Chain and cochain complexes of animae

Construction 6.2.12 (Chain and cochain complexes). Let \(A \in \D (\Z )\) be a complex. Since \(\D (\Z )\) admits all small limits and colimits by Corollary 6.1.29, the universal property of \(\An \) from Corollary 1.8.10 implies that evaluation at the point induces equivalences \[ \ev _{\pt }\colon \Fun ^{\colim }(\An ,\D (\Z )) \iso \D (\Z ) \qquadtext {and} \ev _{\pt }\colon \Fun ^{\lim }(\An \catop ,\D (\Z )) \iso \D (\Z ). \] In particular, the object \(A \in \D (\Z )\) determines two functors \[ C_*(-;A)\colon \An \to \D (\Z ) \qquadtext {and} C^*(-;A) \colon \An \catop \to \D (\Z ), \] characterized uniquely by the properties that \(C_*(-;A)\) preserves colimits, \(C^*(-;A)\) preserves limits, and \(C_*(\pt ;A) \simeq A \simeq C^*(\pt ;A)\).

For a commutative ring \(R\), the same construction in \(\D (R)\) defines chains and cochains with coefficients in any \(A \in \D (R)\). Restriction of scalars along \(\Z \to R\) identifies their underlying complexes in \(\D (\Z )\) with the constructions above.

For a pointed anima \((X,x) \in \An _*\), we define the reduced chain complex and reduced cochain complex as \[ \widetilde {C}_*(X;A) := \cofib (A \simeq C_*(\pt ;A) \xrightarrow {x_*} C_*(X;A)) \] and \[ \widetilde {C}^*(X;A) := \fib (C^*(X;A) \xrightarrow {x^*} C^*(\pt ;A) \simeq A). \]