This is the note taken at CHARMS2024
Additive, Abelian, and Exact Categories
Additive Categories
Definition 1. An additive category is a category \(\mathcal{A}\) such that
(Add-1) \(\mathcal{A}\) has a zero object \(0\) which is both initial and terminal.
(Add-2) For any two objects \(X\) and \(Y\) in \(\mathcal{A}\), the hom-set \(\mathrm{Hom}(X,Y)\) has an abelian group structure and composition of morphisms is bilinear \[\circ : \mathrm{Hom}(Y,Z) \times \mathrm{Hom}(X,Y) \to \mathrm{Hom}(X,Z).\]
(Add-3) \(\mathcal{A}\) has biproducts, i.e. for any \(X_1,X_2\in \mathcal{A}\), there is an object \(X = X_1\oplus X_2\) and morphisms \[X_1\xleftarrow{\pi_1} X \xrightarrow{\pi_2} X_2\] \[X_1\xrightarrow{\sigma_1} X \xleftarrow{\sigma_2} X_2\] such that \[\pi_i \sigma_j = \delta_{ij} \mathrm{id}_{X_i},\quad \sigma_1\pi_1 + \sigma_2\pi_2 = \mathrm{id}_X.\]
Remarks 1.
The zero element in the abelian group \(\mathrm{Hom}_\mathcal{A}(X,Y)\) equals the unique morphism \[0:X\to 0\to Y.\]
\((X,\sigma_1,\sigma_2)\) is a coproduct of \(X_1\) and \(X_2\) and \((X,\pi_1,\pi_2)\) is a product of \(X_1\) and \(X_2\).
The group structure on \(\mathrm{Hom}_\mathcal{A}(X,Y)\) is intrinsic and there is no additional structure on \(\mathcal{A}\). For example, given \[X\xrightarrow{f,g}Y\]
The three underlying morphisms can be constructed using only universal properties of the (co)products and the fact that we have a zero object.
Abelian Categories
Let \(\mathcal{A}\) be a category with zero object and \(f:X\to Y\in \mathcal{A}\) a morphism.
Definition 2.
\((K, K\xrightarrow{i} X)\) is a kernel of \(f\) if for all \(T\xrightarrow{t} X\) with \(ft=0\), then \(T\) factors uniquely through \(K\), there is a unique \(T\xrightarrow{t'} K\) such that \(t = it'\).
A cokernel \((C, Y\xrightarrow{p} C)\) of \(f\) is defined dually, i.e. if there is a map that composes to \(0\) with \(f\) on the left side, then it factors uniquely through \(C\).
The image of \(f\) is the kernel of the cokernel of \(f\), \[\mathrm{Im}(f) = \ker(Y\to \mathrm{coker}(f)).\]
The coimage of \(f\) is the cokernel of the kernel of \(f\), \[\mathrm{Coim}(f) = \mathrm{coker}(\ker(f)\to X).\]
Remarks 2. The existence of kernel can also be formulated as the functor \[\ker(h_X(\cdot) \to h_Y(\cdot)) : \mathcal{A}\to {\bf Ab}\] is representable \(=h_K\) for some object \(K\) in \(\mathcal{A}\).
Definition 3. An abelian category is an additive category \(\mathcal{A}\) such that
(Ab-1) Every morphism has a kernel and a cokernel.
(Ab-2) The canonical map \(\overline{f}:\mathrm{coim}(f)\to \mathrm{Im}(f)\) is always an isomorphism. (This is in fact abstracting the first isomorphism theorem.)
Example 1. Examples of abelian categories include
The module category \({}_{R}\mathbf{Mod}\) for any ring \(R\).
The category of quasi-coherent sheaves \({\bf QCoh}(X)\) on a scheme.
(Non-example) The category of finitely generated free abelian groups is additive but not abelian, we leave the exercise for reader to check that \(\mathcal{A}\) has all kernels and cokernels and isomorphisms hold, but \(\mathcal{A}\) is not abelian.
Exact Categories
Definition 4. An exact category is an additive category \(\mathcal{A}\) together with a class of kernel-cokernel pairs \((i, p)\) called conflations (inflacion, deflation) such that
(Ex-0) \(\mathrm{id}_0\) is a deflation.
(Ex-1) Compositions of deflations are deflations.
(Ex-2) Any deflation \(p\) and morphism \(f\), there exists a pullback square
and \(p'\) is a deflation.
(Ex-2’) Dually, for any inflation \(i\) and morphism \(f\), there exists a pushout square
and \(i'\) is an inflation.
Remarks 3.
The dual statements \((Ex-0')\) and \((Ex-1')\) can be derived from the above axioms
For every isomorphism \(\varphi\), the diagram
is a pullback square, hence every isomorphism \(\varphi\) is a deflation. (Also inflations by the dual argument.)
Example 2.
Every additive category has an exact structure given by \[\{\text{Conflations}\} = \{ \text{All kernel cokernel pairs isomorphic to } X\to X\oplus Y\to Y\}.\]
An abelian category can have different exact structures, for example given yb all short exact sequences, or split short exact sequences.
Let \(\mathcal{B}\subset \mathcal{A}\) be a full and extension closed subcat of an abelian cat \(\mathcal{A}\), with conflations induced by all short exact sequences in \(\mathcal{A}\) that has objects in \(\mathcal{B}\). Then \(\mathcal{B}\) is an exact category. Every small exact category is of this form.
Derived category of an exact category
let \(\mathcal{A}\) be an exact category, we write \(C(\mathcal{A})\) as the category of cochain complexes. A morphism of cochain complexes \(f^\bullet : X^\bullet \to Y^\bullet\) is a collection of morphisms \(f^n : X^n \to Y^n\) such that the squares with differentials commute \(f^{n+1}d_X^n = d_Y^n f^n\). \(f\) is null-homotopic if there exists a collection of morphisms \(s^n : X^n \to Y^{n-1}\) such that \(f^n - 0 = d_Y^{n-1}s^n + s^{n+1}d_X^n\).
