# why haven't certain well-researched classes of mathematical object been framed by category theory?

Category theory is doing/has done a stellar job on Set, FinSet, Grp, Cob, Vect, cartesian closed categories provide a setting for $\lambda$-calculus, and Baez wrote a paper (Physics, Topology, Logic and Computation: A Rosetta Stone) with Mike Stay about many of the interconnections between them.

But there are mathematical objects that aren't thought of in a category-theoretic fashion, at least the extant literature doesn't tend to treat them as such. For instance nobody talks about Series, Products, IndefInt as being categories in their own right. (infinite series, infinite products, and indefinite integrals, respectively). (google searches for the phrase "the category of infinite series" in both the web and book databases have no hits whatsoever). I suppose my question is: why not?

• One of the first things to learn about category theory is that not everything in mathematics is a category. (-: Apr 23, 2010 at 15:02
• "To a man with a hammer, everything looks like a nail." Mark Twain Apr 23, 2010 at 15:19
• Some version of this question is asked here on a monthly basis. The category-theorists always respond in a reasonable way, explaining that their subject, like any other, has limitations. To be blunt: no amount of adjoint-this or colimit-that will tell you whether a sequence converges or whether a PDE has a solution. Huge swathes of current mathematics depends on proving convergence and solving PDE. Apr 23, 2010 at 15:48
• @Yemon: Maybe someone whose only tool is a claw hammer would try to use it to unscrew things, despite the absurdity of even trying. The mathematical analogue is evident. Apr 23, 2010 at 15:50
• Tim Perutz's comment reminds me of a point made in the preface of Evans's PDE book. "PDE is not a branch of functional analysis. ... [T]he insistence on an overly abstract viewpoint, and consequent ignoring of deep calculus and measure theoretic estimates, is ultimately limiting." Even functional analytifying is often too much in analysis and differential equations, let alone categorifying. Apr 23, 2010 at 16:28

Fundamentally I agree with Mike Shulman's comment and I do not really want to claim the following fancy language is at all necessary to answer this question, but you may (or may not) find it illuminating.

From the standpoint of higher category theory, categories (i.e., 1-categories) are just one level among many in a family of mathematical structures. Typically a mathematical object will "naturally" exist as an n-category for some particular n. For example, Set is naturally a 1-category, while Cat is naturally a 2-category. Your examples Series and so on seem to just be 0-categories, i.e., sets, since as Pete explained in his answer, there is no obvious natural notion of morphism between infinite series. Asking why Series is not a 1-category is like asking why Set is not a 2-category; these are just not the natural categorical levels that these objects live at.

To turn a class into a category, you need a notion of morphisms between objects in the class. That's the long and short of it.

Consider for instance the class of infinite real series, say viewed as the set $S = \mathbb{R}^{\aleph_0}$ of sequences of real numbers. (There is often some notational and "ontological" confusion between the terms of an infinite series, its associated sequence of partial sums, and its sum, if it has one. Which one of these "is" the series? But such considerations are not relevant here and indeed are usually viewed as antithetical to the categorical point of view.) To get a category, you need to identify a set of morphisms between any two elements of this set. This can certainly be done in any number of ways -- for instance one could use the ordering induced from the standard ordering on $\mathbb{R}$ and the lexicographic ordering of the sequence, and then $S$ is a totally ordered set. We could then define a category by having $\operatorname{Hom}(s,t)$ to be a one point set if $s \leq t$ and the empty set otherwise (and then take the unique composition law of morphisms, defined when $s \leq t \leq u$).

But the question is: what does this category have to do with any aspect of the theory of infinite series? Apparently nothing. You could create any number of other categories with underlying set $S$ but you run into the same problem: the very old and extremely well-developed area of mathematics which studies the convergence and divergence of real infinite series simply does not have anything evident to do with any notion of "morphisms" between infinite series.

Similarly for the other examples you mention. Categorical structure is a very fundamental kind of mathematical structure; it's a great way of thinking and unifies and conceptualizes the study of many kinds of mathematical objects in highly disparate fields. But it doesn't explain everything, and it is frankly a bit weird to think it should.

