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Stream: learning: questions

Topic: Globular versus coglobular vector spaces

John Baez (May 10 2025 at 16:12):

I'm puzzled by something, so this question is going to be rather vague - sort of like "what's going on here?"

But the phenomenon I'm asking about is precise enough.

First a summary. There's a standard way to get a chain complex from a globular vector space, but I seem to be getting one from something else, which I will temporarily called a 'coglobular vector space'. I don't like this, so I'd like to convert it to a globular vector space. I could do this by taking duals, but I don't really want to, because I'm afraid that will mess up other things.

Next let me give a detailed explanation.

John Baez (May 10 2025 at 16:15):

Suppose I have vector spaces $V_2, V_1, V_0$ and linear maps

$s, t: V_2 \to V_1, \qquad s,t : V_1 \to V_0$

This setup is an example of a globular vector space if

$s s = s t, \qquad t s = t t$

which is the usual identities obeyed by source and target maps in a 2-category. I'll call these the globular identities.

If I then define

$d = s - t$

then I get a chain complex because

$d^2 = (s - t)(s - t) = s^2 - st - ts + t^2 = 0$

where in the last step we use the globular identities.

John Baez (May 10 2025 at 16:27):

But here's another way to get a chain complex! Suppose I have a vector space $X$ with two subspaces $V, W \subseteq X$. I then get a subspace $V \cap W$, and two obvious inclusions

$i_V, i_W : V \cap W \to V \oplus W$

There are also two obvious quotient maps

$p_V, p_W : V \oplus W \to X$

The first maps $(v,w) \in V \oplus W$ to $v \in V$ regarded as an element of $X$. The second maps it to $w \in W$ regarded as an element of $X$.

I can then define a chain complex

$V \cap W \xrightarrow{d} V \oplus W \xrightarrow{d} X$

where

• $d: V \cap W \to V \oplus W$ is given by $i_V + i_W$
• $d: V \oplus W \to X$ is given by $p_V - p_W$

Why is $d^2 = 0$ this time? It's because we have

$p_V i_V = p_W i_V, \qquad p_V i_W = p_W i_W$

(Please check this, I find it so unnerving that I feel it could be wrong though it seems obviously correct!)

Using these identities we get

$d^2 = (p_V - p_W)(i_V + i_W)$
$= p_V i_V - p_W i_V + p_V i_W - p_W i_W = 0$

John Baez (May 10 2025 at 16:32):

Now, this second situation smells a lot like the first one. But these identities

$p_V i_V = p_W i_V, \qquad p_V i_W = p_W i_W$.

are not the globular identities, as far as I can tell. They're a special case of the co-globular identities, which are the globular identities in reverse:

$s s = t s, \qquad s t = t t$.

So: what's going on here?

David Corfield (May 10 2025 at 16:48):

You have $d$ is given by $i_V + i_W$ at one point, and then later $d^2 = (p_V - p_W)(i_V - i_W)$. What's with the sign change?

John Baez (May 10 2025 at 16:50):

Error, probably just a typo. Let me try to fix it.

John Baez (May 10 2025 at 16:51):

Thanks, I think it's okay now!

By the way, the correct formula is supposed to be massively reminiscent of a certain bit of the Mayer-Vietoris exact sequence, and this sort of calculation is why the composite of two maps in that bit equals zero. So it's not something I cleverly made up.

Mike Shulman (May 10 2025 at 18:55):

Shouldn't it be $p_V i_V = p_W i_W$ and $p_V i_W = 0 = p_W i_V$?

Adittya Chaudhuri (May 10 2025 at 18:58):

John Baez said:

• $d: V \cap W \to V \oplus W$ is given by $i_V + i_W$
• $d: V \oplus W \to X$ is given by $p_V - p_W$

Why is $d^2 = 0$ this time?

I think by your definition,

• $d: V \cap W \to V \oplus W$, $x \mapsto (x,0) + (0,x)=(x,x)$, and

• $d: V \oplus W \to X$, $(v,w) \mapsto v- w$ .

Hence, $d^{2}(x)=x - x=0$. So, it is a chain complex.

However, if I am not making any mistake, then, for $x \in V \cap W$, $p_V i_{W}(x)=p_{v}(0,x)=0$ but, $p_{W} \circ i_{w}(x)=p_{w}(0,x)=x$. Hence, $p_{V} i_{W}\neq p_{W} i_{W}$, unless $V \cap W=0$. I think the same is true for the other identity.

David Egolf (May 10 2025 at 19:17):

I had assumed that $i_W:V \cap W \to V \oplus W$ acted by $x \mapsto (x,x)$. I might be wrong about that though! [EDIT: As clarified below, this is not correct.]

John Baez (May 10 2025 at 19:33):

I wanted $i_W(x) = (0,x)$ - the symbol $i_W$ was intended as short for "inclusion of $V \cap W$ into the $W$ summand". Similarly $i_V(x) = (x,0)$.

I just had a beer so I won't attempt to deal with Mike and Adittya's corrections now - if I didn't get something right before, there's no way I'll get it right now! But thanks, folks, for helping.

John Baez (May 11 2025 at 11:56):

Mike Shulman said:

Shouldn't it be $p_V i_V = p_W i_W$ and $p_V i_W = 0 = p_W i_V$?

Yes, you're right! These are the same relations that hold for the inclusions and projections when you have the biproduct of $V$ and $W$.

Somehow I convinced myself that very different relations hold when we have subspaces $V , W \subseteq X$ and

$i_V: V \cap W \to V \oplus W , \qquad i_V(x) = (x,0)$
$i_W: V \cap W \to V \oplus W, \qquad i_W(x) = (0,x)$

$p_V: V \oplus W \to X , \qquad p_V(v,w) = v$
$p_W: V \oplus W \to X, \qquad p_W(v,w) = w$

but in fact the same relations hold.

Mike Shulman (May 11 2025 at 16:20):

You mentioned the Mayer-Vietoris sequence, so you probably know that this construction is pretty standard, but maybe it would be helpful to mention that more generally it's also a standard thing in triangulated category theory (and stable $\infty$-category theory), to make a distinguished triangle (= fiber sequence = cofiber sequence) from a homotopy pullback square (= homotopy pushout square)

John Baez (May 11 2025 at 17:02):

Thanks. I've always been scared of triangulated category (though slightly less so when the management informed me it was secretly stable $\infty$-category theory). So I didn't know this.

I was just hoping that all the chain complexes I need to work with in my project arise from globular objects via the $d = s - t$ construction. So far it ain't looking good.