In this paper, we consider a conilpotent coalgebra $C$ over a field $k$. Let $\Upsilon :\ C{{-\mathsf{Comod}}}\longrightarrow C^*{{-\mathsf{Mod}}}$ be the natural functor of inclusion of the category of $C$-comodules into the category of $C^*$-modules, and let $\Theta :\ C{{-\mathsf{Contra}}}\longrightarrow C^*{{-\mathsf{Mod}}}$ be the natural forgetful functor. We prove that the functor $\Upsilon$ induces a fully faithful triangulated functor on bounded (below) derived categories if and only if the functor $\Theta$ induces a fully faithful triangulated functor on bounded (above) derived categories, and if and only if the $k$-vector space $\textrm {Ext}_C^n(k,k)$ is finite-dimensional for all $n\ge 0$. We call such coalgebras “weakly finitely Koszul”.

Clans are categorical representations of generalized algebraic theories that contain more information than the finite-limit categories associated to the locally finitely presentable categories of models via Gabriel–Ulmer duality. Extending Gabriel–Ulmer duality to account for this additional information, we present a duality theory between clans and locally finitely presentable categories equipped with a weak factorization system of a certain kind.

Building on our previous work on enriched universal algebra, we define a notion of enriched language consisting of function and relation symbols whose arities are objects of the base of enrichment $\mathcal {V}$. In this context, we construct atomic formulas and define the regular fragment of our enriched logic by taking conjunctions and existential quantification of those. We then characterize $\mathcal {V}$-categories of models of regular theories as enriched injectivity classes in the $\mathcal {V}$-category of structures. These notions rely on the choice of an orthogonal factorization system $(\mathcal {E},\mathcal {M})$ on $\mathcal {V}$ which will be used, in particular, to interpret relation symbols and existential quantification.

Let G be a finite group whose order is not divisible by the characteristic of the ground field $\mathbb {F}$. We prove a decomposition of the Hochschild homology groups of the equivariant dg category $\mathscr {C}^G$ associated with the action of G on a small dg category $\mathscr {C}$ which admits finite direct sums. When, in addition, the ground field $\mathbb {F}$ is algebraically closed this decomposition is related to a categorical action of $\text {Rep}(G)$ on $\mathscr {C}^G$ and the resulting action of the representation ring $R_{\mathbb {F}}(G)$ on $HH_\bullet (\mathscr {C}^G)$.

Let $F:\; {\mathscr {C}} \to {\mathscr {E}} \ $ be a functor from a category $\mathscr {C} \ $ to a homological (Borceux–Bourn) or semi-abelian (Janelidze–Márki–Tholen) category $\mathscr {E}$. We investigate conditions under which the homology of an object $X$ in $\mathscr {C}$ with coefficients in the functor $F$, defined via projective resolutions in $\mathscr {C}$, remains independent of the chosen resolution. Consequently, the left derived functors of $F$ can be constructed analogously to the classical abelian case.

Our approach extends the concept of chain homotopy to a non-additive setting using the technique of imaginary morphisms. Specifically, we utilize the approximate subtractions of Bourn–Janelidze, originally introduced in the context of subtractive categories. This method is applicable when $\mathscr {C}$ is a pointed regular category with finite coproducts and enough projectives, provided the class of projectives is closed under protosplit subobjects, a new condition introduced in this article and naturally satisfied in the abelian context. We further assume that the functor $F$ meets certain exactness conditions: for instance, it may be protoadditive and preserve proper morphisms and binary coproducts—conditions that amount to additivity when $\mathscr {C}$ and $\mathscr {E}$ are abelian categories.

Within this framework, we develop a basic theory of derived functors, compare it with the simplicial approach, and provide several examples.

We extend the Auslander–Iyama correspondence to the setting of exact dg categories. By specializing it to exact dg categories concentrated in degree zero, we obtain a generalization of the higher Auslander correspondence for exact categories due to Ebrahimi–Nasr-Isfahani (in the case of exact categories with split retractions).

