I'm reading Goldblatt's "Topoi The Categorial Analysis of Logic". In it he introduces products early on, using a diagram like, $$\begin{array}{center} \; d \\ \; f \swarrow \downarrow m \searrow g \\ \; a \longleftarrow \; a \times b \longrightarrow b \\ \end{array}$$ $$\begin{array}{center} \; pr_a && pr_b \end{array}$$ where $$ m=<f,g> $$ is unique. (the arrow m is denoted by a broken arrow for uniqueness, and the diagram would probably look better if the diagonal arrows were longer and I knew how to do spaces, but I didn't know how to do all that this being my first Q here on these forums, much less my first diagram).

(Before continuing, I like to call the object at top (i.e. d in my diagram above), the "auxilary object". Which afaik could be ANY object in our category that has arrows to the "factor objects": And for two given objects in defining their product, I call them "factor objects" (in reference to the class of isomorphic product objects and their respective projections.) So a and b are factor objects in the diagram above.)

As he continues, it looks as though the unique arrow, called the "product arrow", is usually denoted as an ordered pair of the given arrows from the auxilary object to each of the factor objects, being called the product of the mentioned arrows. So for example, again $$ m=<f,g> $$

The author mentions that a product of two objects is not necessarily unique, and in fact goes on to prove that all products are isomorphic. In the proof he flips the diagram above yielding, $$\begin{array}{center} \; a \times b \\ \; pr_a \swarrow \downarrow n \searrow pr_b \\ \; a \longleftarrow \; d \longrightarrow b \\ \end{array}$$ where this time the auxilary object is the product $$ a \times b $$ and this time an arbitrary product object d, of a and b shows up instead in the middle bottom. But now the product arrow n, is $$ n=<pr_a,pr_b> $$ again staying with the notation that the product arrow is written as an ordered pair of the auxilary object's given arrows to the given factor objects.

In one of the following exercises the author uses the same notation in asking the reader to prove that $$ <pr_a,pr_b>=id_{a\times b} $$ for the product-defining diagram in which both the auxilary object and the product are the same, $$ a \times b $$ for: $$\begin{array}{center} \; a \times b \\ \; pr_a \swarrow \downarrow m \searrow pr_b \\ \; a \longleftarrow \; a \times b \longrightarrow b \\ \end{array}$$ $$\begin{array}{center} \; pr_a && pr_b \end{array}$$ Ok I understand the proof and was able to prove the exercise, but neither tells me:

OK NOW my question:

Is this merely an abuse of notation, using the same symbol for possibly distinct arrows in different diagrams?


Despite the definition of product being "up to isomorphism", it actually happens to always be unique? Iow, products actually ARE unique, and this uniqueness is implied by the category axioms and the defintion of product (which initially is at least up to iso)? In which case I missed the other point of the exercise and didn't actually finish it (and still can't either way, considering I'm asking this now).

  • 3
    $\begingroup$ Products are more than "unique up to isomorphism". They are "unique up to unique isomorphism." That said, it is a bit of abuse of notation, but the philosophy in category theory is that an object is determined by what it "does" rather than what it "is". A product is characterized by the universal property, so it doesn't matter what it actually "is" or what we call it, what matters is that it satisfies the universal property. Any object that satisfies the universal property can be "canonically identified" with any particular instance of the product, so the abuse is rather mild. $\endgroup$ – Arturo Magidin Apr 18 '12 at 4:06
  • $\begingroup$ Welcome to Mathematics stackexchange Chris $\endgroup$ – magma Apr 18 '12 at 11:55
  • $\begingroup$ yes it is very much an abuse of notation. I am going to write a more complete answer later today $\endgroup$ – magma Apr 18 '12 at 12:18
  • $\begingroup$ You may be interested in the notion of an "anafunctor". But it's a bit of an aside: I've never actually seen the notion of anafunctor used for any purpose other than to point out it captures the notion of being defined up to isomorphism. $\endgroup$ – Hurkyl May 22 '12 at 9:03

It is an abuse of notation, of the most common kind in mathematics: the notation does not include all the parameters it depends on. Let $a \times b$ be some chosen product of $a$ and $b$, with chosen projections $pr_a$ and $pr_b$, and let $f : d \to a$ and $g : d \to b$ be two morphisms. When you write $\langle f, g \rangle$ for the morphism $d \to a \times b$ determined by $f$ and $g$, the notation hides the dependence on $(a \times b, pr_a, pr_b)$. I'm sure you'll agree we'd get tired of writing something like $\langle f, g \rangle_{(a \times b, pr_a, pr_b)}$ very quickly. This type of notation that is not fully explicit, in that it shows only the parameters you want to emphasize and lets the rest be determined by the context, is extremely common in mathematics and I'd even say it's necessary to keep things manageable.