Definition 5. The homotopy category \(K(\mathcal{A})\) of the underlying additive category of \(\mathcal{A}\) is \[K(\mathcal{A}) = C(\mathcal{A})/\{\text{Null-homotopic morphisms}\}.\]
Remarks 4. For an additive category \(\mathcal{A}\), \(C(\mathcal{A})\) is an exact category with conflations given by component-wise split short exact sequences of complexes. Moreover, \(C(\mathcal{A})\) is a Frobenius category with projectives and injectives both given by constructible complexes.
Definition 6. A complex \(X\in C(\mathcal{A})\) is called acyclic if there are factorizations \[d^{i-1}:X^{i-1}\to Z^i\to X^i\] such that \(Z^i\to X^i \to Z^{i+1}\) is a conflation. A morphism is a quasi-isomorphism if its mapping cone \(\mathrm{Cone(f)}\) is isomorphic to an acyclic complex.
Remarks 5. So it turns out that the definition of derived catesory \(\mathcal{D}(\mathcal{A})\) does not depend on the exact structure of \(C(\mathcal{A})\), only depend on the exact structure of \(\mathcal{A}\).
Introduction to Gentle Algebras
Gentle Algebras
Assem and Skowronski in 1987 introduced ’gentle algebra’ to study ’iterated tilted algebra’ of type \(A_n\) (combinatorial data)
Definition 7. A quiver pair \((Q,I)\) is gentle if
for any \(v\in Q_0\), there are at most two arrows \(\alpha\in Q_1\), such that \(s(\alpha)=v\), and at most \(2\) arrows \(\beta\) such that \(t(\beta) = v\).
Triangulated Categories
Exact category / homotopy, the notion of long exact sequences.
Definition 8. A triangulated category consists of the following data
An additive category \(\mathcal{C}\).
An autoequivalence \([1]=\Sigma:\mathcal{C}\to \mathcal{C}\).
A class of triangles \(\Delta\) called exact triangles.
A triangle is a diagram \[A\xrightarrow{f} B\xrightarrow{g} C\xrightarrow{h} A[1]\] they must satisfy the following axioms of triangulated categories
(TR-1) \(\Delta\) is closed under isomorphisms, \(1_X\) gives a triangle, and every morphism \(f:X\to Y\) fits into a triangle.
(TR-2) (Rotation) \(A\xrightarrow{f} B\xrightarrow{g} C\xrightarrow{h} A[1]\) is a triangle if and only if \(B\xrightarrow{g} C\xrightarrow{h} A[1]\xrightarrow{-f[1]} B[1]\) is a triangle.
(TR-3) Given a diagram with exact rows
there exists a morphism \(h\) such that the diagram commutes.
(Octahedral axiom) (...)
Remarks 6. Iterated rotations will give you a long sequence \[A \to B \to C \to A[1] \to B[1] \to C[1] \to A[2] \to \cdots\]
Example 3. Homotopy category of an exact category is a triangulated category. Derived category of an exact category is a triangulated category. Derived categories and stable category of Frobenius category are triangulated.
Frobenius categories
Running example, let \(A\) be a finite dimensional algebra and consider exact structures on the category of modules \({}_{A}\mathbf{Mod}\).
\(S_{split}\) consists of all split short exact sequences.
\(S_{short}\) consists of all short exact sequences.
Definition 9. Let \((\mathcal{F},S)\) be an exact category, and \(I\in \mathcal{F}\) is injective if \(\mathcal{F}(\cdot, I)\) sends conflations to exact sequences. And \(\mathcal{F}\) has enough injectives if for every object \(X\in \mathcal{F}\), there exists a conflation \(X\to I\to X[1]\) with \(I\) injective.
Example 4. In \(({}_{A}\mathbf{Mod}, S_{split})\) every object is projective, in \(({}_{A}\mathbf{Mod}, S_{short})\) this is the usual notion of projective and injective modules.
Definition 10. An exact category \((\mathcal{F},S)\) is called Frobenius if the following conditions hold
\(\mathcal{F}\) has enough projectives.
\(\mathcal{F}\) has enough injectives.
The projectives and injectives coincide.
Example 5.
\(({}_{A}\mathbf{Mod}, S_{split})\) is a Frobenius category, this is the trivial case.
In general \(({}_{A}\mathbf{Mod}, S_{short})\) is not Frobenius, but \(({}_{A}\mathbf{Mod}, S_{short})\) is Frobenius if and only if \(A\) is self-injective.
\(\mathcal{A}\) exact \(\Rightarrow (C(\mathcal{A}), C(S_{split}))\) is Frobenius.
Definition 11. Let \((\mathcal{F},S)\) be a Frobenius category, the stable category \(\underline{\mathcal{F}}\) is the category with objects in \(\mathcal{F}\) and morphisms given by \[\underline{\mathcal{F}}(X,Y) = \mathcal{F}(X,Y)/\{\text{Morphisms that factor through projectives}\}.\]
Remarks 7. Let \(P\) be projective then \(1_P\) factors through \(P\), so \(1_P=0\) in \(\underline{\mathcal{F}}\), which makes \(P\cong 0\) a zero object in \(\underline{\mathcal{F}}\).
Theorem 1. Let \((\mathcal{F},S)\) be a Frobenius category, then \(\underline{\mathcal{F}}\) is a triangulated category.
Proof. Sketch: We need \(\Sigma, \Delta\) to be defined. \(\Sigma\) is the shift functor, and \(\Delta\) is the class of triangles in \(\mathcal{F}\). For every object \(A\) we can pick a conflation \[A\to P\to A[1], \quad I\text{ injective}\] Schanuel’s Lemma gives \(\Sigma A\) is unique up to injective objects. Direct summands gives \(\Sigma A\) is unique up to isomorphisms in \(\underline{\mathcal{F}}\). This allows us to turn \(\Sigma\) into a functor on \(\underline{\mathcal{F}}\). We can pick another conflation \[\Omega A \to P \to A, \quad P\text{ projective}\] by a similar process we can turn \(\Omega\) into a functor on \(\underline{\mathcal{F}}\) which is the quasi-inverse of \(\Sigma\). The conflation \(A\to I \to \Sigma A\) actually shows in \(\underline{\mathcal{F}}\), \(A\cong \Omega \Sigma A\) since \(I\) is projective.
