The n-Category Café

Sorry, I’ve been away, otherwise I would have chipped in. I too certainly knew the answer to Puzzle 1, and would have enjoyed having a go at the other bits. It’s very much up my street.

Actually I think of spans and bimodules as being related by both being special cases of enriched profunctors(!!). I think that I learnt this off Tom Leinster – that’s to say, if it’s true then I learnt it off Tom and if it’s not then I made it up.

So it ought to work as follows. Pick your favourite monoidal category $\mathcal{V}$ to enrich in, which if you’re in an impoverished mood could be Set or Vect. Then you have a bicategory $\mathcal{V}$-MOD whose objects are $\mathcal{V}$-categories, where the morphisms from $C$ to $D$ are $\mathcal{V}$-functors $C^{op}\otimes D \to \mathcal{V}$ and where the 2-morphisms are something appropriate.

Then if you look at the full sub-bicategory of Set-MOD where the objects are Set-categories (ie, categories) with only identity morphisms (ie, sets) then you get the bicategory of spans.

On the other-hand if you look at the full sub-bicategory of Vect-MOD where the objects are Vect-categories with only one object (ie, algebras) then you get the bicategory of bimodules.

While I’m here, does anyone know of a standard reference for $\mathcal{V}$-MOD? Enriched bimodules/profunctors/distributeurs only get a one-sentence mention in Kelly’s book.

The unit requires a map from the empty set to your monoid object, which is only possible if the monoid object is also the empty set!

No. There’s always a unique map from the empty set to any set. The empty set is the initial object.

… or How I Learned to Stop Worrying and Love $f(x)$:

When I was a beginning undergrad, I liked writing $f(x)$ for functions and didn’t like writing just $f$. (What am I supposed to take $f$ of?) Then at some point, I got the point, and then thought that writing $f$ was better than $f(x)$, and might have even secretly looked down on people who did it the high-school way. But then much later, I realized that if you just declare that $x$ denotes the identity function, then you have $f=f\text{ o } x = f(x)$. (The o is supposed to denote composition, but I couldn’t get it to display right.) So you can say, “Consider the functions $f$ and $x^2+1$ and their composition $f(x^2+1)$,” with no worries about inconsistency in notation.

On the other hand, then you can’t ever let $x$ be a real number.

James wrote:

On the other hand, then you can’t ever let $x$ be a real number.

“A foolish consistency is the hobgoblin of little minds…”

Back when I was seriously into the philosophical conundra that arise from obsessive attempts at mathematical rigor, I used to wonder: when we write down $x = 5$ on the blackboard, how long is this equation good for? Until the next paragraph? Until the end of class? We “set $x$ equal to 5”, but we never seem to un-set it later, to free it up for future use! We just hope the audience will be sensible enough to not derive a contradiction when we later change our minds and say that $x = 6$.

Of course, computer scientists have real reason to worry about these issues, since computers are too dumb to forget things unless told! That’s why computer scientists need to talk about ‘global variables’ versus ‘local variables’.

Anyway: if we work hard enough, we can surely figure out a way to make standard practice count as mathematically rigorous. For example, instead of taking $x$ to be the identity map from $\mathbb{R}$ to $\mathbb{R}$, how about letting it be a variable which is allowed to stand for any set to $\mathbb{R}$? Then when we set $x = 6$ we are declaring that it’s the map from the 1-point set to $\mathbb{R}$ sending the one point to $6$.

The identity map from $\mathbb{R}$ to $\mathbb{R}$ is just nice because it’s the terminal object in the category of functions taking values in $\mathbb{R}$: any other function taking values in $\mathbb{R}$ factors through this one. So, it’s a way of remaining ‘maximally uncommitted’ about $x$.

Of course, this is the idea behind the Yoneda embedding.

I certainly didn’t wish to suggest that catgeory theorists are less carefully, although, looking back, I did seem to write this. Sorry.

I think all I’m trying to say is that I was confused by John’s answer “A Set”. Something like “A Set, with the obvious and unique maps to form a Monodial Object” would have been more enlightening. However, John says in a comment below that he didn’t mean this to be an easy or elementary example, so fair enough.

Btw, I’m a, I guess, Operator Algebraist, but one with interests ranging from Banach spaces to Topological Quantum Groups.

Tom wrote:

The implication here — that less care is exercised by category theorists — is unfair, I think. My perception is that category theorists are just as careful as practitioners of other subjects. That is to say, in all subjects there are some who are careful and some who are careless; and I don’t think the distribution in category theory is very different from that in other subjects.

Yeah, it’s funny – I’m tempted to take it further and say that I think category theorists are, by the nature of their profession, forced to be on the whole more careful than mathematicians in many other fields. A rough reason why is that the great generality achieved by category theory must be compensated by a tremendous and sometimes almost fanatical precision and care with language. In fact, this is one thing I find deeply attractive about category theory: the high level of linguistic precision and control. This is especially visible to me in categorical logic.

