Category Theory
Zulip Server
Archive

You're reading the public-facing archive of the Category Theory Zulip server.
To join the server you need an invite. Anybody can get an invite by contacting Matteo Capucci at name dot surname at gmail dot com.
For all things related to this archive refer to the same person.

Stream: learning: questions

Topic: signs in the dg-nerve

Tim Hosgood (May 02 2021 at 16:01):

I've always taken the signs in the definition of the dg-nerve as in Lurie, i.e.

$$\partial f_I = \sum_{1\leq j\leq k-1} (-1)^j \big(f_{I\setminus\{i_j\}} - f_{\{i_j<\ldots<i_k\}}\circ f_{\{i_0<\ldots<i_j\}}\big).$$

However, Faonte disagrees, and gives

$$\partial f_I = \sum_{1\leq j\leq k-1} (-1)^{j-1} f_{I\setminus\{i_j\}} +(-1)^{k(j-1)+1} f_{\{i_j<\ldots<i_k\}}\circ f_{\{i_0<\ldots<i_j\}}.$$

If it were just a problem of a global sign, then I wouldn't mind so much, but there's a problem of different "internal" signs for values of $k\geq2$. Does anybody know which definition is "the" good one?

Tim Hosgood (May 05 2021 at 13:10):

to be honest, I'd even be mildly content with an abstract reason for why these two should be the same (i.e. why the initial parts of the derivations of these formulas are equivalent)