ScalingStacks

Suppose $𝒯$ is a triangulated category with set indexed coproducts and let $\mathrm{\Sigma }:𝒯\to 𝒯$ denote its suspension functor. Hoshino, Kato and Miyachi, in [11], show that a natural $t$-structure is induced on $𝒯$ by a suitably nice compact object of $𝒯$. In particular, they consider a compact object $S$ of $𝒯$ which satisfies the following two conditions:

$(1)$ ${\text{Hom}}_{𝒯}(S,{\mathrm{\Sigma }}^{i}S)=0\text{ for all }i>0$;

$(2)$ $\{{\mathrm{\Sigma }}^{i}S|i\in \mathbb{Z}\}$ is a generating set for $𝒯$.

If $S$ is a compact rigid object of $𝒯$ and $\{{\mathrm{\Sigma }}^{i}S|i\in \mathbb{Z}\}$ is a generating set for $𝒯$, then the two halves of the $t$-structure obtained in [11] are given by

This situation bears resemblance to the example of a chain differential graded algebra (DGA) $R$ in its derived category $𝒟(R)$, whose objects are the differential graded modules over $R$ (DG $R$-modules). Introductions to the theory of DGAs, their derived categories and DG modules can be found in [1], [6] and [10].

Recall that a DGA $R$ is called a chain DGA if $H^i(R)=0$ for all $i>0$. Moreover, given a DG $R$-module $M$ we have

Hence, the object $S$ considered in [11] is analogous to a chain DGA and the two halves of the $t$-structure it induces are analogous to the full subcategories of DG $R$-modules whose cohomology vanishes in positive and negative degree, respectively.

In the theory of DGAs, when one has a construction for chain DGAs it is natural to ask: what is the dual construction for cochain DGAs? Likewise, it is natural to ask, what is the structure induced by a compact object of a triangulated category which behaves like a cochain DGA? Recall that a DGA $R$ is called a cochain DGA if $H^i(R)=0$ for .

Unfortunately, it is well known in the theory of DGAs that constructing a viable dual theory for cochain DGAs is often difficult. In fact, at present the construction of a viable dual theory for DGAs always requires the additional assumption that the DGA $R$ is simply connected in the following sense: $H^0(R)$ is a division ring and $H^1(R)=0$. This lack of symmetry between the chain and cochain theories occurs throughout the theory of DGAs and in algebraic topology, see [2], for example. Thus, we shall consider the case of a compact object of a triangulated category which behaves like a simply connected cochain DGA.

The structure which is induced by such an object is not a $t$-structure, but it turns out to be almost dual to the notion of a $t$-structure, and as such we call it a co-$t$-structure. Both $t$-structures and co-$t$-structures provide examples of torsion theories in triangulated categories in the sense of Iyama and Yoshino, [12]. Co-$t$-structures have also been introduced by Bondarko in [7] where they are called weight structures. They are studied in [7] in connection with the theory of motives and stable homotopy theory.

The paper is organised as follows: in section 1, we recall the concepts of preenvelopes and precovers and set up the notation of perpendicular categories. We then recall the notion of a $t$-structure and introduce the new definition of a co-$t$-structure, about which we prove some elementary properties and compare and contrast this new notion with the existing notion of a $t$-structure. We also say why it is almost dual to a $t$-structure but not exactly dual. In addition, we introduce the definition of the coheart of a co-$t$-structure.

In section 2, we look at a canonical example of a co-$t$-structure appearing in the setting of the homotopy category of an additive category. We then look at the simple motivating examples of the $t$-structure induced by a chain DGA and the co-$t$-structure obtained by a simply connected cochain DGA on their respective derived categories. Note again that in order to consider a viable cochain analogue we need to impose the simply connected hypothesis. We present a brief exposition of Hoshino, Kato and Miyachi’s theorem, which is obtained in [11], which generalises and abstracts the example of the $t$-structure induced on the derived category of a chain DGA. Hoshino, Kato and Miyachi’s theorem is presented here as Theorem 2.3.

The remainder of the paper is devoted to proving the simply connected cochain analogue of Hoshino, Kato and Miyachi’s theorem, which is presented as Theorem 2.4, the first half of the proof appearing in sections 3 and 4. In addition to inducing a $t$-structure, Hoshino, Kato and Miyachi also prove that the heart of the induced $t$-structure, which is well known to be admissible abelian (see [3] and [13]), is equivalent to the module category of the endomorphism algebra of the object inducing the $t$-structure on $𝒯$. Here, we are able to prove a similar result regarding the coheart of the induced co-$t$-structure. It is known that the coheart of a co-$t$-structure is not always abelian, and may be very rarely so. Indeed, a specific example whose coheart is not abelian is constructed by Bondarko in [7]. However, the example of a co-$t$-structure which we present in this paper has an abelian coheart by virtue of its equivalence to a module category; this is the subject of section 5.