For the triangles, let \(A\to B=f \in \mathcal{F}\). Let \(A\xrightarrow{L} I\xrightarrow{p} \Sigma A\) be a conflation with \(I\) injective. \[A \xrightarrow{t} B \xrightarrow{(\mathrm{id}, 0)} B\oplus_A I \xrightarrow{(0, p)} I \xrightarrow{g} \Sigma A\] ◻
Definition 12. Let \(\mathcal{A}\) be an abelian cat, a full subcat \(W\) is called wide if it is closed under direct summands and any two objects in an short exact sequence are in \(W\) then so is the third. This is a stronger condition than being a Serre subcat, it adds the kernels and cokernels.
Definition 13. Let \(\mathcal{X}\) be a full subcat of \(\mathcal{A}\) and we define \[\mathcal{X}^\perp = \{Y\in \mathcal{A}: \mathrm{Ext}^1(\mathcal{X},Y) = 0\}\] \[{}^\perp \mathcal{X}= \{Y\in \mathcal{A}: \mathrm{Ext}^1(Y,\mathcal{X}) = 0\}\] A pair \((\mathcal{X},\mathcal{Y})\) of full subcategories is a cotorsion pair if \(\mathcal{X}^\perp = \mathcal{Y}\) and \({}^\perp \mathcal{Y}= \mathcal{X}\).
\((\mathcal{X},\mathcal{Y})\) is functorially complete if every \(A\in \mathcal{A}\) admits functorial short exact sequence \[0 \to Y \to X \to A\to 0\] \[0 \to A\to Y' \to X'\to 0\] with \(X,X'\in \mathcal{X}\) and \(Y,Y'\in \mathcal{Y}\).
\((\mathcal{X},\mathcal{Y})\) is hereditory if \[\mathrm{Ext}^{\ge 1}(\mathcal{X}, \mathcal{Y}) = 0\]
Differential Graded (dg) Categories
Let \(k\) be a commutative ring.
Definition 14. A differential graded category is a category that is enriched over complexes \(\mathcal{C}k\). Where \(\mathcal{C}k\) is the complex of \(k\)-modules \(\dots\to V^i\xrightarrow{d^i} V^{i+1}\to \dots\).
There is a monoidal structure on \(\mathcal{C}k\), \((V^\bullet, d^\bullet_V)\), \((W^\bullet, d^\bullet_W)\) \[(V\otimes W)^n = \bigoplus_{i+j=n} V^i\otimes W^j\] \[d^n_{V\otimes W} = d_V\otimes \mathrm{id}_W + (-1)^{|v|} \mathrm{id}_V\otimes d_W\] It is symmetric, \[V\otimes W \cong W\otimes V, \quad v\otimes w\mapsto (-1)^{|v||w|} w\otimes v.\] Observation: the functor \((\cdot)\otimes V\) has a right adjoint, this functor is \(\mathrm{Hom}^\bullet_k(V,\cdot)\), which is defined as (curious execise: derive this formula by adjunction) \[\mathrm{Hom}^\bullet_k(V,W)^n = \mathrm{Hom}_k(V,W[n]) = \prod_{i\in \mathbb{Z}} \mathrm{Hom}_k(V^i, W^{i+n}).\] this is a complex whose differential is given by \[d^n (f^\bullet) := d_W\circ f^\bullet - (-1)^{|f|} f^\bullet \circ d_V.\] There is also a translation functor \(V[1]\) defined by \((V[1])^n = V^{n+1}\) and \(d^{n}_{V[1]} = -d^{n+1}_V\).
Definition 15. A dg category \(\mathcal{A}\) consists of a class of objects, for all \(X,Y\in \mathcal{A}\), \(\mathcal{A}(X,Y)\in \mathcal{C}k\) together with composition morphism (which gives Leibniz rule) \[\mathcal{A}(Y,Z)\otimes \mathcal{A}(X,Y) \to \mathcal{A}(X,Z)\] that is associative and unital.
Example 6.
\(\mathcal{C}_{dg} k = (\mathcal{C}k, \mathrm{Hom}^\bullet_k)\) is a dg category. Be aware of the funny formula \(f\otimes g(x\otimes y) = (-1)^{|g||x|}f(x)\otimes g(y)\), \(d_{V\otimes W} = 1_V\otimes d_W + 1_V\otimes d_W\).
\(B\) a \(k\)-algebra, then the category of right \({}_{}\mathbf{Mod}_B\) modules, \(\mathcal{C}_{dg}{}_{}\mathbf{Mod}_B = \mathcal{C}_{dg} B\) is a dg category of complexes of \(B\)-modules.
A dg algebra is equivalent to a dg category with one object \(\{*\}\), \(A = \mathrm{Hom}_\mathcal{A}(*, *)\).
A dg quiver, for example \(\dots\)
Given a dg category \(\mathcal{A}\), we can form its opposite category \(\mathcal{A}^{op}\), with \(f\circ^{op} g = (-1)^{|f||g|} g\circ f\) which is also a dg category.
Tensor product of two dg cats, \(\mathcal{A}\otimes \mathcal{B}= (\mathcal{A}\times \mathcal{B}, ((x,y),(x',y'))\mapsto \mathcal{A}(x,x')\otimes \mathcal{B}(y,y'))\). \[(g\otimes g')\circ (f\otimes f') = (-1)^{|g'||f|} (g\circ f)\otimes (g'\circ f').\]
Definition 16. A dg-functor of dg categories \(\mathcal{A}\) and \(\mathcal{B}\) is a functor \(F:\mathcal{A}\to \mathcal{B}\) such that \[F:\mathcal{A}(x,y) \to \mathcal{B}(Fx, Fy)\] is a morphism of complexes compatible with composition and units. This turns \(\mathrm{Hom}_{\mathrm{Cat}}(\mathcal{A},\mathcal{B})\) a category of df-functors.