Your other question is whether products are actually unique and not just unique up to isomorphism. They most definitely are not! Consider the category of sets: if $a$ is a two element set and $b$ is a three element set, then for any set $c$ of six elements you can define appropriate functions $pr_a : c \to a$ and $pr_b : c \to b$ that show that $c$ is a possible product of $a$ and $b$ (in fact, given a fixed 6 element set $c$, there are $6! = 720$ choices of $(pr_a, pr_b)$ that work).

The kind of uniqueness that products (and other things defined by universal properties) is that they are unique up to a unique structure-preserving isomorphism (the "structure-preserving" is often understood and not mentioned explicitly). For example, what this means for products is that if $(c,pr_a,pr_b)$ and $(c',pr'_a,pr'_b)$ are two products of $a$ and $b$, then

  • $c$ and $c'$ are isomorphic, and

  • although there may be lots of isomorphisms $f : c \to c'$, exactly one of them will preserve the structure of "being a product of $a$ and $b$", that is, only one will satisfy $pr_a = pr'_a \circ f$ and $pr_b = pr'_b \circ f$.

In the example above (the category is the category of sets, $a$ has two elements, $b$ has three), given any two products $c$ and $c'$, they both are forced to be six element sets, so there are isomorphisms between them, in fact $6! = 720$ isomorphisms. Of these $720$, exactly one commutes with the projections to $a$ and $b$.

  • $\begingroup$ Omar's excellent answer (+1 from me) is very similar to the one I was going to write. These kinds of notational abuses are very common in category theory, but are rarely mentioned in standard texts. As I mentioned in this question math.stackexchange.com/q/130527/19609 I am developing a Mathematica package for category theory. It is when you start programming a PC to do abstract symbolic calculations that you realize how imprecise/simplified the traditional notation is and the need to use/invent a more rigorous one. $\endgroup$ – magma Apr 19 '12 at 17:27

To add on to Omar's response:

Not only is it an abuse in notation, but it is one you are going to have to internalize for the rest of your stay here in Math Land. Almost every object is defined (at least categorically) as some kind of universal arrow, object, colimit, or limit (not that these are mutually exclusive) which means they come with two, three, or infinitely many paramaters. For example, if we are thinking categorically the kernel of the group map $f:\mathbb{Z}\to\mathbb{Z}/2\mathbb{Z}$ is $(2\mathbb{Z},i)$ where $i$ is the inclusion $2\mathbb{Z}\hookrightarrow \mathbb{Z}$.

Let me see if I can (most likely poorly) explain why we have this unfortunate situation.

It is a literal mantra (one made precise by Yoneda's lemma) that one should focus on arrows instead of objects--that objects are just glorified placeholders for where arrows start and end (also, look in other books on category theory that define things "object free" interms of the domain and codomain function). Intuitively this thinking says that an objects interaction with other objects determine it entirely. Thus, if we really want to "categorically" define anything with any level of oomph we need to define it with arrows.

This all said, humans being creatures of set theory (at least historically) it's hard to us to divorce our predefined notions of groups, topological spaces, etc. from set/object based thinking. Thus, all of the things we actually want to end up defining categorically are objects! Thus, we find ourselves wanting to define objects categorically which, by previous statement, must mean defining them in terms of arrows. Thus, to define a product in a category we normally work in, say $\mathbf{Grp}$, we are really after an object and the arrows are just along for a ride. Thus, even though categorically a product of $G$ and $H$ is a triple $G\overset{p_1}{\leftarrow}P\overset{p_2}{\rightarrow}H$ with certain properties we secretly (regardless of whether or no this is a good thing) are focused on $P$. In fact, most of the time we really only care if the group $P$ can be put into such a special triple! Of course, this is a terrible idea since anything we should want to prove about $P$ should literally not involve $P$ at all and just properties of the maps $p_1,p_2$.

So, I guess my point is that never including the arrows in a universal construction is just the plight of being an arrow oriented person living in an object oriented world.


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