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Stream: learning: questions

Topic: Weighted (co)limits without (co)ends

Fawzi Hreiki (Dec 12 2020 at 13:04):

I'm trying to learn some enriched category theory, but it's a little difficult to wrap my head around ends and coends. I've seen the various definitions in the unenriched case (universal extranatural transformation, weighted (co)limit, etc..) and the weighted (co)limit approach is most intuitive to me. The problem, as I understand it, is that in the V-enriched case, we can only define weighted (co)limits once we have enriched functor categories, which themselves depend on V-valued ends.

Basically, my question is this: what is the proper hierarchy of concepts here? When developing enriched category theory, are (co)ends necessary from the get go or is is possible to define enriched functor categories and weighted (co)limits without them? If (co)ends are in fact necessary, is there some sort of intuition as to why that is?

John Baez (Dec 12 2020 at 17:29):

You can use weighted colimits to define coends or you can use coends to define weighted colimits. When James Dolan taught me this stuff he argued that it's more intuitive to start with weighted colimits.

John Baez (Dec 12 2020 at 17:32):

If you like analysis (as I do), weighted colimits are a natural concept, since they are analogous to integrals

$\int f(x) d\mu(x)$

where you need a function and a measure. The diagram you're taking the weighted colimit of is analogous to the function, while the weight is analogous to the measure.

Fawzi Hreiki (Dec 12 2020 at 17:46):

How are weighted (co)limits defined in the enriched setting without first defining (co)ends though?

Fawzi Hreiki (Dec 12 2020 at 17:47):

To define weighted (co)limits, we need representability and therefore enriched presheaf categories.

John Baez (Dec 12 2020 at 17:50):

This doesn't exactly answer your question but it should help you understand what's going on:

https://golem.ph.utexas.edu/category/2007/02/day_on_rcfts.html#c007688

Fawzi Hreiki (Dec 12 2020 at 18:19):

So the colimit of $f: X \rightarrow L$ weighted over $w: X^{op} \rightarrow V$ is defined to be the (usual, conical) colimit of the composite $X^{op} \times X \xrightarrow{w \times f} V \times L \xrightarrow{\otimes} L$? (here $X$ and $L$ are $V$-enriched categories with $L$ tensored over $V$)

Fawzi Hreiki (Dec 12 2020 at 19:24):

Ok no that's false - it's the coend of that composite.

Matteo Capucci (he/him) (Dec 14 2020 at 12:11):

I consider the 'Motivation...' paragraph here https://ncatlab.org/nlab/show/weighted+limit to be quite englightening (disclaimer: I wrote that paragraph)

John Baez (Dec 15 2020 at 21:31):

Okay, thanks! The page "weighted limit" also satisfies @Fawzi Hreiki's request for a definition of weighted limits in the enriched setting that does not mention ends.

John Baez (Dec 15 2020 at 21:34):

Here it is:

Let $V$ be a closed symmetric monoidal category, and let $K$ and $C$ be $V$-enriched categories.

A weighted limit of the $V$-functor called the diagram

$F : K \to C$

with respect to the $V$-functor called the weight

$W : K \to V$

is an object that represents the $V$-functor from $C^{\mathrm{op}}$ to $V$ given on objects $c \in C$ by

$c \mapsto K,V$

and doing the obvious thing on homs.

John Baez (Dec 15 2020 at 21:35):

So that's pretty nice and quick!

John Baez (Dec 15 2020 at 21:45):

Expanding a bit: given $c \in C$ we have two $V$-functors from $K$ to $V$, the weight

$W : K \to V$

and the $V$-functor involving our diagram $F$:

$C(c, F(-)) : K \to V$

John Baez (Dec 15 2020 at 21:49):

Both these give functors from $C^{\mathrm{op}}$ to the $V$-category $[K,V]$. (The first depends trivially on $c \in C$, i.e. not all, while the second depends contravariantly on $c \in C$.)

