Category of chain complexes

The category of chain complexes $𝖢 𝗁 𝖠$ consists of chain complexes $(𝐶 •, 𝜕 •)$ in $𝖠$ as objects and chain maps $𝑓 • : 𝐶 • \to 𝐶' •$ as morphisms, with composition given by $(𝑔 𝑓) 𝑘 = 𝑔 𝑘 𝑓 𝑘$ . #m/def/homology For notational convenience, we will often use $𝐶$ to refer to $(𝐶 •, 𝜕 •)$ , and $𝑓$ to refer to $𝑓 •$ . Furthermore, for a ring $𝑅$ we write $𝖢 𝗁 𝑅 = 𝖢 𝗁 𝑅 𝖬 𝗈 𝖽$ .

Limits and colimits

Both Initial and terminal objects are the trivial complexes.
The coproduct is given by the sum of constituent modules

Proof of (co)limits

Consider the trivial chain complex $(0, 0)$ of trivial modules with trivial homomorphisms between them. Clearly for any chain complex $(𝐶 •, 𝜕 •) \in 𝖢 𝗁 𝑅$ the following diagram commutes

Moreover no other vertical morphisms are definable, let alone commuting. Therefore $(0, 0)$ is the initial and terminal object of $𝖢 𝗁 𝑅$ .

Let $(𝑋 1 •, 𝜕 1 •)$ and $(𝑋 2 •, 𝜕 2 •)$ be chain complexes. The sequence $(𝑋 •, 𝜕 •) = (𝑋 1 • \oplus 𝑋 2 •, 𝜕 1 • \oplus 𝜕 2 •)$ is a well defined chain complex, since

𝜕 𝑘 𝜕 𝑘 + 1 = (𝜕 1 𝑘 \oplus 𝜕 2 𝑘) (𝜕 1 𝑘 + 1 \oplus 𝜕 2 𝑘 + 1) = 𝜕 1 𝑘 𝜕 1 𝑘 + 1 \oplus 𝜕 2 𝑘 𝜕 2 𝑘 + 1 = 0

The following diagram commutes with unique vertical morphisms due to the coproduct in $𝖬 𝗈 𝖽 𝑅$ .

Hence $(𝑋 •, 𝜕 •)$ is the coproduct of $(𝑋 1 •, 𝜕 1 •)$ and $(𝑋 2 •, 𝜕 2 •)$ .

Homology functor

$𝐻 𝑘 : 𝖢 𝗁 𝐾 \to 𝖬 𝗈 𝖽 𝐾$ becomes a functor for each $𝑘 \in 𝐾$ via induced homomorphisms (see chain map), where for $𝑓 : 𝐶' \to 𝐶$ we have

𝐻 𝑘 𝑓 : 𝐻 𝑘 𝐶' \to 𝐻 𝑘 𝐶 [𝑏'] \mapsto [𝑓 𝑘 (𝑏')]

This functor preserves initial and terminal objects in a trivial fashion, as well as coproducts. #to/prove

Proof of functor

Consider the identity chain map $i d 𝐶$ . Then $i d 𝐻 𝑘 (𝐶 •, 𝜕 •)$ already has the property that that $i d 𝐻 𝑘 𝐶 [𝑏] = [i d 𝐶 𝑘 (𝑏)] = [𝑏]$ , hence $i d 𝐻 𝑘 𝐶 = 𝐻 𝑘 i d 𝐶$ . Now consider chain complex $(𝐶 •, 𝜕 •)$ , $(𝐶' •, 𝜕' •)$ , and $(𝐶 ″ •, 𝜕 ″ •)$ with chain maps $𝑓 : 𝐶 ″ \to 𝐶'$ and $𝑔 : 𝐶' \to 𝐶$ . Then

(𝐻 𝑘 𝑔) (𝐻 𝑘 𝑓) [𝑏 ″] = 𝐻 𝑘 𝑔 [𝑓 𝑘 (𝑏 ″)] = [𝑔 𝑘 𝑓 𝑘 (𝑏 ″)] = [(𝑔 𝑓) 𝑘 (𝑏 ″)] = 𝐻 𝑘 (𝑔 𝑓) (𝑏 ″)

as required.

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