# Pretriangulated A ∞-categories We take,. A homomorphism is defined to be a cocycle in this complex. From this, or from a direct argument, one obtains the following corollary which will be applied often implicitly. The concept of a polymodule is derived from the more natural notion of a differential comodule over several coalgebras in C o g.

From this point of view, we have taken a backwards approach by defining the polymodule first, as the structure maps and definitions of morphisms are more transparent in the comodule setting. Nevertheless, we continue along our path full circle towards a realization of this structure as the bar construction of a polymodule.

We define its differential b P as. Then it follows from the defining equation 2. For our purposes, this is not enough as we wish to keep track of the length filtration throughout. In other words, morphism composition respects the total filtration on the tensor product.

In this section we define tensor products and inner homs of polymodules. To do this effectively, it is helpful to have a picture in mind as well as the appropriate notation associated to this picture. We abbreviate the differential coalgebra. As usual, this product is given by first passing through the bar construction, applying the cotensor product and then recognizing the result as the bar construction of a polymodule.

The details of this are now given.

## A∞-bimodules and Serre A∞-functors | SpringerLink

Then we let. Then we see that there is an isomorphism of graded vector spaces:. Recall that this is the kernel of. It is extended to the bar construction by tensoring with the remaining coalgebras. As this differential is a square zero comodule coderivation, it is induced by its composition with the projection.

We will preserve this filtration in the definition and write. Thus, we have obtained the DG functor 2. It will be useful to have notation for filtered quotients in this setting. For this, we write. As expected, the tensor product of a given polymodule with the diagonal polymodule yields a quasi-equivalent polymodule. However, the filtration is added structure which will be exploited later in the paper.

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For now, we simply define the natural quasi-equivalence and its inverse. Then we define the natural equivalences.

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As a consequence, we obtain the following basic lemma which is instructive as to the bar construction of a module. Then the lemma follows from the basic observation that, by definition,.

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Using the natural equivalence in 2. By the compactness assumption, this implies that tensor products of finitely generated projective polymodules are finitely generated projective polymodules. To define the internal Hom, we again follow the approach for the tensor product and pass to coalgebras and comodules. There is an additional notion needed here from classical homotopy theory, that of a twisting cochain which we recall here.

We take a moment to recall this construction for the case of a left module. It follows from Eqs. The structure map. Again we define the natural transformations. In this section we observe the classic adjunction between tensor product and internal Hom for polymodules. We will be concerned with preserving the lattice filtrations naturally throughout. To state the theorem, we need to specify the gluing data between three categories of polymodules. A gluing cycle can be represented graphically as a directed graph with three vertices.

This is depicted in Fig. With this notation, we can prove the following classic adjunction:. This is simply an exercise in the definitions of the last section and the observation that g r l f is a closed category. Recall that closed means that there is an internal Hom and tensor with the usual adjunction. To complete the proof, one must show that the isomorphisms above respect the differentials, which follows immediately from the definitions.

We apply this theorem to a simple gluing cycle to obtain the following corollary. Then the filtered adjunction 2. For formal algebras concentrated in degree zero, the above corollary is the elementary fact that flat dimension is less than or equal to projective dimension. We will observe conditions for which this inequality is an equality below. We start with an elementary lemma for perfect polymodules over compact algebras. Every perfect A module P has a finite dimensional minimal model P min defined uniquely up to isomorphism.

The following proposition, which was observed early in homological algebra, is stated below in terms of polymodules. This gives. In this section we lift many of the definitions and theorems of the dimension theory for triangulated categories to the pretriangulated setting. We take a moment to recall some definitions and notation from . With this notation in hand, the following definitions can be stated. We also use the same language and notation as above for level, generation time and dimension. Before proceeding with this discussion, we take a moment to illustrate this theorem with some examples. In general, one can mutate an exceptional collection to obtain a new exceptional collection and thereby a new generator. Below we examine a generator which does not arise from such mutations and in fact is not defined as the direct sum of objects in an exceptional collection.

### 1. Introduction

The map T 1 on morphisms is composition and T k is defined using higher multiplication. Clearly, ev G factors through the Yoneda embedding and can be thought of as evaluation of Yoneda at the point G. In a moment, we will examine the right hand side of Eq. However, all of these categories and functors fit into the diagram of categories below. The kernel i. It is an additional structure introduced by the choice of generator which provides homological information relative to G. We will now establish the link between generation time and filtration length.

go The following lemma is straightforward, but we supply a proof to establish some notation. The analogue of the length filtration in 2. These classes of maps will be useful when defining homotopies. We leave the elementary proof of the following properties to the reader. More concretely,. Pretriangulated A ∞-categories Pretriangulated A ∞-categories
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