Proposition 1. The category of dg-cats \((\mathrm{Cat}_{dg}, \otimes)\) is a symmetric monoidal category. with an internal hom \[(\cdot)\otimes \mathcal{A}\to \mathrm{Hom}_{\mathrm{Cat}_{dg}}(\mathcal{A}, \cdot)\]
Definition 17. Let \(\mathcal{A}\) be a dg-category, the \(0\)-truancation is the dg-cat with the same objects but with every morphism space truncated to degree \(\le 0\) i.e. it has morphism spaces \[\tau_{\le 0} \mathcal{A}(x,y)\] The \(0\)-cocycle of \(\mathcal{A}\) are category with the same objects and morphisms \[Z^0\mathcal{A}(x,y) = \ker(d^0_{\mathcal{A}(x,y)})\] The \(0\)-cohomology category or homotopy category of \(\mathcal{A}\) is the category with the same objects and morphisms \[H^0\mathcal{A}(x,y) = Z^0\mathcal{A}(x,y)/B^0\mathcal{A}(x,y)\]
Remarks 8. There are dg-Functors \[H^0\mathcal{A}\leftarrow \tau_{\le 0}\mathcal{A}\hookrightarrow \mathcal{A}\]
Definition 18. \(F:\mathcal{A}\ to \mathcal{B}\) is a quasi-equivalence if the induced functor on \(0\)-cohomology categories \[H^0F:H^0\mathcal{A}\to H^0\mathcal{B}\] is an equivalence.
Example 7.
For \(B\) a \(k\)-algebra, \(Z^0\mathcal{C}_{dg} B = \mathcal{C}{}_{B}\mathbf{Mod}\), \(H^0\mathcal{C}_{dg} B = K({}_{B}\mathbf{Mod})\) are the usual category of complexes and its homotopy category.
For \(\mathcal{A}\) a dg-algebra, \(Z^0\mathcal{A}= Z^0A\) and \(H^0\mathcal{A}= H^0A\) are the usual notions of cycles and cohomology.
In the example of dg-quiver,
Homological Mirror Symmetry of Gentle Algebras (B-side)
A motivating example, consider the projective line \(\mathbb{P}^1\), whose derived category is equivalent to that of the path algebra of the Kronecker quiver. Goal is to generalize this in two ways: gluing several \(\mathbb{P}^1\)s, and consider the weighted projective lines.
Geometric Model for \(\mathcal{D}^b(\mathcal{A})\) for a graded gentle algebra
Let \(A = kQ/I\) be a gentle algebra. Define the grading \(Q_1\to \mathbb{Z}\), we can consider \(A\) as a differential graded algebra with the \(0\) differential.
Abelian Model Categories
Motivation
Consider \(R\) a commutative ring, \(\Sigma\subset R\) is multiplicatively closed set. We can form the ring of fractions \(R[\Sigma^{-1}]\) written as \(r/s=rs^{-1}\). If \(R\) is not commutative, then you can still form this ring but \(\Sigma\) must satisfy some properties for it to be written as the fraction form. In general it is only written in a product form \(r_1\dots r_n\) where some \(r_i\in \Sigma\) and some \(r_i\in R\).
Theorem 2 (Gabriel-Zisman). Let \(\mathcal{C}\) be a category and \(S\subset \mathrm{Mor}(\mathcal{C})\), there is a construction of a category \(\mathcal{C}[S^{-1}]\) called the localizing category with a canonical functor \(\mathcal{C}\to \mathcal{C}[S^{-1}]\) sending \(s\) into isomorphisms, such that for any functor \(F:\mathcal{C}\to \mathcal{D}\) that sends \(S\) to isomorphisms, there exists a unique functor \(\mathcal{C}[S^{-1}]\to \mathcal{D}\) that makes the diagram commute.
Such categories exist but the morphisms are very hard to control, we can have long sequences of elements in \(rsrsr\) not reducible to a fraction (roof). Also, the hom sets may not be sets but proper classes. The concept of Model categories is a way to control the morphisms in the localization that solves these problems.
Example 8. Consider \(R\) a ring and \(\mathcal{A}\) abelian, if we localize with quasi-isoms, \[C(R)\to C(R)[\text{qiso}^{-1}]=D(R)\] this is a hard step an in general we pass to \(K(R)\) and then to \(D(R)\) by calculus of fractions.
Model Categories
Definition 19. Let \(\mathcal{A}\) be a category and take two morphisms \(f:X\to Y, g: A\to B\) in \(\mathcal{A}\). We say that \(f\square g\), if for any CD there exists at least one dashed arrow making the diagram commute.
A pair \((\mathcal{L},\mathcal{R})\) of subclasses of morphisms of \(\mathcal{A}\) is called a weak factorizatoin system (WFS) if
\(\mathcal{L},\mathcal{R}\) are closed under retarcts,
\(\mathcal{L}\square \mathcal{R}\),
for every \(f\), there is \(f = r \circ l\) with \(l\in \mathcal{L}, r\in \mathcal{R}\). If this is determined functorially, we say that the WFS is functorial.
Let \(\mathcal{A}\) be a category and we take three subclasses cof, fib, weq of \(\mathrm{Mor}(\mathcal{A})\), we say that these subclasses define a model structure on \(\mathcal{A}\) if
weq is closed under retractions (in a sense they contain all isomorphisms),
weq satisfy the 2-out-of-3 property, i.e. if \(f\) and \(g\) are composable and any two of \(f,g,fg\) are in weq, then so is the third.
\((\mathrm{cof}, \mathrm{fib}\cap \mathrm{weq})\) and \((\mathrm{cof}\cap \mathrm{weq}, \mathrm{fib})\) are WFS. They are called trivial cofibrations and trivial fibrations.
Let \(\mathcal{A}\) be a model category with initial object \(\varnothing\) and terminal object \(*\)
\(X\in\mathcal{A}\) is called cofibrant if \(\varnothing\to X\) is a cofibration.
\(X\in\mathcal{A}\) is called fibrant if \(X\to *\) is a fibration.
For any object \(X\in\mathcal{A}\), there exists factorizations
Homotopy Category
To speak about homotopy categories we need to define homotopies.
Definition 20. Let \(\mathcal{A}\) be a model category and \(f,g:X\to Y\) be two morphisms. We say \(f\) is left homotopic to \(g\) if there exists a cylineder object \[X \to X\sqcup X\]
Enhancements
Extriangulated categories
Exact dg categories, after Xiaofa Chen
Enhancements
Let \(k\) a commutative graded ring (eg a field), \(\mathcal{A}\) a dg \(k\)-category (which could be a \(k\)-algebra, identified with a dg-cat with one object, or proj A, \(C^b_{dg}(proj(A))\)) \(\mathcal{C}\mathcal{A}\) is the cat of dg \(\mathcal{A}\)-modules \(\{ M : \mathcal{A}^{op}\to \mathcal{C}k\}\). (eg, \(\mathcal{A}= A\), \(\mathcal{C}\mathcal{A}= \mathcal{C}A = \mathcal{C}({}_{}\mathbf{Mod}A)\)) \(\mathcal{K}\mathcal{A}\) is the cat module homotopy of dg \(\mathcal{A}\)-modules.
