, of course, this is impossible since we would then have that $2(1,1)=(0,2)$ is in $\text{im }j$, nonzero, and not invertible, which contradicts that $\text{im }j$ is a field.

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The above also illustrates the important fact that subcategories (even full ones!) do not preserve products. That said, it’s obvious from the above example that if you have a category $\mathcal{C}$ with a subcategory $\mathcal{S}$ and if $p$ is a product of $x$ by $y$ in $\mathcal{S}$ and $p'$ is a product of $x$ by $y$ in $\mathcal{C}$ then we are guaranteed a $\mathcal{C}$-arrow $p\to p'$ which is not necessarily an isomorphism.

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More generally, one can define arbitrary products in a given category. Namely, suppose that we have some set $\mathcal{S}$, regarded as a small discrete category, then we have for any category $\mathcal{C}$ the constant functor $\Delta:\mathcal{C}\to\mathcal{C}^\mathcal{S}$ defined by taking an object $x$ in $\mathcal{C}$ to the functor $F_x:I\to\mathcal{C}$, which takes every object to $x$ and every arrow $1_x$, and which takes an arrow $x\xrightarrow{f}y$ to the obvious natural transformation $\eta:F_x\to F_y$ given by defining $\eta_z=f$ for every $z$ an object of $\mathcal{S}$. This is easily seen to be equivalent, thinking about elements of $\mathcal{C}^\mathcal{S}$ as being $\mathcal{S}$-tuples, taking an element $x$ to it’s corresponding $\mathcal{S}$-tuple and taking an arrow $x\xrightarrow{f}y$ to the $\mathcal{S}$-tuple with $f$ in each coordinate.

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Then, given a $\mathcal{S}$– indexed set $(x_s)$ of objects in $\mathcal{S}$, thought of as a functor $F:\mathcal{S}\to\mathcal{C}$, we define the product over $(x_s)$, to be a universal arrow from $\Delta$ to $F$. Of course, it’s easy to see that the information given by such a universal arrow is equivalent to a $\mathcal{C}$-object $p$ along with arrows $p\xrightarrow{\pi_s}$x_s\$ for each $s\in\mathcal{S}$ such that given any other $\mathcal{C}$-object $z$ and a set of arrows $z\xrightarrow{j_s}x_s$ there exists a unique arrow $z\xrightarrow{j}p$ such that $\pi_s\circ j=j_s$.

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Note that while a category may have the product two objects, say, it needn’t have arbitrary products. As a silly example, consider that the category $\mathbf{FinVect}_k$ of finite dimensional vector spaces over some field $k$ is certainly closed under finite products, but not under arbitrary ones.

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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] Mitchell, Barry. Theory of Categories. New York: Academic, 1965. Print.

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

January 24, 2012 -

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