dg category

Remarks

Example: the dg category $C_{dg}(k)$ has as objects the cochain complexes $X$ of $k$-modules, with $\Hom^n(X, Y) = \prod_{i} \Hom_k(X^i, Y^{i+n})$. For $f$ of degree $n$ the differential is $d(f) = d_Y \circ f - (-1)^n f \circ d_X$. One checks $d^2(f) = d_Y^2 \circ f - f \circ d_X^2 = 0$, because the two middle terms $\pm d_Y \circ f \circ d_X$ cancel, and the Leibniz rule follows from the same sign computation. The closed morphisms of degree $0$ are the chain maps, and the exact ones are the null-homotopic chain maps.

Non-example: over $k = \Q$, use the same graded maps but drop the sign, so $d(f) = d_Y \circ f + f \circ d_X$. Then $d^2(f) = 2\, d_Y \circ f \circ d_X$. Let $C$ be the complex with $C^0 = C^1 = \Q$ and differential the identity, and let $f \in \Hom^{-1}(C, C)$ be the identity map $C^1 \to C^0$. Then $d^2(f)$ is $2 \ne 0$ on $C^0$, so $\Hom(C, C)$ is not a complex.

A dg category with one object is the same as a differential graded algebra.