Q-shaped derived category
By Henrik Holm (Copenhagen) arXiv 2212.12524
Examples
\(k\) be a hereditory noetherian commutative ring, \(A\) a \(k\)-algebra, (field, path algebra, \(\mathbb{Z}\)-algebra (ring))
Let \(Q=Q^{cpx}\) is the \(k\)-pre-additive category given by the quiver \(\dots \to 2\to 1 \to 0 \to -2 \to \dots\) with relations that two consequtive arrows compose to zero. \[{}_{Q,A}\mathbf{Mod} = \{k-\text{linear functors }Q\to {}_{A}\mathbf{Mod}\}=C(A)\] You can see this is just chain complexes of \(A\)-modules.
We have weak equivalences, think them as quasi-isomorphisms. We invert them \[\mathrm{weq}^{-1}({}_{Q,A}\mathbf{Mod}) = \mathrm{qiso}^{-1}C(A) = D(A)\]
Let now \(Q = Q^{N,cpx}\) is the \(k\)-pre-additive category given by the same quiver with the previous one, but with the relations that \(N\) consequtive compsitions give to zero. Now \[{}_{Q,A}\mathbf{Mod} = \{k-\text{linear functors }Q\to {}_{A}\mathbf{Mod}\}=C_N(A)=\{N-\text{complexes}\}\] \[\mathrm{weq}^{-1}({}_{Q,A}\mathbf{Mod}) = \mathrm{qiso}^{-1}C_N(A) = D_N(A)\]
You could also replace \(Q\) by some different diagram, let now \(Q = Q^{m-per}\) defined by the \(m\)-cycle quiver with relations that \(2\) consequtive compositions give to zero. (you can also use \(m\) instead of \(2\)). \[{}_{Q,A}\mathbf{Mod} = \{k-\text{linear functors }Q\to {}_{A}\mathbf{Mod}\}=C_{m-per}(A)\] \[\mathrm{weq}^{-1}({}_{Q,A}\mathbf{Mod}) = \mathrm{qiso}^{-1}C_{m-per}(A) = D_{m-per}(A)\]
Setup
Let \(Q\) be a category, satisfying
\(Q\) is \(k\)-pre-additive,
\(Q\) has each hom set \(Q(p,q)\) finitely generated projective over \(k\),
\(Q\) locally bounded, if you look at the set of \(p\) inside \(Q_0=\mathrm{ob}(Q)\) such that \(Q(p,q)\neq 0\), this set is finite. The other way holds as well. \[\{p\in Q_0 : Q(p,q)\neq 0\}<\infty\] \[\{q\in Q_0 : Q(p,q)\neq 0\}<\infty\]
\(Q(q,q)=k 1_q + r_q\) where \(r_q\circ r_q \subset r_q\) and \(Q(p,q)\circ Q(q,p) \subset r_q\) for \(p\neq q\).
There’s an ideal \(r(p,q)=r_q\) if \(p=q\) and \(r(p,q)=Q(p,q)\) if \(p\neq q\). We must have \(r^N=0\) for some \(N\).
Serre functor, a \(k\)-linear equivalence \(S:Q\to Q\) such that Serre duality formulas holds \[Q(p,q)\cong Q(q,S(p))^\vee=\mathrm{Hom}_k(Q(q,S(p)),k)\]
Example 9. Let \(k\) be a field and \(Q\) be given by a quiver with relations over \(k\) such that the corresponding bound quiver algebra \(\Lambda\) is self-injective. Q will actually be recovered in the following way, \[Q\cong \mathrm{inde.proj}(\Lambda^{op})\] Actually we can give the Serre functor by the Nakayama-functor \[S(\cdot) = (\cdot) \otimes_\Lambda D\Lambda\] where \(D\Lambda\) is the dual of \(\Lambda\).
\[{}_{Q,A}\mathbf{Mod} = \{k-\text{linear functors }Q\to {}_{A}\mathbf{Mod}\}\] \[{}_{Q}\mathbf{Mod} = \{k-\text{linear functors }Q\to {}_{k}\mathbf{Mod}\}\] there is an obvious forgetful functor \({}_{Q,A}\mathbf{Mod}\to {}_{Q}\mathbf{Mod}\).
They are pretty nice Grothendieck abelian categories, have enough projectives and injectives, and have a Set indexed limits and colimits.
\(\mathrm{Proj}{Q,A}\) is the full subcat of \({}_{Q,A}\mathbf{Mod}\) consisting of projective objects. \(\mathrm{Inj}{Q,A}\) is the full subcat of \({}_{Q,A}\mathbf{Mod}\) consisting of injective objects. Within then you have \(\mathrm{Hom}_{Q,A}\), \(\mathrm{Ext}^i_{Q,A}\) in \({}_{Q,A}\mathbf{Mod}\), and \(\mathrm{Hom}_Q\), \(\mathrm{Ext}^i_Q\) in \({}_{Q}\mathbf{Mod}\).
Homology
The over all thought is that \({}_{Q,A}\mathbf{Mod}\) has two abelian model cat structures, and we can get the derived category by these model structures.
If we have \(q\in Q_0\), for \(S\langle q\rangle=Q(q,\cdot)/r(q,\cdot):{}_{Q}\mathbf{Mod}\), \[S\langle q\rangle(p)=k1_{p=q}\]
Fractionally Calabi Yau Posets Corroborating a conjecture by Chapoton
Combinatorial sequences lined to representation of finite dimensional algebras, linked to some symplectic geometry.
A Lattice of Categories of An Algebra
Let \(A\) be a finite dimensional algebra over some field \(k\), and \({}_{A}\mathbf{Mod}\) of the category of f.g. modules, and \(\tau\) is the Auslander Reilen transformation of the module category.