John Baez (Dec 15 2020 at 21:50):

So, we get a $V$-functor from $C^{\mathrm{op}}$ to $V$ given on objects by

$c \mapsto [K,V](W, C(c,F(-))]$

John Baez (Dec 15 2020 at 21:53):

And we can ask if this is representable: is there an object $v \in V$ such that there's a natural isomorphism

$C(c,v) \cong [K,V](W,C(c,F(-))] \; ?$

If there is, we say $v$ is the weighted limit of the diagram $F$ with the weight $W$.

John Baez (Dec 15 2020 at 21:55):

If this seems confusing, it's probably good to start with some examples where $V = \mathsf{Set}$!

Fawzi Hreiki (Dec 15 2020 at 22:15):

Thanks a lot for this - I'm slowly starting to get to grips.

Dan Doel (Dec 15 2020 at 22:28):

Wasn't the original complaint with this that $K,V$ is actually defined to be an end?

Dan Doel (Dec 15 2020 at 22:29):

At least, sometimes you see that.

Dan Doel (Dec 15 2020 at 22:31):

In fact, if you click through, that is how the nlab defines it.

Dan Doel (Dec 15 2020 at 22:36):

So you need to bootstrap yourself with ends in $V$, or unpack those into the definition of the $V$-enriched functor category.

Fawzi Hreiki (Dec 15 2020 at 23:09):

Thats true, although once worked out, the definition of the enriched functor category 'in components' is actually fairly understandable. Pedagogically, it should be possible to develop enriched category theory that way, only defining (co)ends once you get to weighted limits - and that's what Borceux does in his chapter on enriched categories.

Nonetheless, reading around, it seems that (co)ends are an extremely useful computational tool. Comparing proofs in Borceux and in Kelly's book, Borceux, in some places, has pages and pages of equalisers and coequalisers that could be reexpressed as (co)ends I think.

Dan Doel (Dec 15 2020 at 23:18):

Yeah, (co)ends are nice. I think I've never used weighted (co)limits.

Dan Doel (Dec 15 2020 at 23:18):

Not that I do anything fancy, of course.

John Baez (Dec 15 2020 at 23:28):

Okay, so you're saying the V-category of V-functors between two V-categories is defined using V-enriched ends.

So then you need to understand at least that one case of ends to understand weighted V-limits.

Luckily this particular case of ends is pretty approachable.

But in the end, I guess you need to understand ends.

Eric Forgy (Dec 15 2020 at 23:31):

image.png

Dan Doel (Dec 15 2020 at 23:31):

I assume you can define the functor category directly, talking about the covariant actions stuff in the nlab section on ends in $V$ to define V-natural transformations. Then define weighted (co)limits in terms of just that.

Dan Doel (Dec 15 2020 at 23:33):

But nlab does the opposite, and basically defines a V-natural transformation to be an end.

Fawzi Hreiki (Dec 15 2020 at 23:41):

No, a V-natural transformation is just a V-natural transformation - no ends or equalisers or whatever required. Where things get complicated is with V-enriched functor categories. You can define the unenriched functor category but the trick is to define an enriched functor category such that its underlying category is the unenriched functor category.

Fawzi Hreiki (Dec 15 2020 at 23:42):

You can do that directly by using limits in the enriching category, or taking a detour via (co)ends.

Dan Doel (Dec 15 2020 at 23:45):

Oh right. Defining $V$-natural transformations doesn't do you good directly, because you need to collect them into an object of $V$.

Dan Doel (Dec 15 2020 at 23:47):

And the end is how they're defining that object.

Fawzi Hreiki (Dec 15 2020 at 23:49):

Yeah, although its a proposition-definition, because you still have to show that this gives the object of natural transformations, but that's not too hard.