Let $\mathcal {A}$ be an abelian length category containing a d-cluster tilting subcategory $\mathcal {M}$. We prove that a subcategory of $\mathcal {M}$ is a d-torsion class if and only if it is closed under d-extensions and d-quotients. This generalises an important result for classical torsion classes. As an application, we prove that the d-torsion classes in $\mathcal {M}$ form a complete lattice. Moreover, we use the characterisation to classify the d-torsion classes associated to higher Auslander algebras of type $\mathbb {A}$, and give an algorithm to compute them explicitly. The classification is furthermore extended to the setup of higher Nakayama algebras.

Let Λ be an artin algebra and $\mathcal{M}$ be an n-cluster tilting subcategory of Λ-mod with $n \geq 2$. From the viewpoint of higher homological algebra, a question that naturally arose in Ebrahimi and Nasr-Isfahani (The completion of d-abelian categories. J. Algebra 645 (2024), 143–163) is when $\mathcal{M}$ induces an n-cluster tilting subcategory of Λ-Mod. In this article, we answer this question and explore its connection to Iyama’s question on the finiteness of n-cluster tilting subcategories of Λ-mod. In fact, our theorem reformulates Iyama’s question in terms of the vanishing of Ext and highlights its relation with the rigidity of filtered colimits of $\mathcal{M}$. Also, we show that ${\rm Add}(\mathcal{M})$ is an n-cluster tilting subcategory of Λ-Mod if and only if ${\rm Add}(\mathcal{M})$ is a maximal n-rigid subcategory of Λ-Mod if and only if $\lbrace X\in \Lambda-{\rm Mod}~|~ {\rm Ext}^i_{\Lambda}(\mathcal{M},X)=0 ~~~ {\rm for ~all}~ 0 \lt i \lt n \rbrace \subseteq {\rm Add}(\mathcal{M})$ if and only if $\mathcal{M}$ is of finite type if and only if ${\rm Ext}_{\Lambda}^1({\underrightarrow{\lim}}\mathcal{M}, {\underrightarrow{\lim}}\mathcal{M})=0$. Moreover, we present several equivalent conditions for Iyama’s question which shows the relation of Iyama’s question with different subjects in representation theory such as purity and covering theory.

We expand the theory of 2-classifiers, that are a 2-categorical generalization of subobject classifiers introduced by Weber. The idea is to upgrade monomorphisms to discrete opfibrations. We prove that the conditions of 2-classifier can be checked just on a dense generator. The study of what is classified by a 2-classifier is similarly reduced to a study over the objects that form a dense generator. We then apply our results to the cases of prestacks and stacks, where we can thus look just at the representables. We produce a 2-classifier in prestacks that classifies all discrete opfibrations with small fibres. Finally, we restrict such 2-classifier to a 2-classifier in stacks. This is the main ingredient of a proof that Grothendieck 2-topoi are elementary 2-topoi. Our results also solve a problem posed by Hofmann and Streicher when attempting to lift Grothendieck universes to sheaves.

In this paper, we investigate locally finitely presented pure semisimple (hereditary) Grothendieck categories. We show that every locally finitely presented pure semisimple (resp., hereditary) Grothendieck category $\mathscr {A}$ is equivalent to the category of left modules over a left pure semisimple (resp., left hereditary) ring when $\mathrm {Mod}(\mathrm {fp}(\mathscr {A}))$ is a QF-3 category, and every representable functor in $\mathrm {Mod}(\mathrm {fp}(\mathscr {A}))$ has finitely generated essential socle. In fact, we show that there exists a bijection between Morita equivalence classes of left pure semisimple (resp., left hereditary) rings $\Lambda $ and equivalence classes of locally finitely presented pure semisimple (resp., hereditary) Grothendieck categories $\mathscr {A}$ that $\mathrm {Mod}(\mathrm {fp}(\mathscr {A}))$ is a QF-3 category, and every representable functor in $\mathrm {Mod}(\mathrm {fp}(\mathscr {A}))$ has finitely generated essential socle. To prove this result, we study left pure semisimple rings by using Auslander’s ideas. We show that there exists, up to equivalence, a bijection between the class of left pure semisimple rings and the class of rings with nice homological properties. These results extend the Auslander and Ringel–Tachikawa correspondence to the class of left pure semisimple rings. As a consequence, we give several equivalent statements to the pure semisimplicity conjecture.

