I have a question about following statement
from ncatlab about derived categories: (https://ncatlab.org/nlab/show/derived+category#idea)
Often in the literature, the term
derived category refers to the
homotopy category, viewed only as a
triangulated category. The loss of
information can often be problematic,
but for many purposes is not important.
I not understand which part of structure
exactly get qualitatively lost if we
regard a certain homotopy theory
"only as a triangulated category"? If $D(A)$ (for certian Abelain category $A$) is our derived category, which homotopy theory $H$ is meant here? $K(A)$?
And is that a kind of application of certain
forgetful functor? But what
gets exactly lost here?
Recall how a derived category is
constructed: we start with an Abelian
category $A$, form the category
category of chain complexes $Com(A)$ with
terms in $A$. Next we can proceed in two
ways (see https://en.wikipedia.org/wiki/Derived_category#Definition):
-
Localize $Com(A)$ directly with respect
the quasi-isomorphisms (= morphisms
inducing isomorphisms between (co)homologies
of two chain complexes) -
Localize firstly $Com(A)$ with respect
homotopies in $Com(A)$ (call this new
homotopy category $K(A)$ ) and then localize
$K(A)$ as in 1 with respect quasi-isomorphisms.
We observe that this two construction
give the same category – call it the derived
category of $A$
$D(A)$ –
because homotopic maps are quasi-isomorphisms. Therefore $D(A)$ has intrinsical structure
of a homotopy category and I'm wondering
how it can be "forgotten" as suggested in
the statement above.
Back to my question: What is exactly in this context
the homotopy theory, which viewed only
as a triangulated category becomes our
derived category $D(A)$? The intermediate
category $K(A)$ from step 2?
It is known that there is a canonical way to endow
$D(R)$ with structure of a triangulated
category, and looking closely in the single steps of the construction one sees
that this construction exploits only the
homotopy structure of $D(R)$ – therefore already
$K(A)$ can be endowed with structure of a
triangulated category verbatim in the same way.
Is this $K(A)$ exactly the homotopy category
which is mentioned in the quoted text which
gives $D(R)$ if we regard it as triangulated
category "only". On the other hand which amount of structure of $K(A)$ gets
then really lost?
Best Answer
In general, if $(\mathcal C, W)$ is a category with weak equivalences, you can construct an $\infty$-category $L_W\mathcal C$ via simplicial localisation. In our case, the category with weak equivalences that we consider is $(\operatorname{Com}(\mathcal A), \mathrm{qiso.})$. The associated simplicial localisation is called the derived $\infty$-category of $\mathcal A$, and the homotopy category of this $\infty$-category is the ordinary derived category $\operatorname{D}(\mathcal A)$.