Section 22.5 (09JM): The homotopy category

22.5 The homotopy category

Our homotopies take into account the $A$ -module structure and the grading, but not the differential (of course).

Definition 22.5.1. Let $(A, \text{d})$ be a differential graded algebra. Let $f, g : M \to N$ be homomorphisms of differential graded $A$ -modules. A homotopy between $f$ and $g$ is an $A$ -module map $h : M \to N$ such that

$h(M^ n) \subset N^{n - 1}$ for all $n$ , and
$f(x) - g(x) = \text{d}_ N(h(x)) + h(\text{d}_ M(x))$ for all $x \in M$ .

If a homotopy exists, then we say $f$ and $g$ are homotopic.

Thus $h$ is compatible with the $A$ -module structure and the grading but not with the differential. If $f = g$ and $h$ is a homotopy as in the definition, then $h$ defines a morphism $h : M \to N[-1]$ in $\text{Mod}_{(A, \text{d})}$ .

Lemma 22.5.2. Let $(A, \text{d})$ be a differential graded algebra. Let $f, g : L \to M$ be homomorphisms of differential graded $A$ -modules. Suppose given further homomorphisms $a : K \to L$ , and $c : M \to N$ . If $h : L \to M$ is an $A$ -module map which defines a homotopy between $f$ and $g$ , then $c \circ h \circ a$ defines a homotopy between $c \circ f \circ a$ and $c \circ g \circ a$ .

Proof. Immediate from Homology, Lemma 12.13.7. $\square$

This lemma allows us to define the homotopy category as follows.

Definition 22.5.3. Let $(A, \text{d})$ be a differential graded algebra. The homotopy category, denoted $K(\text{Mod}_{(A, \text{d})})$ , is the category whose objects are the objects of $\text{Mod}_{(A, \text{d})}$ and whose morphisms are homotopy classes of homomorphisms of differential graded $A$ -modules.

The notation $K(\text{Mod}_{(A, \text{d})})$ is not standard but at least is consistent with the use of $K(-)$ in other places of the Stacks project.

Lemma 22.5.4. Let $(A, \text{d})$ be a differential graded algebra. The homotopy category $K(\text{Mod}_{(A, \text{d})})$ has direct sums and products.

Proof. Omitted. Hint: Just use the direct sums and products as in Lemma 22.4.2. This works because we saw that these functors commute with the forgetful functor to the category of graded $A$ -modules and because $\prod$ is an exact functor on the category of families of abelian groups. $\square$

Comments (3)

Comment #288 by arp on September 06, 2013 at 07:56

Typo: In the paragraph following Definition 22.5.1 (TAG 09JN), I think it should say that $h$ defines a morphism $h : M \to N[-1]$ (i.e. a $-$ sign is missing). By the way, it seems a bit strange to say a homotopy is "compatible with the grading," but I know what you are saying.

Comment #1057 by Kian Abolfazlian on October 03, 2014 at 16:47

In Lemma 22.5.2, the morphism h must be $h:L \to M[-1]$ .

Comment #1062 by Johan on October 05, 2014 at 16:24

Thanks for pointing this out. Fixed in this commit. In Definition 22.5.1 we decided to think of a homotopy $h : L \to M$ as an $A$ -module map homogeneous of degree $-1$ . Of course, this also means it is a graded $A$ -module map $h : L \to M[-1]$ . If in Lemma 22.5.2 we want to write $h : L \to M[-1]$ we should also change the wording in the definition.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$ ). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

The Stacks project

22.5 The homotopy category

Comments (3)

Post a comment