Basic definitions and backgrounds
A pair of modules (and a projective) \((M,P)\) is \(\tau\)-rigid if \(M\) has no non-zero homomorphisms \(\mathrm{Hom}(M,\tau M)=0, \mathrm{Hom}(P,M)=0\). \(\tau\)-tilting, if additionally, \(|M|+|P|=|A|\) (number of direct summand of indecomps).
Theorem 3. there is a poset isomoprhism between the set of \(\tau\)-tilting pairs and functorialy finite torsion classes of \(A\)-modules, two term tileing objects in \({}_{A}\mathbf{Mod}\).
\(A=kQ/I\) graded gentle gives a ribbon graph \(\Delta\) which in turn gives a inter polygon and a dual graph \(\Delta^*\) and disection.
Theorem 4 (Opper-Plauoudou-S18+new version, Qiu-Zhang-Zhou 22). We have \((S,\Delta,\Delta^*)\) gives a geometric model of \(\mathcal{D}^b(\mathcal{A})\). Enough to know the indecomposable objects, which B-M tells us are in bijection with graded curves \((Y,f)\).
The ingredients are, \(\Gamma\) is a homotopy class of curves and some funny stuffs, which gives a comibnatorial winding number which encodes both the shift and fct and the grading structure of S (LP 20).
A basis of morphism space between indecomposables given by intersections of curves. Mapping cones given y resolutions of crossings.
Homology
Idea: aim for the definitions of weak equivalences. For \(q\in Q_0 =\mathrm{ob}(Q)\) and \(i\ge 1\), consider the functor \[\mathbb{H}^i_{[q]}(\cdot) = \mathrm{Ext}^i(S\langle q\rangle, \cdot) :{}_{Q,A}\mathbf{Mod}\to {}_{A}\mathbf{Mod}\]
Exact objects \[\begin{aligned} \mathcal{E}&= \left\{ X\in {}_{Q,A}\mathbf{Mod} : \mathbb{H}^1_{[q]}(X)=0 \text{ for } q\in Q_0\right\}\\ &= \left\{ X\in {}_{Q,A}\mathbf{Mod} : \mathbb{H}^i_{[q]}(X)=0 \text{ for } q\in Q_0, i\ge 1\right\} \end{aligned}\] by varing \(q\) we can get all classical homological functors.
Weak equivalences
This is gonna be all the morphisms in \[\begin{aligned} \mathrm{weq} &= \left\{\varphi\in \mathrm{Mor}({}_{Q,A}\mathbf{Mod}) : \mathbb{H}^{1,2}_{[q]}(\varphi) \text{ is an isomorphism for all } q\in Q_0 \right\}\\ &= \left\{\varphi\in \mathrm{Mor}({}_{Q,A}\mathbf{Mod}) : \mathbb{H}^{i}_{[q]}(\varphi) \text{ is an isomorphism for all } q\in Q_0, i\ge 1 \right\} \end{aligned}\] Here note that we need not just \(\mathbb{H}^1_{[q]}\) but also \(\mathbb{H}^2_{[q]}\) be zero.
\[\mathcal{D}_Q(A) = {}_{Q,A}\mathbf{Mod}[\mathrm{weq}^{-1}]\] Let us see how \(\mathcal{E}\) be come exact complexes and \(\mathrm{weq}\) be come quasi-isomophisms in examples.
Example 10. Let \(Q=Q^{cpx}\), remember that objects in \({}_{Q,A}\mathbf{Mod}\) are chain complexes of \(A\)-modules. If you look the definiotion of \(\mathbb{H}^i\), we can compute them in two different ways, projective resolve the left term, or injective resolve the right term. We will use the projective resolution.
Now \[S\langle 0\rangle = \dots \to 0\to k\to 0\to \dots\] we take the representable
Last Lecture of Q-shaped derived category
Recall that we have \[\mathcal{D}_Q(A) = {}_{Q,A}\mathbf{Mod}[\mathrm{weq}^{-1}] \cong \frac{\mathcal{E}^\perp}{{}_{Q,A}\mathrm{Inj}} \cong \frac{{}^\perp \mathcal{E}}{{}_{Q,A}\mathrm{Prj}}\] The machinary of these isomoprhisms requivers varieous approximations. Today we want to choose a different \(Q\) and produce completely different results. Let \(Q = Q^{N-cpx}\) be the following category \[\dots 2\to 1 \to 0 \to -1 \to -2 \to \dots, \quad \partial^N=0.\] Consider a \(P\) which is given as (at degree \(0\)) \[P= \dots 0\to 0\to A\to 0\to 0\to \dots \in {}^\perp\mathcal{E}\]
Cotorsion Pairs
The following are hereditary, functorially complete cotorsion pairs in \({}_{Q,A}\mathbf{Mod}\),
\(({}^\perp\mathcal{E}, \mathcal{E})\), \(({}_{Q,A}\mathrm{Prj}, {}_{Q,A}\mathbf{Mod})\), these are \[{}^\perp\mathcal{E}\cap \mathcal{E}= {}_{Q,A}\mathrm{Prj}\cap {}_{Q,A}\mathbf{Mod} = {}_{Q,A}\mathrm{Prj}\]
\((\mathcal{E}, \mathcal{E}^\perp)\), \(({}_{Q,A}\mathbf{Mod}, {}_{Q,A}\mathrm{Inj})\), these are \[\mathcal{E}\cap \mathcal{E}^\perp = {}_{Q,A}\mathrm{Inj}\cap {}_{Q,A}\mathbf{Mod} = {}_{Q,A}\mathrm{Inj}\]
(Recall that cotorsion pair \((\mathcal{X},\mathcal{Y})\) means \(\mathcal{X}^\perp = \mathcal{Y}\) and \({}^\perp \mathcal{Y}= \mathcal{X}\), hereditary means \(\mathrm{Ext}^{\ge 1}(\mathcal{X},\mathcal{Y})=0\), functorially complete means for every object \(M\) there exists a functorial (in \(M\)) short exact sequence \[0\to Y\to X\to M\to 0, \quad 0\to M\to Y'\to X'\to 0\] with \(X,X'\in \mathcal{X}\) and \(Y,Y'\in \mathcal{Y}\).)