By way of contrast, let me quote something (perhaps tendentiously, but without the slightest intent to ‘diss’ another field) from the book Stratified Morse Theory, by Goresky and MacPherson (section 1.8: Remarks on Geometry and Rigor):

As is shown by the above history of Morse theory or by the history of stratification theory, there is often a creative tension between geometry and rigor. Rigor follows the initial conception with a much greater time delay in geometry than it does in algebra. Also, when it comes, true geometers often feel its language misses the essential geometric ideas. Language is not well adapted to describing geometry, as the facilities for language and geometry live on opposite sides of the human brain. This perhaps accounts for the presence in the current literature on singularities of expressions like “using the isotopy lemma, it can be shown” without the forty pages of geometric constructions and estimates needed to apply the isotopy lemma.

But that’s just the mathematical side. It looks like Doormat (and maybe Tom) are referring more to the sociological aspects of ordinary mathematical discourse. Here I would side with Tom, that there is probably no difference on average in the degree of sloppiness (or pedantry) between category theorists and other mathematicians – how much rigor is given or needed is all in the context.

For example, even if we formally define a group as an ordered quadruple, does that mean we think, when we talk with someone at the blackboard, of an ordered quadruple when he says ‘group’? Generally not – for anyone used to it, the notion of ‘group’ is a kind of gestalt where we don’t necessarily itemize the individual structures, unless we really need to.

I can imagine several social reasons why John didn’t mention the comonoid structure on a set: (1) he wanted to create a kind of snappy effect; (2) he wanted to say no more about it, to give people a chance to chew on it a bit more; (3) he momentarily forgot or didn’t realize that there might be people who would be confused by what he was saying; (4) he was in a hurry, etc. Whereas in addressing a class, he might have found it more appropriate to add for clarity’s sake words like “where the comultiplication is…, and where the counit is…”

Discussions on blogs are often pretty loose anyway, maybe closer to coffeehouse conversations than papers, depending on who’s doing the “talking”.

stub asked:

The function in the answer to puzzle 2 is a function from where to where? Is this function supposed to be the identity on the factor M?

I don’t want to take away the pleasure of discovery from you, so I’ll just ask if you’ve written out what the underlying map in the definition of a module is and if you’ve figured out what the unit axiom is.

If you’re after hints, it’s helpful if you say how far you’ve actually got.

stub wrote:

If you have two objects, which are isomorphic like ‘a set’ and ‘a set with an unique extra structure’ (i.e. if they are the same in your sense, if I understood correctly) and if you want to make use only of the set without the extra structure then how do you express this in category theory?

First of all, it’s very odd wanting to make use ‘only of the set without the extra structure’ in this case — because if there’s a unique way to put on this extra structure, then there’s no real way to tell when you’re using it to do something!

But here’s a way to think about this sort of stuff. Let $Set$ be the category of sets and let $C$ be the category of sets equipped with some extra structure. Then there’s a functor

$$F : C \to Set$$

which forgets that extra structure.

If we have a construction that systematically takes a set with extra structure and builds an object of some other category $D$, then we have a functor

$$G : C \to D$$

If we have a construction that does not use the extra structure, just the underlying set, then we have a functor

$$H : Set \to D$$

Given a construction of the first sort, we can wonder if it makes essential use of the extra structure. Maybe the structure wasn’t really necessary for what we did.

And the nice thing is that there’s a way to make this question precise! We say the construction $G : C \to D$ does not make essential use of the extra structure if we can find some functor $H : Set \to D$ such that

$$G \cong H \circ F$$

In other words: doing $G$ is naturally isomorphic to first forgetting the extra structure and then doing $H$.

Now, if there always exists a unique way to put the extra structure on any set, $F$ is an equivalence of categories. So, it has a weak inverse $F^{-1}$. So, we can choose

$$H = G \circ F^{-1}$$

So, in this case no construction $G: C \to D$ makes essential use of the extra structure: we can always accomplish the same effect by first forgetting the structure and then doing $H$.

And the point is, these are questions that can be settled by computation in examples! It’s not just talk, it’s math.

Stub wrote:

The function in the answer to puzzle 2 is a function from where to where?

Like Simon, I don’t want to rob you of the pleasure of discovering the answer for yourself… even if it kills you.

So, I’ll just emphasize that I meant what I said: functions are the same as modules of monoid objects in $Set^{op}$.

stub wrote:

What are the -as John put it- “usual equations” for a morphism being an action?

Suppose $M$ is a monoid object in some monoidal category, with multiplication

$$m : M \otimes M \to M$$

and identity

$$i: I \to M$$

where $I$ is the unit object of our monoidal category.

Then an action of $M$ consists of an object $X$ together with a morphism

$$a : M \otimes X \to X$$

such that

$$(M \otimes M) \otimes X \stackrel{m \otimes 1_X}{\to} M \otimes X \stackrel{a}{\to} X$$

equals

$$(M \otimes M) \otimes X \stackrel{\sim}{\to} M \otimes (M \otimes X) \stackrel{1_M \otimes a}{\to} M \otimes X \stackrel{a}{\to} X$$

and

$$X \stackrel{\sim}{\to} I \otimes X \stackrel{i \otimes 1_X}{\to} M \otimes X \stackrel{a}{\to} X$$

equals

$$X \stackrel{1_X}{\to} X$$

In the category of sets with $\otimes = \times$, we often write $a(m,x)$ simply as $m x$ and write the image of $i$ as $1 \in M$. Then the two equations above say

$$(m_1 m_2)x = m_1 (m_2 x)$$

and

$$1 x = x$$

These are the usual equations for an action of a monoid on a set.

Please, someone: teach Wikipedia this stuff!