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Categorified Koszul duality of algebras

This paper develops a categorified generalization of Koszul duality for monoidal stable $\infty$ -categories, establishing duality results for module $\infty$ -categories associated with Artin algebras and algebras over the little 2-discs operad that connect to complete t-structures and Ind-coherent modules.

Original authors: Isamu Iwanari

Published 2026-08-13

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: Isamu Iwanari

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine the universe of mathematics as a giant, bustling library where every book is a set of rules for building things. In the "Algebra" section of this library, mathematicians have long known a magical trick called Koszul duality. Think of it as a special translator that takes a complicated set of instructions for building a tower (an algebra) and instantly converts it into a completely different set of instructions for building a bridge (its dual). Surprisingly, if you translate the bridge back, you get the original tower right back again. This trick has been a superpower for connecting different areas of math, from geometry to physics, because it reveals that two things looking totally different are actually secret twins.

But what if we stop looking at single books and start looking at entire libraries of rules? What if the "algebra" isn't just a list of numbers, but a whole universe of shapes and movements? This is the world of stable $\infty$ -categories. If a normal algebra is like a single Lego instruction manual, a stable $\infty$ -category is like the entire Lego factory, complete with all the machines, the workers, and the infinite ways you can combine bricks. The big question mathematicians have been asking is: Does our magical translator still work when we upgrade from a single manual to the whole factory? Can we translate an entire Lego factory into a different kind of factory and get a perfect mirror image back?

This paper, written by Isamu Iwanari, says yes, but with a twist that makes the story even more interesting. The author develops a "categorified" version of Koszul duality, which means they upgraded the translator to handle these massive, complex factories (monoidal stable $\infty$ -categories) instead of just simple instruction manuals.

Here is what the paper discovers. The author focuses on a specific type of algebraic structure called an Artin algebra (think of these as very well-behaved, finite Lego sets). They take the "factory" of modules (the things built using these rules) and run it through their new, upgraded translator. The result is a new factory that looks like a collection of Ind-coherent sheaves. To use a metaphor: if the original factory was a perfectly organized warehouse of specific, finite Lego sets, the translated factory is a massive, infinite warehouse that contains every possible way those sets could be stretched, combined, or expanded, while still keeping the core structure intact.

The paper proves two main things. First, if you take this new, massive factory and translate it back using the same tool, you get the original factory back, but only after you "right complete" it. In our analogy, "right completion" is like making sure the factory has all the necessary safety rails and infinite storage space to handle the most complex builds. Once you add those, the translation is a perfect match: the original factory and the translated-back factory are identical twins.

Second, the paper reveals that this new factory (the Ind-coherent sheaves) isn't just a random mess; it has a very specific, beautiful structure. It turns out that this translated factory is exactly the same as the factory of "Ind-coherent sheaves" on the original algebra. This is a big deal because it connects two seemingly different worlds: the world of algebraic modules and the world of geometric sheaves (which are like maps describing how shapes fit together). The author shows that the "dual" of a module factory is actually a sheaf factory, and they prove exactly how the rules of one turn into the rules of the other.

The paper is very careful about what it doesn't claim. It doesn't say this works for every single type of algebra in existence; it specifically proves it for Artin algebras (and related complete algebras). It also notes that the translation isn't always a perfect, instant swap without some extra steps; you often have to perform that "right completion" (adding the safety rails) to make the pieces fit perfectly. The author doesn't just guess or simulate this; they provide a rigorous mathematical proof that these connections hold true.

In the end, this work is like finding a new, universal language that allows mathematicians to talk between the "algebra" side of the library and the "geometry" side without losing any meaning. It shows that the deep, hidden connections Koszul duality found in simple algebras are actually part of a much grander pattern that governs entire universes of mathematical objects. By proving that these massive, complex factories can be translated back and forth with precision, the paper opens the door to using these powerful tools in new areas, like studying how these mathematical structures can change or "deform" over time, which is a hot topic in modern math.

Technical Summary: Categorified Koszul Duality of Algebras

1. Problem and Context

Koszul duality is a classical duality between augmented associative algebras, establishing deep connections between seemingly distinct algebraic objects. In the classical setting, for an augmented associative $k$ -algebra $A$ , the Koszul dual $D_1(A)$ is defined as the derived endomorphism algebra $\text{End}_A(k)$ . Under favorable conditions, the biduality map $A \to D_1(D_1(A))$ is an equivalence, and a Fourier–Morita-type equivalence exists between the categories of modules over $A$ and $D_1(A)$ .

This paper addresses the categorification of this duality. The central problem is to generalize Koszul duality from algebras to monoidal stable $\infty$ -categories. In this framework, monoidal stable $\infty$ -categories play the role of associative algebras, and module $\infty$ -categories play the role of modules. Specifically, the paper investigates the duality phenomena among monoidal stable $\infty$ -categories associated with Artin algebras and algebras over the little 2-discs operad ( $E_2$ -algebras).

2. Methodology and Framework

The author employs the language of $(\infty, 1)$ -categories (quasi-categories) and higher algebra, utilizing the theory of operads and stable $\infty$ -categories.

2.1. Categorified Augmentation and Dual

Let $k$ be a base field and $\text{Mod}_k$ the stable $\infty$ -category of $k$ -module spectra. The paper considers a monoidal $\infty$ -category $\mathcal{A}$ equipped with a monoidal functor $\mathcal{A} \to \text{Mod}_k$ , viewed as a categorified analog of an algebra augmentation $A \to k$ .
The Koszul dual of $\mathcal{A}$ is defined as the endomorphism monoidal $\infty$ -category:
$D(\mathcal{A}) = \text{End}_{\mathcal{A}}(\text{Mod}_k)$
This category classifies $\mathcal{A}$ -module functors $\text{Mod}_k \to \text{Mod}_k$ , with the monoidal structure induced by composition.