Model Structures
There is a hereditary Hovey triple \((\mathcal{C}_p, \mathcal{W}_p, \mathcal{F}_p) = ({}^\perp\mathcal{E}, \mathcal{E}, {}_{Q,A}\mathbf{Mod})\) in \({}_{Q,A}\mathbf{Mod}\). This gives the projective model category structure on \({}_{Q,A}\mathbf{Mod}\) consisting of \((\mathrm{weq}_p, \mathrm{cof}_p, \mathrm{fib}_p)\) with
\(\mathrm{weq}_p\) consists of compositions \(X\hookrightarrow Y\twoheadrightarrow Z\) where the first arrow have cokernel in \(\mathcal{W}_p\) and the second arrow have kernel in \(\mathcal{W}_p\).
\(\mathrm{cof}_p\) consists of monomorphisms with cokernels in \(\mathcal{C}_p = {}^\perp\mathcal{E}\).
\(\mathrm{fib}_p\) consists of epimorphisms with kernels in \(\mathcal{F}_p = \mathcal{E}\).
You also get \(\mathcal{C}_p\cap \mathcal{F}_p= {}^\perp\) is a Frobenius category with projectives-injectives \(\mathcal{C}_p\cap \mathcal{W}_p \cap \mathcal{F}_p = {}_{Q,A}\mathrm{Prj}\). And \[\frac{\mathcal{C}_p\cap \mathcal{F}_p}{\mathcal{C}_p\cap \mathcal{W}_p\cap \mathcal{F}_p} = \mathrm{weq}_p^{-1}({}_{Q,A}\mathbf{Mod}) = \mathcal{D}_Q(A)\] i.e. \[\frac{{}^\perp \mathcal{E}}{{}_{Q,A}\mathrm{Prj}} = \mathcal{D}_Q(A)\] The injective version : \[(\mathcal{C}_i, \mathcal{W}_i, \mathcal{F}_i) = ({}_{Q,A}\mathbf{Mod}, \mathcal{E},\mathcal{E}^\perp)\]
Perfect Objects
(Bounded complexes of finitely generated projective modules, ’compact’ in the derived category) The compact objects are \[\mathcal{D}_Q^c(A) =\left\{ C\in \mathcal{D}_Q(A) : \mathrm{Hom}_{\mathcal{D}_Q(A)}(C, \cdot) \text{respect set indexed} \coprod\right\}\] (this is a strong form of being finitely generated)
Theorem 5 (Neeuman). \[\mathcal{D}^c(A) = \{C: C\cong \text{perfect complex}\}\]
Definition 21. Strictly perfect objects are the \(K\in {}_{Q,A}\mathbf{Mod}\) such that
\(|\{q\in Q_0: K(q)\neq 0\}|<\infty\)
\(K(q)\) is f.g. proj.
These can be viewed as the full subcategory \(\mathcal{D}_Q^{sperf}(A)\) inside \(\mathcal{D}_Q(A)\).
Theorem 6. In general \(\mathcal{D}_Q^{s.perf}(A)\not\subset \mathcal{D}_Q^c(A)\). For example take \(A=k[x,y]/(x^2,xy)\) where \(k\) is a field, and \(Q\) the Jordan quiver with relation \(\partial^2=0\).
Theorem 7. \(\mathcal{D}_Q(A)\) is compactly generated by \[\{S\langle q\rangle \otimes_k A : q\in Q_0\}\]
Exact dg-categories after Xiaofa Chen
Homotopy Kernels and Cokernels
Exact dg-categories definitions and first properties
Let \(K\) be a commutative ring, let’s fix \(\mathcal{A}\) a dg-category, then we have all that we have seen the notion of three term complexes, that’s just a diagram in \(A\) of the following shape \[X\xrightarrow{i} Y\xrightarrow{p} Z\] whose composition is not zero but homotopy to zero, i.e. \(h:X\to Z\) with \(|h|=-1\) and \(dh=p\circ i\). Let me introduce an auxiliary category \(T\) which is a dg-path \(k\)-category of the dg quiver whose representations are these diagrams \(1\xrightarrow{i} 2\xrightarrow{\pi} 3\) where \(|\eta|=1\), \(d\eta=\pi\circ i\). I don’t want to view them as dg-functors but dg-bimodules, just a change of view point. \(3\)-term homotopy complex are right representable dg-bimodules \(M\in \mathcal{D}(\mathcal{A}\otimes_k T^{op})\). The bimodule depends covariantly on the second argument. \(M(,i)\) is representable in \(\mathcal{A}\), for \(i=0,1,2\).
Definition 22. H3t = Homotopy category of 3-term complexes = full subcat of \(\mathcal{D}(..)\) whose objects are right representable dg-bimodules \(M\in \mathcal{D}(\mathcal{A}\otimes_k T^{op})\).
Remarks 9. Equivalently H3t = \(H^0(\mathrm{Fun}_{A_\infty}(T,\mathcal{A})) = H^0(\mathrm{Fun}_{A_\infty}(1\to \dots 2\dots\to 3,\mathcal{A}))\).
Definition 23. An exact structure on \(\mathcal{A}\) is a class of homotopy kernel-cokernel pairs \[X\xrightarrow{i} Y\xrightarrow{p} Z, h\]
(Ex-0) \(1_0\) is an inflation.
(Ex-1) Inflations are stable under composition.
(Ex-2) Any cospan
admits a homotopy pushout where \(i'\) is an inflation.
(Ex-2op) Any span
admits a homotopy pullback where \(p\) is a deflation.
Example 11.
If \(\mathcal{A}=\mathcal{A}^o\) then exact structure on \(\mathcal{A}\) coincides with the Quillen exact structure on \(\mathcal{A}^o\).