Condensed mathematics, developed by Clausen and Scholze over the last few years, proposes a generalization of topology with better categorical properties. It replaces the concept of a topological space by that of a condensed set, which can be defined as a sheaf for the coherent topology on a certain category of compact Hausdorff spaces. In this case, the sheaf condition has a fairly simple explicit description, which arises from studying the relationship between the coherent, regular, and extensive topologies. In this paper, we establish this relationship under minimal assumptions on the category, going beyond the case of compact Hausdorff spaces. Along the way, we also provide a characterization of sheaves and covering sieves for these categories. All results in this paper have been fully formalized in the Lean proof assistant.

The proof of Theorem 3.14 contains an unsubstantiated claim. To overcome this problem, we add a hypothesis to the statement of 3.14 and we provide a new valid proof. We adjust Theorem 3.15, Corollary 3.16, Proposition 4.23, Theorem 4.26, Corollary 4.29, and Corollary 4.32 accordingly.

We extend the notion of regular coherence from rings to additive categories and show that well-known consequences of regular coherence for rings also apply to additive categories. For instance, the negative K-groups and all twisted Nil-groups vanish for an additive category, if it is regular coherent. This will be applied to nested sequences of additive categories, motivated by our ongoing project to determine the algebraic K-theory of the Hecke algebra of a reductive p-adic group.

Given any toric subvariety Y of a smooth toric variety X of codimension k, we construct a length k resolution of ${\mathcal O}_Y$ by line bundles on X. Furthermore, these line bundles can all be chosen to be direct summands of the pushforward of ${\mathcal O}_X$ under the map of toric Frobenius. The resolutions are built from a stratification of a real torus that was introduced by Bondal and plays a role in homological mirror symmetry.

As a corollary, we obtain a virtual analogue of Hilbert’s syzygy theorem for smooth projective toric varieties conjectured by Berkesch, Erman and Smith. Additionally, we prove that the Rouquier dimension of the bounded derived category of coherent sheaves on a toric variety is equal to the dimension of the variety, settling a conjecture of Orlov for these examples. We also prove Bondal’s claim that the pushforward of the structure sheaf under toric Frobenius generates the derived category of a smooth toric variety and formulate a refinement of Uehara’s conjecture that this remains true for arbitrary line bundles.

We extend the Kechris–Pestov–Todorčević correspondence to weak Fraïssé categories and automorphism groups of generic objects. The new ingredient is the weak Ramsey property. We demonstrate the theory on several examples including monoid categories, the category of almost linear orders and categories of strong embeddings of trees.

Dualities of resolving subcategories of module categories over rings are introduced and characterized as dualities with respect to Wakamatsu tilting bimodules. By restriction of the dualities to smaller resolving subcategories, sufficient and necessary conditions for these bimodules to be tilting are provided. This leads to the Gorenstein version of both the Miyashita’s duality and Huisgen-Zimmermann’s correspondence. An application of resolving dualities is to show that higher algebraic K-groups and semi-derived Ringel–Hall algebras of finitely generated Gorenstein-projective modules over Artin algebras are preserved under tilting.

Let $p \geq 5$ be a prime number, and let $G = {\mathrm {SL}}_2(\mathbb {Q}_p)$. Let $\Xi = {\mathrm {Spec}}(Z)$ denote the spectrum of the centre Z of the pro-p Iwahori–Hecke algebra of G with coefficients in a field k of characteristic p. Let $\mathcal {R} \subset \Xi \times \Xi $ denote the support of the pro-p Iwahori ${\mathrm {Ext}}$-algebra of G, viewed as a $(Z,Z)$-bimodule. We show that the locally ringed space $\Xi /\mathcal {R}$ is a projective algebraic curve over ${\mathrm {Spec}}(k)$ with two connected components and that each connected component is a chain of projective lines. For each Zariski open subset U of $\Xi /\mathcal {R}$, we construct a stable localising subcategory $\mathcal {L}_U$ of the category of smooth k-linear representations of G.

