The n-Category Café

Here are a few comments that might help you a little when it comes to ends, coends and enriched categories.

First, you’ll notice that Kelly’s book discusses ‘ends’ shortly before discussing ‘indexed limits’. This is because the two concepts, while distinct, are interchangeable, in the sense that they’re both special cases of each other.

Kelly tackles ends first, and then indexed limits. You’ll see on page 37 that he says: “… is the limit precisely when … is an end.” Here is reducing the theory of indexed limits to the theory of ends.

Personally I think it’s easier to go the other way: first learn about indexed limits, then, if necessary, learn about ends.

In fact, I prefer to dualize, and think about indexed colimits and coends instead of indexed limits and ends.

The reason is this: the theory of indexed colimits is a straightforward categorification of the theory of integrals.

If you’re trying to categorify inner products, path integrals and other integrals, this is good to know!

Here’s how it goes, in a nutshell.

If we have a finite set $X$ we can think of a measure on $X$ as just a map

$$w: X \to \mathbb{C}$$

assigning to each point its ‘weight’. Suppose $L$ is any vector space. We can integrate any vector-valued function

$$f: X \to L$$

with respect to the measure $w$ as follows:

$$\int_X f d w = \sum_{x \in X} w(x) f(x)$$

Kelly’s theory of indexed colimits — usually called the theory of weighted colimits — categorifies this idea!

The replacement for our set $X$ is a category $X$.

The replacement for $\mathbb{C}$ is a cocomplete symmetric monoidal category $V$. The colimits of $V$ are the categorified version of ‘addition’, while the tensor product in $V$ is the categorified version of ‘multiplication’.

The replacement for our vector space $L$ is a category $L$ that’s enriched over $V$ and tensored over $V$. ‘Enriched over $V$’ says that given objects $x,y \in L$, we have $hom(x,y) \in V$. ‘Tensored over $V$’ means that given $x \in L$ and $v \in V$, we can define $v \otimes x \in L$. There is of course more to the definitions of these concepts that I’m giving here, but this is enough to get the idea.

The replacement for the vector-valued function we’re integrating is a functor

$$f: X \to L$$

The replacement for our measure is what Kelly calls an ‘indexing type’, but most people prefer to call a ‘weight’: it’s a functor

$$w : X^{op} \to V$$

Note the all-important ‘op’!

The weighted colimit, when it exists, is an object

$$\int_X f d w \in L$$

In the simple case where $X$ is a discrete category (basically just a set), we have

$$\int_X f d w = \sum_{x \in X} w(x) \otimes f(x)$$

See? It’s just like what we had before!

Another simple case is when we use the most boring possible weight

$$w : X \to V$$

namely the one with

$$w(x) = 1$$

for every object $x \in X$, and $w(f) = 1_1$ for every morphism $f$ in $X$.

If we use this weight, our weighted colimit reduces to an ordinary colimit:

$$\int f d w = colim_{x \in X} f(x)$$

So, the theory of weighted colimits generalizes the theory of colimits much as the theory of integrals generalizes the theory of sums.

But, in the simple examples I just gave, we don’t see the need for the ‘op’ in the weight

$$w : X^{op} \to V$$

If you already know and love coends, you can guess why this ‘op’ shows up. From our weight and our functor

$$f: X \to L$$

we can form a functor

$$X^{op} \times X \stackrel{w \times f}{\to} V \times L \stackrel{\otimes}{\to} L$$

where the second arrow uses the fact that $L$ is tensored over $V$.

This kind of functor

$$X^{op} \times X \to L$$

is precisely the sort of thing we take a coend of, getting an object of $L$!

But, if you don’t already know and love coends, you now see why we need them: precisely to do weighted colimits.

By the way, you might complain my use of the term ‘integral’ is overblown, since the only integrals I’m really considering are weighted sums. That’s a valid complaint. So, you might prefer it if I changed my main message to this:

The theory of weighted colimits is a straightforward categorification of the theory of weighted sums.

I ended the above entry with the question:

(Is Brian Day maybe just talking about reconstructing a tensor category from its fusion ring?)

On second reading, I think the answer is yes.

A promonoidal category $A$ is supposed to be like an ordinary monoidal category, but $V$-enriched and with functors replaced by profunctors.

So in particular, the tensor product functor $$\otimes : A \times A \to A$$ is taken to be a $V$-profunctor, i.e. a $V$-functor $$p : A^\mathrm{op} \times A^\mathrm{op} \times A \to V$$ (where “$\times$” is the $V$-tensor product).

Just like monoidal categories are just bicategories with a single object, promonoidal categories are just probicategories with a single object.

