# nLab internal hom of chain complexes

### Context

#### Homological algebra

homological algebra

and

nonabelian homological algebra

diagram chasing

### Theorems

#### Monoidal categories

monoidal categories

## With traces

• trace

• traced monoidal category?

# Contents

## Idea

A natural internal hom of chain complexes that makes the category of chain complexes into a closed monoidal category.

## Definition

Let $R$ be a commutative ring and $𝒜=R$Mod the category of modules over $R$. Write ${\mathrm{Ch}}_{•}\left(𝒜\right)$ for the category of chain complexes of $R$-modules.

###### Definition

For $X,Y\in {\mathrm{Ch}}_{•}\left(𝒜\right)$ any two objects, define a chain complex $\left[X,Y\right]\in {\mathrm{Ch}}_{•}\left(𝒜\right)$ to have components

$\left[X,Y{\right]}_{n}:=\prod _{i\in ℤ}{\mathrm{Hom}}_{R\mathrm{Mod}}\left({X}_{i},{Y}_{i+n}\right)$[X,Y]_n := \prod_{i \in \mathbb{Z}} Hom_{R Mod}(X_i, Y_{i+n})

(the collection of degree-$n$ maps between the underlying graded modules) and whose differential is defined on homogeneously graded elements $f\in \left[X,Y{\right]}_{n}$ by

$df:={d}_{Y}\circ f-\left(-1{\right)}^{n}f\circ {d}_{X}\phantom{\rule{thinmathspace}{0ex}}.$d f := d_Y \circ f - (-1)^{n} f \circ d_X \,.

This defines a functor

$\left[-,-\right]:{\mathrm{Ch}}_{•}\left(𝒜{\right)}^{\mathrm{op}}×{\mathrm{Ch}}_{•}\left(𝒜\right)\to {\mathrm{Ch}}_{•}\left(𝒜\right)\phantom{\rule{thinmathspace}{0ex}}.$[-,-] : Ch_\bullet(\mathcal{A})^{op} \times Ch_\bullet(\mathcal{A}) \to Ch_\bullet(\mathcal{A}) \,.

## Properties

###### Proposition

The collection of cycles of the internal hom $\left[X,Y\right]$ in degree 0 coincides with the external hom functor

${Z}_{0}\left(\left[X,Y\right]\right)\simeq {\mathrm{Hom}}_{{\mathrm{Ch}}_{•}}\left(X,Y\right)\phantom{\rule{thinmathspace}{0ex}}.$Z_0([X,Y]) \simeq Hom_{Ch_\bullet}(X,Y) \,.

The chain homology of the internal hom $\left[X,Y\right]$ in degree 0 coincides with the homotopy classes of chain maps.

###### Proof

By Definition 1 the 0-cycles in $\left[X,Y\right]$ are collections of morphisms $\left\{{f}_{k}:{X}_{k}\to {Y}_{k}\right\}$ such that

${f}_{k+1}\circ {d}_{X}={d}_{Y}\circ {f}_{k}\phantom{\rule{thinmathspace}{0ex}}.$f_{k+1} \circ d_X = d_Y \circ f_k \,.

This is precisely the condition for $f$ to be a chain map.

Similarly, the boundaries in degree 0 are precisely the collections of morphisms of the form

${\lambda }_{k+1}\circ {d}_{X}+{d}_{Y}\circ {\lambda }_{k}$\lambda_{k+1} \circ d_X + d_Y \circ \lambda_k

for a collection of maps $\left\{{\lambda }_{k}:{X}_{k}\to {Y}_{k+1}\right\}$. This are precisely the null homotopies.

## References

A standard textbook account is

Revised on August 26, 2012 23:34:01 by Urs Schreiber (89.204.139.66)