This is the second installment in a series of papers applying descriptive set theoretic techniques to both analyze and enrich classical functors from homological algebra and algebraic topology. In it, we show that the Čech cohomology functors on the category of locally compact separable metric spaces each factor into (i) what we term their definable version, a functor taking values in the category $\mathsf {GPC}$ of groups with a Polish cover (a category first introduced in this work’s predecessor), followed by (ii) a forgetful functor from $\mathsf {GPC}$ to the category of groups. These definable cohomology functors powerfully refine their classical counterparts: we show that they are complete invariants, for example, of the homotopy types of mapping telescopes of d-spheres or d-tori for any $d\geq 1$, and, in contrast, that there exist uncountable families of pairwise homotopy inequivalent mapping telescopes of either sort on which the classical cohomology functors are constant. We then apply the functors to show that a seminal problem in the development of algebraic topology – namely, Borsuk and Eilenberg’s 1936 problem of classifying, up to homotopy, the maps from a solenoid complement $S^3\backslash \Sigma $ to the $2$-sphere – is essentially hyperfinite but not smooth.

Fundamental to our analysis is the fact that the Čech cohomology functors admit two main formulations: a more combinatorial one and a more homotopical formulation as the group $[X,P]$ of homotopy classes of maps from X to a polyhedral $K(G,n)$ space P. We describe the Borel-definable content of each of these formulations and prove a definable version of Huber’s theorem reconciling the two. In the course of this work, we record definable versions of Urysohn’s Lemma and the simplicial approximation and homotopy extension theorems, along with a definable Milnor-type short exact sequence decomposition of the space $\mathrm {Map}(X,P)$ in terms of its subset of phantom maps; relatedly, we provide a topological characterization of this set for any locally compact Polish space X and polyhedron P. In aggregate, this work may be more broadly construed as laying foundations for the descriptive set theoretic study of the homotopy relation on such spaces $\mathrm {Map}(X,P)$, a relation which, together with the more combinatorial incarnation of , embodies a substantial variety of classification problems arising throughout mathematics. We show, in particular, that if P is a polyhedral H-group, then this relation is both Borel and idealistic. In consequence, $[X,P]$ falls in the category of definable groups, an extension of the category $\mathsf {GPC}$ introduced herein for its regularity properties, which facilitate several of the aforementioned computations.

The main objective of the present paper is to present a version of the Tannaka–Krein type reconstruction theorems: if $F:{\mathcal B}\to {\mathcal C}$ is an exact faithful monoidal functor of tensor categories, one would like to realize ${\mathcal B}$ as category of representations of a braided Hopf algebra $H(F)$ in ${\mathcal C}$. We prove that this is the case iff ${\mathcal B}$ has the additional structure of a monoidal ${\mathcal C}$-module category compatible with F, which equivalently means that F admits a monoidal section. For Hopf algebras, this reduces to a version of the Radford projection theorem. The Hopf algebra is constructed through the relative coend for module categories. We expect this basic result to have a wide range of applications, in particular in the absence of fiber functors, and we give some applications. One particular motivation was the logarithmic Kazhdan–Lusztig conjecture.

We consider the homology theory of étale groupoids introduced by Crainic and Moerdijk [A homology theory for étale groupoids. J. Reine Angew. Math. 521 (2000), 25–46], with particular interest to groupoids arising from topological dynamical systems. We prove a Künneth formula for products of groupoids and a Poincaré-duality type result for principal groupoids whose orbits are copies of an Euclidean space. We conclude with a few example computations for systems associated to nilpotent groups such as self-similar actions, and we generalize previous homological calculations by Burke and Putnam for systems which are analogues of solenoids arising from algebraic numbers. For the latter systems, we prove the HK conjecture, even when the resulting groupoid is not ample.