Hence, more generally, we have lots of objects $x,y,z,\cdots$ and $V$-$\mathrm{Hom}$-objects $$A(x,y)$$ etc. and horizontal composition is a $V$-profunctor $$\circ : A(x,y) \times A(y,z) \to A(x,z)$$ hence an ordinary $V$-functor $$p : A(x,y)^\mathrm{op} \times A(y,z)^\mathrm{op} \times A(x,z) \to V \,.$$

Next, when considering morphisms (2-functors) $F :A \to B$ between promonoidal categories, we want composition be respect ed by enforcing something like $$F(a)\circ F(b) \simeq F(c)$$ for $c$ the composite of $a$ and $b$.

In the probiotic language – er, I mean: in the probicategory language, this amounts to demanding $$\int^c p(a,b,c) \otimes F(c) \simeq F(a) \circ F(b) \,,$$ where the integral sign denotes a coend.

As far as I am beginning to understand the point of Brian Day’s paper here, he emphasizes that this composition law has a very important and familiar incarnation in the context of Moore-Seiberg data of RCFTs, namely in the context of semisimple braided tensor categories.

There, we consider abelian monoidal (=”tensor”) categories with a finite number of isomorphism classes of simple objects $U_i$, such that $$U_i \otimes U_j \simeq \oplus_k U_k^{\oplus N_{i j}^k} \,.$$

Brian Day notes that if we set $V = \mathrm{Vect}_K$ for $K$ our ground field and moreover set $$p(i,j,k) = K^{\oplus N_{i j}^k}$$ then this is a special case of the above probiotic composition law $$U_i \otimes U_j \simeq \oplus_k p(i,j,k)\otimes U_k \,.$$

As far as I understand. (So somehow the coend reduces to a mere coproduct here.)

Moreover, Day indicates how we can, this way, understand every such semisimple tensor category for given fusion data $N_{ij}^k$ as the image of a suitable probifunctor from something like the universal such category to a suitable codomain.

Roughly, at least. Somehow Kan extensions play a role, too, and we don’t just take the domain to be $A$, but $[A,\mathrm{Vect}]$.

I need to better understand this. But so much for now.

Grrr… I just spent a long time composing a comment, hit preview, and was told there was an internal server error and lost all this work. Time to follow Jeff Morton’s advice, and write out posts in advance on a word editor!

In the meantime, there have been a number of posts and it looks like Urs has got things mostly or maybe completely sorted out by now, but I’ll throw out a few comments anyway, and hope some of them stick.

Most (probably all) of the coends which appear in Brian Day’s note are really tensor products of modules – that really is a prototypical application of coends, and gives a great way of understanding how weighted colimits work generally.

Let’s first consider modules over a ring $R$: let $A$ be a right $R$-module and $B$ a left $R$-module. Their tensor product is a coequalizer of a pair of maps

$$A \otimes R \otimes B \stackrel{\to}{\to} A \otimes B$$

where one arrow is $A$ tensor the action on $B$ and the other is the action on $A$ tensor $B$. Now think of $R$ as a one-object category enriched in Ab (I’ll call its object ‘1’). A left module amounts to an enriched functor $B: R \to Ab$, and a right module to an enriched functor $A: R^{op} \to Ab$. In other words, these functors are tantamount to actions of the hom-object $R = hom(1, 1)$, and the tensor product can be expressed as the coequalizer of a diagram written

$$A(1) \otimes \hom(1, 1) \otimes B(1) \stackrel{\to}{\to} A(1) \otimes B(1).$$

Much more generally now, take any category $X$ enriched in any symmetric monoidal closed category $V$ (assume $V$ complete and cocomplete), and let $F: X \to V$ be a covariant action of $X$, meaning we have maps

$$F(x) \otimes \hom(x, y) \to F(y)$$

expressing functoriality of $F$ (I should say ‘enriched’, but I get tired of saying that – please insert as appropriate!). Similarly, let $G: X^{op} \to V$ be a contravariant action. Then the tensor product $F \otimes_X G$ is the coequalizer of the evident pair of maps built out of the actions:

$$\sum_{x, y \in Ob(X)} F(x) \otimes \hom(x, y) \otimes G(y) \stackrel{\to}{\to} \sum_x F(x) \otimes G(x)$$

and this is also called the colimit of $F$ weighted by $G$, also denoted

$$\int^x F(x) \otimes G(x).$$

Similarly, if one has two modules $F, G: X \to V$, one can form their module-hom $\hom_X(G, F)$ as an object of $V$, as an equalizer of a pair of maps again built from the actions, again directly generalizing the case of modules over a ring. This is a limit of $F$ weighted by $G$.

More generally still, given a $V$-category $C$, a $V$-functor $F: X \to C$, and a weight $G: X^{op} \to V$, one can define the notion of colimit of $F$ weighted by $G$, as an object of $C$. The trick is to define it universally, by stipulating that all contravariant hom-functors $\hom(-, c)$ carry this weighted colimit to the appropriate weighted limit in $V$. [It is not generally correct to try to define it as a colimit of a diagram in the underlying (ordinary) category of $C$ – instead, bring it back to hom-base (“home base”) $V$, where you are safe.]

