van Kampen statement about book

[mikeshulman]

What is "Pi_1"?

On Mon, Mar 25, 2013 at 4:29 PM, Daniel R. Grayson <[email protected]

wrote:

The statement of van Kampen will be unsatisfying to mathematicians,
because it involves the "code" fibration, whose definition is complicated.
Example 7.5.8 attempts to rectify this in the special case where A=1 by
relating it to the free product of groups. But what about rephrasing
vanKampen so the statement is that Pi_1 transforms a pushout of spaces into
a pushout of groupoids, and then specialize that to the case where A is
connected by discussing free product with amalgamation? A lemma to prove
would then be that the codes fibration is really giving the pushout of
groupoids, represented by words.

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[DanGrayson]

It's the notation for the fundamental groupoid, the 0-truncation of the identity type.

I see now that the free product with amalgamation is mentioned in the next section, where the "naive" theorem is replaced by one with "base points". But it mystifies me what base points have to do with it, or why we have to wait until the next section. Well, maybe I'm not so mystified -- it could be said that the introduction of base points makes the relation with diagrams of fundamental groups easy.

[mikeshulman]

It would be tricky to phrase anything as a pushout of groupoids, since at
this point in the book we have not yet defined groupoids.

On Mon, Mar 25, 2013 at 4:38 PM, Daniel R. Grayson <[email protected]

wrote:

It's the notation for the fundamental groupoid, the 0-truncation of the
identity type.

I see now that the free product with amalgamation is mentioned in the next
section, where the "naive" theorem is replaced by one with "base points".
But it mystifies me what base points have to do with it, or why we have to
wait until the next section.

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[mikeshulman]

We have to wait until the next section because it's in the next section
that we deal with base points. And we need a set of base points if we want
to express the result purely in terms of set-level data, since otherwise
the space A (including all its higher homotopy) will enter into the
construction.

I'll think about how to explain this better.

On Mon, Mar 25, 2013 at 4:38 PM, Daniel R. Grayson <[email protected]

wrote:

It's the notation for the fundamental groupoid, the 0-truncation of the
identity type.

I see now that the free product with amalgamation is mentioned in the next
section, where the "naive" theorem is replaced by one with "base points".
But it mystifies me what base points have to do with it, or why we have to
wait until the next section.

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[DanGrayson]

Okay. It's not clear to me what "set-level" data means, by the way.

[mikeshulman]

It just means "in terms of sets".

On Mon, Mar 25, 2013 at 4:48 PM, Daniel R. Grayson <[email protected]

wrote:

Okay. It's not clear to me what "set-level" data means, by the way.

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[DanGrayson]

I see your latest changes in 16b4283 and 06659f7 but I still wonder. Consider this sentence:

Specifically, note that even for fixed elements $b:B$ and $c:C$, the definition of the set $\code(i(b),j(c))$ involves arbitrary elements of the type $A$, which might contain arbitrary higher homotopy.

and compare it to the theorem:

\Pi_1P(u,v) \simeq \code(u,v)

The left hand side is a set by definition, so the right hand side is a set, too. Because of that, the phrase "might contain arbitrary higher homotopy" fails to impart a sense of urgency to me.

[mikeshulman]

Yes, as I said, I'm still thinking about the best way to explain this.

On Mon, Mar 25, 2013 at 5:12 PM, Daniel R. Grayson <[email protected]

wrote:

I see your latest changes in 16b428316b428328d54250a9b3d7a329736a2eb06ebed57and
06659f706659f7d2ffccfc0bf7c3955c138e8e0f15a3180but I still wonder. Consider this sentence:

Specifically, note that even for fixed elements $b:B$ and $c:C$, the
definition of the set $\code(i(b),j(c))$ involves arbitrary elements of the
type $A$, which might contain arbitrary higher homotopy.

and compare it to the theorem:

\Pi_1P(u,v) \simeq \code(u,v)

The left hand side is a set by definition, so the right hand side is a
set, too. Because of that, the phrase "might contain arbitrary higher
homotopy" fails to impart a sense of urgency to me.

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[mikeshulman]

See the changes in d7eb4bb, and reopen if you're not satisfied. I don't think I want to try to write out the definition of a pushout of groupoids.

[DanGrayson]

Looks good. See 9dd2034 for some minor
modifications to what you wrote.

On Wed, Mar 27, 2013 at 3:17 PM, Mike Shulman [email protected]:

See the changes in d7eb4bbhttps://github.com/HoTT/book/commit/d7eb4bb3d07,
and reopen if you're not satisfied. I don't think I want to try to write
out the definition of a pushout of groupoids.

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