Suppose we have a complex without a homological grading (i.e. an abelian group). We can speak meaningfully about there being a differential $d$ on this group such that $d^2=0$.
Is this creature then still technically called a chain complex, or is it commonly known by some other name?
For such ungraded complexes, we can define a chain map as a map $f:A\to B$ that commutes with $d$, where $A,B$ are abelian groups. We can also define a homotopy to be a map $h:A\to B$ such that $f-g=hd+dh$.
Do we get that the familiar homological properties still hold with these creatures?
Eg: if $f\sim g$, and $p \sim q$, then $fp \sim gq$; being homotopic/homotopy equivalent is an equivalence relation; composing homotopy equivalences yields a homotopy equivalence.
It seems like they must hold as one should be able to recast the ungraded complex $A$ to get an (actual) homologically-graded chain complex with a homological grading given by $…\to A\to A \to A…$, and we can move back and forth between the two without losing information.
Further, do these properties continue to hold for ungraded complexes, when no restriction is placed on the value of $d^2$. Again, this last thing holds for homologically-graded chain complexes, but I’m unsure about it holding for ungraded complexes.