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added illustrating diagram to transfinite composition
I also renamed the resulting composite morphism into . Hope I did this consistently.
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Another instance of transfinite composition was related to me by Jim Stasheff; it's in Milnor's proof that fiber bundles have the homotopy lifting property. At some point I'll see if I can record it in the Lab.
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<p>That would great. I am not actually aware of how that proof works.</p>
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I will now try to jot down some scattered memories of the conversation with Stasheff from ages ago at Milnor slide trick.
I have tried to make the entry transfinite composition more readable:
I included a quick self-contained definition of ordinals and limit ordinals (for the case with LEM). For if any reader really needs to be reminded, then presently clicking on ordinal sends him or her off onto a long, long chase, until the definition is fully assembled. (Eventually all these entries on XY-orders could be streamlined for public consumption, but I won’t do this right now.)
I changed the name of the transfinite composition diagram from $F$ to $X_\bullet$. That allowed to remove various clauses on how notation is to be matched and simply have the transfinite composite be labeled $X_0 \to X_\alpha$.
I reordered the two pieces of the definition. Now it first states what the diagram is, then it says how the transfinite composite itself is the given by the colimit. (Previously it was a highly nested sentence starting with “is the colimit” and only then beginning to say of which diagram subject to which conditions).
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