Timeline for Is the logic of the ($\infty$-?)topos of simplicial sets "contradictory up to homotopy"?
Current License: CC BY-SA 4.0
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| Oct 14 at 8:22 | comment | added | Peter LeFanu Lumsdaine | (2) the square $(\mathrm{id},m) : (\mathrm{id} : X \to X) \to (m : X \to A)$ is a homotopy pullback; (3) every homotopy-fiber of $m$ is either empty or contractible; (3') (when $m$ is a fibration) every fiber of $m$ is either empty or contractible; (4) over each connected component of $A$, $m$ is either empty or an equivalence. // The main thing throughout is to be careful how you mix strict and ∞-categorical notions. A priori, they can be totally independent; some good interactions hold, especially with fibrations, but don’t expect an interaction to work until it’s proven. | |
| Oct 14 at 8:22 | comment | added | Peter LeFanu Lumsdaine | Yes. I’d phrase your refinement as “any equality $mx=my$ can be lifted to an equality $x=y$,” and also this lifting must be continuous/functorial in $x$, $y$, and $m$. It’s a good intuition, but often technically awkward (it’s simple in the internal language, but unwieldy when working diagrammatically); fortunately there are many equivalent conditions for $m : X \to A$ to be ∞-mono, which are nice in different contexts: (1) the (∞-)diagonal $\Delta : X \to X \times^\infty_A X$ is an equivalence; (1') (when $m$ is a fibration) the 1-categorical diagonal is an equivalence; [cont’d] | |
| Oct 14 at 6:06 | history | edited | მამუკა ჯიბლაძე | CC BY-SA 4.0 |
added 107 characters in body
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| Oct 14 at 5:59 | history | answered | მამუკა ჯიბლაძე | CC BY-SA 4.0 |