Thirteen Ways of Looking at a Topological Group

I.
Among twenty topological spaces
The only one that composed
Was the topological group.

II.
I took three loops
They composed
Up to homotopy associativity, formed a group.

III.
Stasheff's associahedra commute, with inverses.
It was enough to look like a group.

IV.
A space and a simplicial set
Are one.
A space and a simplicial set as a group
Are one.

V.
I do not know which to prefer
The beauty of a topological group
Or the beauty of functors from T_G
Preserving products
On spaces.

VI.
Badzioch filled the Annals
With homotopy algebras.
The shadow of the strict ones
Only up to homotopy.
The structure
Up to weak equivalence
Was strong enough.

VII.
Oh simplicial homotopy theorists
Why do you despair?
Do you not see how simplicially
Strict algebras are
Defined just as well?

VIII.
I know up to homotopy
We can again rigidify
But I know, too,
That simplicial sets work well
For what we already know.

IX.
When some of our morphisms flew out of sight
It left the operation
Still held by I\Delta^{op}.

X.
At the sight of Segal groups,
Defined in terms of spaces,
Simplicial sets cry out,
To protest they work well too.

XI.
Bousfield changed projections,
In the \Delta^{op}.
To gain his compositions,
Between the spaces
He ended up with inverses,
Hence groups.

XII.
Groups come from spaces.
Hence Bousfield simplicial spaces.

XIII.
The theories worked so well
Take spheres
And take the maps between
Badzioch, Chung, and Voronov,
Say you'll still get a group.

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