2.2. Focus on Artin and $E_2$ -Algebras

The study focuses on the case where $\mathcal{A} = \text{LMod}_A$ , the stable $\infty$ -category of left modules over an augmented $E_2$ -algebra $A$ (or an Artin algebra). The paper utilizes:

$E_n$ -Koszul Duality: The classical duality $D_n$ for augmented $E_n$ -algebras.
Ind-coherent Sheaves: The $\infty$ -category $\text{Ind}(\text{LCoh}(A))$ of Ind-coherent sheaves on $A$ , which generalizes coherent sheaves to the non-connective and derived setting.
$t$ -structures: A detailed analysis of left and right complete $t$ -structures, particularly the "standard" and "dual standard" $t$ -structures on module categories and categories of Ind-coherent sheaves.
Algebraization: The use of the functor $\Theta_k$ (left adjoint to the endomorphism algebra functor $E_k$ ) to relate monoidal $\infty$ -categories back to algebras.

3. Key Contributions and Results

3.1. Identification of the Koszul Dual with Ind-coherent Sheaves

The paper establishes a canonical equivalence between the Koszul dual of the category of modules over an $E_2$ -Koszul dual algebra and the category of Ind-coherent sheaves.

Theorem 1.2 (Theorem 3.6): If $A$ is an Artin $E_2$ -algebra, there is a canonical equivalence of monoidal $\infty$ -categories:
$D(\text{LMod}_{D_2(A)}) \simeq \text{Ind}(\text{LCoh}(A))$
Here, $D_2(A)$ is the $E_2$ -Koszul dual of $A$ . This result identifies the monoidal $\infty$ -category of Ind-coherent sheaves as the categorified Koszul dual of the $\infty$ -category of ordinary left $D_2(A)$ -modules. The monoidal structure on $\text{Ind}(\text{LCoh}(A))$ is interpreted as the composition of functors $\text{Mod}_k \to \text{Mod}_k$ .

3.2. Biduality and Right Completion

The paper investigates the biduality map for module categories. Unlike the classical case where $A \simeq D_1(D_1(A))$ holds directly, the categorified version requires completion.

Theorem 1.1 (Theorem 4.13): If $A$ is Artin, the biduality map
$\delta_{\text{LMod}_A} : \text{LMod}_A \to D(D(\text{LMod}_A))$
becomes an equivalence after passing to the right completion of the target. Specifically, if $D_t^+$ denotes the composition of the Koszul dual functor $D$ followed by right completion, then:
$\text{LMod}_A \simeq D_t^+(D_t^+(\text{LMod}_A))$
This is a categorical generalization of the classical biduality $A \simeq D_1(D_1(A))$ .

3.3. Left Completion and $E_2$ -Modules

The paper relates the category of modules over the $E_2$ -Koszul dual to the left completion of the original module category.

Theorem 1.3: There exists an equivalence of monoidal $\infty$ -categories:
$\widehat{b}_L(\text{LMod}_{D_2(A)}) \simeq D_t^+(D_t^+(\text{LMod}_{D_2(A)}))$
where $\widehat{b}_L$ denotes the left completion with respect to the standard $t$ -structure. The paper notes that the standard $t$ -structure on $\text{LMod}_{D_2(A)}$ is generally not left complete.

3.4. Structural Features and $t$ -structures

The results reveal subtle structural features of categorified duality:

The Koszul dual $D(\text{LMod}_A)$ is related to the left completion of $\text{LMod}_{D_2(A)}$ .
The paper distinguishes between the "standard" $t$ -structure (determined by the connective part) and the "dual standard" $t$ -structure (determined by the smallest subcategory containing the unit and closed under colimits).
It is shown that for certain Artin algebras (e.g., trivial square zero extensions), the $t$ -structure on the dual category is not right separated, and thus not right complete, necessitating the right completion functor in the biduality theorem.

4. Significance and Claims

The paper claims that its results provide a categorical generalization of classical Koszul duality for modules. By replacing modules with categories and sheaves with sheaves of categories, the work extends the duality paradigm to the realm of stable $\infty$ -categories.

Key aspects of the significance highlighted in the text include:

Geometric Interpretation: The equivalence $D(\text{LMod}_{D_2(A)}) \simeq \text{Ind}(\text{LCoh}(A))$ provides a geometric interpretation of the monoidal structure of Ind-coherent sheaves as functor composition.
Refinement of Duality: The results demonstrate that categorified Koszul duality possesses more subtle features than the duality between $A$ and $D_2(A)$ . Specifically, the dual of the module category is not simply the module category of the dual algebra, but rather the category of Ind-coherent sheaves.
Connection to Deformation Theory: The paper notes that these results are motivated by and connected to categorical invariants of Landau–Ginzburg models (e.g., matrix factorizations) and the deformation theory of stable $\infty$ -categories. The right adjoint functor $\kappa$ constructed in the introduction plays a role in associating Hochschild cohomology classes to deformations of stable $\infty$ -categories with right complete $t$ -structures.
Generalization: The framework is extended to pointed formal $E_2$ -stacks (Section 6), suggesting that the biduality equivalence holds for categorified structure sheaves on these stacks.

The paper maintains a modest tone regarding applications, stating that the application to deformation theory (specifically involving non-left complete $t$ -structures) "will be discussed in future work." The primary contribution remains the rigorous establishment of the duality equivalences and the identification of the necessary completions (left and right) for these equivalences to hold in the $\infty$ -categorical setting.