Post 8: Correction to Perverse Sheaves Part 3

05 Jan 2024

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You don’t need to read this post if you read Part 3 after January 5 2024.

In my last post, Perverse Sheaves Part 3, I mistakenly gave this as the definition of equivalence of roof diagrams:

• Two roof diagrams from $X$ to $Y$, $(f:A\to Y, s:A\to X)$ and $(g:A'\to Y,t:A'\to X)$, are equivalent if there are morphisms $a:A''\to A$ and $b:A''\to A’$ in $S$ such that $s\circ a = t\circ b$ and $f\circ a=g\circ b$.

However, it has now been updated to say:

• Two roof diagrams from $X$ to $Y$, $(f:A\to Y, s:A\to X)$ and $(g:A'\to Y,t:A'\to X)$, are equivalent if there are morphisms $a:A''\to A$ and $b:A''\to A’$ such that $s\circ a = t\circ b$ is in $S$ and $f\circ a=g\circ b$.

I’m not 100% certain that these aren’t equivalent, but just in case, I decided to correct it.

Thankfully, these conditions are equivalent when $S$ is the collection of quasi-isomorphisms in the homotopy category, which is why I made this mistake in the first place.