Day’s note is written in the well-known style of hardcore Australian category theory: powerful medicine, but not always gentle on the system. But here’s a way of understanding this probicategory and p(a, b, c) business: a probicategory enriched in $V$ is really a bicategory internal to the bicategory of $V$-categories and $V$-profunctors between them. (A $V$-profunctor from $C$ to $D$ is just a bimodule from $C$ to $D$, i.e., a covariant-$C$ contravariant-$D$ functor $C \otimes D^{op} \to V$. These are composed by taking tensor products of bimodules, as in the discussion above.) Thus, a probicategory consists of a collection of objects $x$, $y$, $z$, …, and hom-categories $\hom(x, y)$ enriched in $V$, and compositions

$$\hom(x, y) \otimes \hom(y, z) \to \hom(x, z)$$

which are bimodules, and so forth and so on. If you unravel all this in nuts-and-bolts terms, you find yourself looking at all these coend thingies that Day writes down.

I am trying to understand the role played by these “Kan extensions” that Day mentions.

It seems that the issue is the following:

For an ordinary bifunctor $$F : A \to B$$ respect for composition is expressed by $$\array{ A(x,y) \times A(y,z) &\stackrel{\circ_A}{\to}& A(x,z) \\ F \times F \downarrow \;\; &\simeq& \;\; \downarrow F \\ B(F x,F y) \times B(F y,F z) &\stackrel{\circ_B}{\to}& B(F x,F y) } \,.$$ But when composition of Hom-categories $A(x,y)$ is just a profunctor, i.e. a functor $$p : A(x,y)^\mathrm{op} \times A(y,z)^\mathrm{op} \times A(x,z) \to V \,,$$ then the top horizontal morphism does not land in $A(x,z)$, but in $[A(x,z),V]$:

$$A(x,y)^\mathrm{op} \times A(y,z)^\mathrm{op} \to [A(x,z),V] \,.$$ The ${}^\mathrm{op}$ is not an issue, but the different right hand side is.

So for morphisms $F$ between probicategories there must be some $\hat F$ such that $$\array{ A(x,y)^\mathrm{op} \times A(y,z)^\mathrm{op} &\stackrel{p}{\to}& [A(x,z),V] \\ F \times F \downarrow \;\; &\simeq& \;\; \downarrow \hat F \\ B(F x,F y) \times B(F y,F z) &\stackrel{\circ_B}{\to}& B(F x,F y) } \,.$$

Under the condition on $p$ that I already mentioned above, these $\hat F$ are given by the coend $$\hat F(f) = \int^c f(c)\otimes F(c) \,,$$ apparently, where $f \in [A(x,z),V]$. With these $\hat F$ then we get a diagram of the form we are familiar from for ordinary bifunctors, but now with all Hom-categories replaced by their categories of maps into $V$: $$\array{ [A(x,y),V] \times [A(y,z),V] &\stackrel{\circ}{\to}& [A(x,z),V] \\ \hat F \times \hat F \downarrow \;\; &\simeq& \;\; \downarrow \hat F \\ B(F x,F y) \times B(F y,F z) &\stackrel{\circ_B}{\to}& B(F x,F y) } \,.$$

If I understand correctly, the “$\circ$” in the top horizontal row here is now that important convolution product

$$f \circ g = \int^{a b} f(a)\otimes g(b)\otimes p(a,b,-) \,.$$ So, using the definition of $\hat F$ from above, the source 1-morphism of the above diagram would be $$\hat F(f \circ g) = \int^c \int^{a b} f(a)\otimes g(b) \otimes p(a,b,c) \otimes F(c) \,,$$ while $$\hat F(f) \times \hat F(g)$$ would be given by $$\int^{a b} f(a)\otimes g(b) \otimes (F(a) \circ F(b)) \,.$$

So, in as far as I can see, this is the reason why we are interested in finding “Kan extensions” of morphisms $F : A^\mathrm{op} \to B$ to morphisms $\hat F : [A,V] \to B$. (This is probably not the right way to say this.)

In prop. 1 (p. 2 ) of Day&Street, Kan Extensions along Promonoidal Functors, something similar appears.

I am still far from having a good understanbding for what is going on here. But I thought I’d share this, maybe in the hope to provoke somebody to help out.

I am afraid I still need to think a little about coends (or bombard you with questions – thanks a lot for all your input, John, Todd and Tom!) (or read more about it) before I feel really comfortable with them. But not today. It’s late already.

But here is something related:

today I was asked to demonstrate that composition of bimodules is, under suitable conditions, weakly associative.

To my great embarrassment, I had to realize that for general bimodules (as opposed to, say, left-induced bimodules or Frobenius bimodules) I had never worked that out myself, nor seen it worked out anywhere, but always taken the standard hint in the literature for granted, which would just assert that there is a canonical associator for bimodule composition.

Now, it may get even more embarrassing, because possibly this is quite easy to show and all I could come up with is the construction given in

these brief notes

I assume that a good exercise to become really familiar with coends, would be to redo this exercise for the general case of profunctors.