• There are morphisms between different series of a given type -- there are variety of <a href="en.wikipedia.org/wiki/Series_acceleration">series transformations</a>, such as Eulerian series acceleration, there's scalar multiplication, which turns one given series into another, and there's the equivalence class of all series that sum to the same thing (pi and e formulas for instance.) Apr 23, 2010 at 20:41
• the link doesn't seem to be working right. The full url is here: en.wikipedia.org/wiki/Series_acceleration Apr 23, 2010 at 20:45
• I notice you made a point of mentioning convergence. On the other hand, if you consider formal series without regard to convergence, then we have some nice category theory going on. For a long time, combinatorics resisted a categorical treatment, but then along came combinatorial species, and lots of interesting combinatorial structures turned out to be functors. And at that point, many properties of formal power series, considered as generating functions, suddenly acquired category theoretical meaning. Resistance is futile. :-) en.wikipedia.org/wiki/Combinatorial_species Apr 23, 2010 at 21:14
• +1 for the Borg reference. I often have the fealing, category theory is assimilating the rest of the mathematical galaxy. ;-) Apr 24, 2010 at 0:06

It might be worth noting that the problems of computing Feynmann integrals in quantum field theory is one that is traditionally phrased as one of analysis, but is now studied by pure mathematicians using categorical techniques (among others).

Paul Taylor's "Abstract Stone Duality" http://www.paultaylor.eu/ASD/ is an attempt to recast elementary real analysis (including sequences) involving categorical ideas.

A Google scholar search for "category theory" "power series" brings up the paper: Elements of Stream Calculus::(An Extensive Exercise in Coinduction). So series can be usefully thought of in a category-theoretic fashion, and although this involves formal series the methods can be used to find solutions to differential equations.

For just another point, we could contemplate the hilarious line "why can't a woman... be more like a man", from you-know-where.

That is, the style of (much) "analysis" reflects as much the personalities of the participants as it does the "mathematical reality"... the latter being self-referentially "defined" by the personalities.

That is, perhaps the issue addressed by the question resides in the implicit assumptions of the questioner? :)

In my own experience, "(naive?) category theory", in the sense of paying attention more to interactions among objects than to their internal structures, has been extremely helpful, at the very least in showing that various seeming "choices" or "constructions" were irrelevant... But I do recognize that other people thing other-ly...

• I would phrase Paul's sentence a bit differently, in saying the notion of for example colimit allows for the description of an internal structure in terms of the way the object is put together from external objects, and indeed that the colimit structure is defined by all its interrelationships with other objects. In a technical sense, proving something is a colimit allows for the elegant proof by "verification of the universal property", which often is the "best" proof, since that is how the notion of colimit is defined. Jan 4, 2013 at 14:28

I really like to think of $\mathbb{N}$-graded $R$-modules as power series $\bigoplus_{n \in \mathbb{N}} M_n \otimes X^{\otimes n}$, where each "coefficient" $M_n$ is an $R$-module and $X$ is the graded module concentrated degree $1$ and which is $R$ there. Therefore, we have a category of power series, where a morphism is just a family of morphisms between the coefficients. Actually this is a symmetric monoidal category - the tensor product is given by some convolution. And the same works if we replace $\mathsf{Mod}(R)$ by any cocomplete symmetric monoidal category. This is spelled out for example in Section 5.4 of my thesis.

In order to get a connection to power series in analysis, we might endow the unit interval $[0,1]$ with the structure of a cocomplete symmetric monoidal category (cf. Example 3.1.6 in loc.cit.): We use the usual ordering to make it a (thin) category and the usual multiplication to make it a symmetric monoidal category. Colimits are given by suprema. Therefore, we get a cocomplete symmetric monoidal category of sequences valued in $[0,1]$. This is again just an order with a multiplication, where we have $(a_n) \leq (b_n)$ iff $a_n \leq b_n$ for all $n$, and $((a_p) \cdot (b_q))_n = \sup_{p+q=n} a_p \cdot b_q$.

Also notice that coends in category theory capture some ideas of (definite) integration. See MO/78471 for some intuition.

• @ Martin: Without any provocative aim, but what for? I am quite categorically-inclined, or like to think that I am, but is this interpretation of any use for - say - proving Boltzano-Weierstrass or Lagrance Mean Value Theorem? If the answer is yes, do you have any reference? Oct 7, 2014 at 21:16
• Unfortunately, I cannot answer that question. (Notice that the question was not about any applications of categories of series, but just about their existence.) But you might be interested in looking for literature about "quantales", which are exactly cocomplete (symmetric) monoidal preorders. Oct 8, 2014 at 8:26