nLab Postnikov system in a triangulated category

Idea

A triangulated category, or at least an enhanced triangulated category, is a model for a stable (infinity,1)-category. The archetypical example of these is the stable (infinity,1)-category of spectra of topological spaces. For topological spaces there is the classical notion of Postnikov system.

A Postnikov system in a general triangulated category is an abstraction of that topological construction to the general case.

Definition

Given a triangulated category $D$ a finite complex over $D$ is any sequence of objects and morphisms in $D$

$$X^\bullet = \{ X^0 \stackrel{d^0}\to X^1 \stackrel{d^1}\to X^2 \stackrel{d^2}\to\ldots \stackrel{d^n}\to X^n\}$$

in which $d_k\circ d_{k-1} = 0$.

A right Postnikov system subordinated to $X^\bullet$ is given by the following data

• a sequence of objects and morphisms

$$Y^0\stackrel{f_1}\leftarrow Y^1\stackrel{f_2}\leftarrow Y^2 \stackrel{f_3}\leftarrow\ldots\stackrel{f_{n-1}}\leftarrow Y^{n-1}\stackrel{f_n}\leftarrow Y^n= X^n[1]$$

• morphisms $j_{k-1}: Y^k\to X^k[n-k+1]$, $k = 0,\ldots,n-1$ with $j_{n-1} = id : Y^n = X^n[1]\to X^n[1]$

• morphisms $i_k : X^k[n-k]\to Y^{k+1}$, $k = 0,\ldots,n-1$

such that

$$j_k\circ i_k = d^k[n-k]: X^k[n-k]\to X^{k+1}[n-k], \,\,\,\,\,\,\,for\,\,\,all\,\,\,\,\,\,\,\, 0\lt k \lt n-1$$

(in particular, $i_{n-1} = d^{n-1}$) and for all $0\leq k\leq n$ the triangles

$$X^k[n-k]\stackrel{i_k}\to Y^{k+1}\stackrel{f_k}\to Y^k \stackrel{j_{k-1}}\to X^k[n-k+1]$$

are distinguished.

A right convolution of complex $X^\bullet$ is any object in $D$ of the form $Y^0[n-1]$ where $Y^0$ is the target end of the sequence as above in some right Postnikov system subordinated to $X^\bullet$.

There is also a left-hand version of Postnikov systems and convolutons; however the classes of left and right convolutions of complex $X^\bullet$ coincide; denote that class $Tot(X^\bullet)$. This class in general may be empty, but it may also contain many nonisomorphic objects. D. Orlov (see below) has proved some criteria for existence and uniquenss of (classes of isomorphic) objects in $Tot(X^\bullet)$.

If $D = K^b(A)$ is the triangulated category of bounded complexes in an additive category $A$ then many examples can be obtained via twisted complexes in that setup, as introduced by Kapranov. Postnikov systems are also used to define generalizations of stability conditions in triangulated setup.

The standard reference (beware typoi!) is chapter 4 (exercise section after 4.2) of

• S. I. Gel’fand, Yu. I. Manin, Methods of homological algebra, Russian Наука 1988, English Springer (2 editions).

Other references

• D. Orlov, Equivalences of derived categories and K3 surfaces, (alg-geom/9606006)

• A. Beilinson, J. Bernstein, P. Deligne, Faisceaux pervers, Astérisque 100 (1982). MR 86g:32015

• G. Hein, D. Ploog Postnikov stability for complexes, Postnikov-stability vs semistability of sheaves

• M. Kapranov, On the derived categories of coherent sheaves on some homogeneous spaces, Inventiones Mathematicae, vol. 92, n. 3, 479–508, 1988 doi:10.1007/BF01393744

• A. L. Gorodentsev, S. A. Kuleshov, A. N. Rudakov, t-stabilities and t-structures on triangulated categories, Izv. Math. 68, 749–781, 2004 doi

• A. L. Gorodentsev, S. A. Kuleshov, On finest and modular t-stabilities, MPIM Bonn preprint