naive van Kampen

[DanGrayson]

In homotopy.tex we read a remark about the naive van Kampen's "code":

"code(u, v) is required to be a set, even though it involves the untruncated type A in its definition. This is why the version of van Kampen we are currently describing is “naive”. We will explain at the end of this subsection why this is an undesirable feature, and remedy it in the next subsection."

But in the next section, where the improved "code" is defined, we read

"code(ib, ib′ ) is now a set-quotient of the type of sequences..."

So the second "code" is also forced to be a set. Why doesn't it suffer from the same defect?

[mikeshulman]

Because, unlike the first one, it doesn't involve an untruncated type in its definition.

[mikeshulman]

I've removed these sentences; they were clearly confusing and didn't really convey any important information.

[DanGrayson]

Do you mean, if S were a set, then the definition of "code" would not
involve an untruncated type? (I'm thinking of the sentences "it turns out
to be unnecessary for our proof to assume that the “set of base-points” is
a set — it might just as well be an arbitrary type" and "If S ≡ A and k is
the identity function, then we will recover the naive van Kampen theorem")

Do you also mean, that if S were a set, then where we define "code(ib, ib′
) is now a set-quotient of the type of sequences..,", we could define it as
an inductive type and it would turn out to be a set? Such a remark might
be important in justifying the examples involving colimits of fundamental
groups.

On Wed, Mar 27, 2013 at 8:10 PM, Mike Shulman notifications@github.comwrote:

Because, unlike the first one, it doesn't involve an untruncated type in
its definition.

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[mikeshulman]

Yes and no, respectively. The sentences that I just removed were
written before we realized that the improved version works just as
well even if S is not a set --- although it's not clear to me at the
moment whether it has applications when S is neither a set nor A
itself.

What is it that you think needs more "justification"?

On Wed, Mar 27, 2013 at 9:00 PM, Daniel R. Grayson
notifications@github.com wrote:

Do you mean, if S were a set, then the definition of "code" would not
involve an untruncated type? (I'm thinking of the sentences "it turns out
to be unnecessary for our proof to assume that the “set of base-points” is
a set — it might just as well be an arbitrary type" and "If S ≡ A and k is
the identity function, then we will recover the naive van Kampen theorem")

Do you also mean, that if S were a set, then where we define "code(ib, ib′
) is now a set-quotient of the type of sequences..,", we could define it as
an inductive type and it would turn out to be a set? Such a remark might
be important in justifying the examples involving colimits of fundamental
groups.

On Wed, Mar 27, 2013 at 8:10 PM, Mike Shulman notifications@github.comwrote:

Because, unlike the first one, it doesn't involve an untruncated type in
its definition.

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[DanGrayson]

Okay. Your latest version is clearer.

On further thought, I'm no longer worried about justifying those examples applying the theorem.

But now I'm wondering about this sentence: "Indeed, the conclusion of Theorem 7.5.4 says nothing at all about π_1(A); the paths in A are “built into the quotienting” in a type-theoretic way that makes it hard to extract explicit information." For some reason I find it hard to understand. I wonder if a more prosaic and satisfactory explanation for the advantage of reformulating van Kampen is that fundamental groups refer to base points, but the naive van Kampen theorem doesn't.

[mikeshulman]

Is that going to be convincing if you are not a hardened classical
algebraic topologist who already cares about basepoints?

On Wed, Mar 27, 2013 at 9:40 PM, Daniel R. Grayson <notifications@github.com

wrote:

Okay. Your latest version is clearer.

On further thought, I'm no longer worried about justifying those examples
applying the theorem.

But now I'm wondering about this sentence: "Indeed, the conclusion of
Theorem 7.5.4 says nothing at all about π_1(A); the paths in A are “built
into the quotienting” in a type-theoretic way that makes it hard to extract
explicit information." For some reason I find it hard to understand. I
wonder if a more prosaic and satisfactory explanation for the advantage of
reformulating van Kampen is that fundamental groups refer to base points,
but the naive van Kampen theorem doesn't.

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[DanGrayson]

Sure. The only way to get a group out of a groupoid is to pick a base
point. And everyone should be interested in groups already.

On Wed, Mar 27, 2013 at 10:16 PM, Mike Shulman notifications@github.comwrote:

Is that going to be convincing if you are not a hardened classical
algebraic topologist who already cares about basepoints?

On Wed, Mar 27, 2013 at 9:40 PM, Daniel R. Grayson <
notifications@github.com

wrote:

Okay. Your latest version is clearer.

On further thought, I'm no longer worried about justifying those
examples
applying the theorem.

But now I'm wondering about this sentence: "Indeed, the conclusion of
Theorem 7.5.4 says nothing at all about π_1(A); the paths in A are
“built
into the quotienting” in a type-theoretic way that makes it hard to
extract
explicit information." For some reason I find it hard to understand. I
wonder if a more prosaic and satisfactory explanation for the advantage
of
reformulating van Kampen is that fundamental groups refer to base
points,
but the naive van Kampen theorem doesn't.

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