If \(\mathcal{A}\) is pretriangulated, then the class of all homotopy ker-cokernel pairs is an exact structure. (Up to isomorphism, there pairs are the sequence \(h\sim X\xrightarrow{f} Y\to \mathrm{Cone}(f)\))
\[\{\text{Exact structures on $\mathcal{A}^{op}$}\} \sim \{\text{Exact structures on $\mathcal{A}$}\}=_\text{trivial} \{\text{Exact structures on $\tau_{\le 0}\mathcal{A}$}\}\] Recall that \(\mathcal{A}\) quasi-equivalence is a dg-functor \(F:\mathcal{A}\to \mathcal{B}\) such that \(F\) is fully faithful (induces quasi-isoms in the morphisms of complexes) \[F:\mathcal{A}(x,y)\to \mathcal{B}(Fx,Fy) \in \text{qiso}\] \(H^0F:H^0\mathcal{A}\to H^0\mathcal{B}\) is an equivalence
If \(F:\mathcal{A}\to \mathcal{B}\) is a quasi-equiv, it induces a bijection \[\{\text{Exact structures on $\mathcal{A}$}\}\leftrightarrow \{\text{Exact structures on $\mathcal{B}$}\}.\] Remark: Positselsk also have a notion of ’exact DG-category’ that is different of complete different motivations and does not have this property. (Preprint 2110.08237 October 21, 140 page)
If \(\mathcal{A}\) is exact dg and \(\mathcal{B}\) is an extension closed (in terms of homotopy kernel and cokernel pairs) full subcat, then \(\mathcal{B}\) inherits an exact structure from \(\mathcal{A}\).
Some basic results
We know that triangulated categories have topological enhancement given by stable infinity categories and algebraic enhavment given by pretriangulated extrianglulated.
Theorem 8 (Chen). If \(\mathcal{A}\) is exact dg, then \(H^0\mathcal{A}\) is canonically extriangulated with \(\mathbb{E}^{H^0\mathcal{A}}\) parametrizing.
images of \(X\xrightarrow{\overline{i}} Y\xrightarrow{\overline{p}} Z\) in \(H^0\mathcal{A}\) of conflations of \(\mathcal{A}\), \(h\sim X\xrightarrow{i} Y\xrightarrow{p} Z\) in \(\mathcal{A}\) should be exactly equal to zero.
Example 12. \(A\) is a \(k\)-algebra, take dg category \(\mathcal{A}= \mathcal{C}_{dg}^{[-1,0]}(\mathrm{proj}A)\subset_\text{full dg} \mathcal{C}_{dg}^b(\mathrm{proj}A)\). With all ses of complexes (component wise split exact sequences). Then \(\mathcal{A}\) is extrianglulated and \(H^0\mathcal{A}= \mathcal{H}^{[-1,0]}(\mathrm{proj}A)\) is the homotopy category of complexes of projective modules.
Link to exact \(\infty\)-cat
\(\mathcal{A}\) dg cat, \(N_{dg}\mathcal{A}\) the dg-nerve of \(\mathcal{A}\) is a simplicial set with \(n\)-simplices are the trictly unital \(A_\infty\) functors \(k[0\to 1\to \dots \to n]\to \mathcal{A}\).
Lurie: \(N_{dg}\mathcal{A}\) is an \(\infty\)-category (quasi-cat).
Faonle : \(\mathcal{A}\) pretriangulated \(\Rightarrow N_{dg}\mathcal{A}\) is stable \(\infty\)-cat.
Suppose we have an extract dg structure on \(\mathcal{A}\), we hope that \(N_{dg}\mathcal{A}\) is a bounded exact dg-cat, this is indeed the case.
Theorem 9 (Chen).
\(N_{dg}\mathcal{A}\) becomes a Barwick exact \(\infty\)-category.
We have a canonical bijection \[\{\text{Exact structures on $\mathcal{A}$}\} \leftrightarrow \{\text{Barwick exact structure on $N_{dg}\mathcal{A}$}\}\] Note that \(N_{dg}\) is completely independent of the \(k\)-linear structure on \(\mathcal{A}\) since it comes from a free \(k\)-module.
Further results: An exact dg category \(\mathcal{A}\) has a good derived category \(\mathcal{D}^b_{dg}\mathcal{A}\), an exact dg categories do reproduce! the poset of all exact structures on \(\mathcal{A}\) is manageable. (Find details in the notes)
Now I would like to conclude by giving an application, a Lower-Auslander correspondence. It works very nicely using this framework.
Lower-Auslander Correspondence
First we have to define what zero Auslander categories should be.
Definition 24. An exact dg cat \(\mathcal{A}\) is zero-Auslander if \(H^0\mathcal{A}\) is zero-Auslander as an extriangulated category, i.e.
\(H^0\mathcal{A}\) has enough projectives and its global dimension \(\le 1\le\) dominate dimension of \(H^0\mathcal{A}\).
Example 13. Let’s take \(\mathcal{A}_1 = \mathcal{C}_{dg}^{[-1,0]}(\mathrm{proj}A)\) eg \(A = k[A_3]\), then \(H^0\mathcal{A}_1 = \mathcal{H}^{[-1,0]}(\mathrm{proj}A)\) is zero-Auslander. (Draw an AR-quiver of projectives and injectives in \(H^0\mathcal{A}_1\), there are no proj-injectives.)
Another example \(\mathcal{A}_2 = \mathcal{A}_2^{0} = \mathcal{C}_{dg}^{[-1,0]}(\mathrm{proj}A)\) concentrated in degree zero, \(A=k[A_3]/(A_3^2)\), then \(H^0\mathcal{A}_2\) is not zero-Auslander.
Another example, let \(\Lambda\) be the prepoje algebra of type \(A_2\), \(\mathcal{A}\) is the module of \(\Lambda\), with a nonstandard exact structure, \(T=P_1\oplus P_2\oplus S_1\) tilting object. \[0\to L\to M\to N\to 0\] is a conflation iff its image under \(\mathrm{Hom}(T,)\) is exact. \(\mathcal{A}\) is concentrated in degree \(0\) and is \(0\)-Auslander.
Theorem 10 (Chen). There is a canonical bijection between connective exact dg categories which are zero-Auslander / equiv, and pairs \((\mathcal{P},\mathcal{J})\) where \(\mathcal{P}\) is a connective additive dg category and \(\mathcal{J}\subset \mathcal{P}\) is a full dg subcategory.
Bijection : \[\mathcal{A}\mapsto (\mathcal{P}=\{\text{proj in }H^0\mathcal{A}\}, \mathcal{J}= \text{proj-inj in $H^0\mathcal{A}$})\]
Remarks 10. As a consequence, Chen proved the algebraic case of a conjecture by Fang-Gorsky-Palu-Plamondon-Pressland.