Bullet notation I'm just trying to make acquaintance with homological algebra. I see there the notation $(A_\bullet,b_\bullet)$ as a short notation for $(\dots,A_{-1},A_0,A_1,\dots,\dots,b_{-1},b_0,b_1,\dots)$. Seemingly $A_\bullet$ stands for $\dots,A_{-1},A_0,A_1,\dots$ and $b_\bullet$ stands for $\dots,b_{-1},b_0,b_1,\dots$, so it seems, that $_\bullet$ represents a sequence. However, I've seen this notation for sequences exclusively in homological algebra, for other sequences not. What is the general rule for using $\bullet$?

It's just a notation. It means "you can plug in an integer here and get something". It's also often used for simplicial sets, for example: a simplicial set $X_\bullet$ is a sequence of sets $(X_0, X_1, X_2, \dots)$ and some maps between them satisfying some identities. Or for example a cosimplicial set $Y^\bullet$ is also a sequence of spaces $(Y^0, Y^1, \dots)$ (plus maps).
For chain complexes one also often finds the notation $(A_*, d_*)$, which basically means exactly the same thing. Similarly for cochain complexes you often find $(B^*, d^*)$ (and generally the differential increase the degree then: $d^n : B^n \to B^{n+1}$ instead of $d_n : A_n \to A_{n-1}$). At the end of the day it's just a matter of preference, and as long as it's clear what you mean from the context it should be okay.