Matteo Capucci (he/him) (Dec 16 2020 at 09:54):

Something I've never seen (not that I've tried too hard to look for it, actually) is a definition of composition for the V-enriched functor categories defined as ends
Let $\mathbf C, \mathbf D$ be V-enriched categories. Then $[\mathbf C, \mathbf D]$ is V-enriched with hom-objects

$$[\mathbf C, \mathbf D](F,G) = \int_c [Fc, Gc]$$

where $[-,-]$ is V's internal hom (secondary question here: the nlab uses $V(-,-)$ in this end, but does it even typecheck?).
Now to define composition $\mathbf C, \mathbf D \otimes \mathbf C, \mathbf D \to \mathbf C, \mathbf D$ one should define a map

$$\left(\int_c [Fc, Gc] \right)\otimes \left( \int_{c'} [Gc', Hc'] \right) \longrightarrow \int_{c''} [Fc'', Hc'']$$

Any ideas on how to do it?

Fawzi Hreiki (Dec 16 2020 at 11:09):

Your square bracket notation $[F, G]$ for the object of natural transformations $\mathscr{C}, \mathscr{D}$ is kind of confusing since it looks like some sort of internal hom, which it isn't (since $\mathscr{C}, \mathscr{D}$ lives in $\mathscr{V}$, not in the enriched functor category $[\mathscr{C}, \mathscr{D}]$ itself).

Matteo Capucci (he/him) (Dec 16 2020 at 11:25):

You're right, let me fix it

Fawzi Hreiki (Dec 16 2020 at 12:45):

Regarding composition, you can check out chapter 2 of http://www.tac.mta.ca/tac/reprints/articles/10/tr10.pdf

fosco (Dec 21 2020 at 09:20):

Matteo Capucci said:

Now to define composition $\mathbf C, \mathbf D \otimes \mathbf C, \mathbf D \to \mathbf C, \mathbf D$ one should define a map

$$\left(\int_c [Fc, Gc] \right)\otimes \left( \int_{c'} [Gc', Hc'] \right) \longrightarrow \int_{c''} [Fc'', Hc'']$$

Any ideas on how to do it?

It's easy if you think about it: at any given object $x$ you can fall from $\left(\int_c [Fc, Gc] \right)\otimes \left( \int_{c'} [Gc', Hc'] \right)$ into $[Fx, Gx]\otimes [Gx, Hx]$, and this family of maps $\alpha_x$ evidently is a wedge for $[F,H]$. Now, by the universal property of the end...

Matteo Capucci (he/him) (Dec 21 2020 at 18:15):

Indeed, that's what I realized when I followed @Fawzi Hreiki's reference, which is also something I should have looked up before posting my question :joy:

Matteo Capucci (he/him) (Dec 21 2020 at 18:16):

@fosco I got so used to think about ends 'synthetically' (i.e. Use only their computational properties) that sometimes I forgot/I refuse to use their universal property!

fosco (Dec 21 2020 at 19:37):

@Matteo Capucci All is forgiven, don't worry :grinning:

Fawzi Hreiki (Dec 21 2020 at 22:39):

@Matteo Capucci wouldn’t ‘synthetically’ mean with their universal property and ‘analytically’ mean computationally?

Dan Doel (Dec 21 2020 at 22:46):

Synthetically should almost certainly include the universal property. Presumably what was meant was various equivalences that probably ultimately can stem from the universal property.

Matteo Capucci (he/him) (Dec 22 2020 at 07:38):

The meaning of 'synthetic' and 'analytic' is debatable to a certain extent, though they usually mean 'axiomatically construed' vs 'assembled from pre-existing things'.
The universal property is a construction from pre-existing things, imo. Compare this with 'cartesian products' vs 'monoidal products'.
I do agree that universal properties themselves can be seen as synthetic version of products, but here we are working a level higher than that.

Nathanael Arkor (Dec 22 2020 at 12:12):

The universal property is the axiomatic definition (essentially by definition). You should be able to derive all the properties of a construction from the universal property, not the other way around.

Matteo Capucci (he/him) (Dec 22 2020 at 12:14):

Yeah, I'm talking one level higher than that. I think I use 'synthetic' as 'formal' in 'formal category theory'