A regular category is said to be normal when it is pointed and every regular epimorphism in it is a normal epimorphism. Any abelian category is normal, and in a normal category one can define short exact sequences in a similar way as in an abelian category. Then, the corresponding $3 \times 3$ lemma is equivalent to the so-called subtractivity, which in universal algebra is also known as congruence 0-permutability. In the context of non-pointed regular categories, short exact sequences can be replaced with “exact forks” and then, the corresponding $3 \times 3$ lemma is equivalent, in the universal algebraic terminology, to congruence 3-permutability; equivalently, regular categories satisfying such $3 \times 3$ lemma are precisely the Goursat categories. We show how these two seemingly independent results can be unified in the context of star-regular categories recently introduced in a joint work of A. Ursini and the first two authors.

We define quasi-locally presentable categories as big unions of a chain of coreflective subcategories that are locally presentable. Under appropriate hypotheses we prove a representability theorem for exact contravariant functors defined on a quasi-locally presentable category taking values in abelian groups. We show that the abelianization of a well generated triangulated category is quasi-locally presentable, and we obtain a new proof of the Brown representability theorem. Examples of functors that are not representable are also given.

We study matrix factorizations of regular global sections of line bundles on schemes. If the line bundle is very ample relative to a Noetherian affine scheme we show that morphisms in the homotopy category of matrix factorizations may be computed as the hypercohomology of a certain mapping complex. Using this explicit description, we prove an analogue of Orlov’s theorem that there is a fully faithful embedding of the homotopy category of matrix factorizations into the singularity category of the corresponding zero subscheme. Moreover, we give a complete description of the image of this functor.

We show that if a generator of a differential Gerstenhaber algebra satisfies certain Cartan-type identities, then the corresponding Lie bracket is formal. Geometric examples include the shifted de Rham complex of a Poisson manifold and the subcomplex of differential forms on a symplectic manifold vanishing on a Lagrangian submanifold, endowed with the Koszul bracket. As a corollary we generalize a recent result by Hitchin on deformations of holomorphic Poisson manifolds.

We explore a somewhat unexpected connection between knot Floer homology and shellable posets, via grid diagrams. Given a grid presentation of a knot $K$ inside $S^3$, we define a poset which has an associated chain complex whose homology is the knot Floer homology of $K$. We then prove that the closed intervals of this poset are shellable. This allows us to combinatorially associate a PL flow category to a grid diagram.

In this article we will define the notions of Hurwitz automorphism and comorphism of the ring of Hurwitz series. A Hurwitz automorphism is the analog of a Seidenberg automorphism of a power series ring when the characteristic of the underlying ring is not necessarily zero. We will show that the sets of all Hurwitz automorphisms, comorphisms, and derivations of the underlying ring are naturally isomorphic to one another.

We produce a cofibrantly generated simplicial symmetric monoidal model structure for the category of (small unital) $\mathrm{C}^*$-categories, whose weak equivalences are the unitary equivalences. The closed monoidal structure consists of the maximal tensor product, which generalizes that of $\mathrm{C}^*$-algebras, together with the Ghez-Lima-Roberts $\mathrm{C}^*$-categories of *-functors, $\mathrm{C}^*(A;B)$, providing the internal Hom’s.

In this article we endow Kontsevich’s triangulated category $\mathrm{KMM}_k$ of noncommutative mixed motives with a nondegenerate weight structure in the sense of Bondarko. As an application we obtain: (1) a convergent weight spectral sequence for every additive invariant (e.g., algebraic $K$-theory, cyclic homology, topological Hochschild homology, etc.); (2) a ring isomorphism between $K_0(\mathrm{KMM}_k)$ and the Grothendieck ring of the category of noncommutative Chow motives; (3) a precise relationship between Voevodsky’s (virtual) mixed motives and Kontsevich’s noncommutative (virtual) mixed motives.

We present an introduction to the equivariant slice filtration. After reviewing the definitions and basic properties, we determine the slice-connectivity of various families of naturally arising spectra. This leads to an analysis of pullbacks of slices defined on quotient groups, producing new collections of slices. Building on this, we determine the slice tower for the Eilenberg- Mac Lane spectrum associated to a Mackey functor for a cyclic p-group. We then relate the Postnikov tower to the slice tower for various spectra. Finally, we pose a few conjectures about the nature of slices and pullbacks.

We relate the construction of groups which realize saturated fusion systems and signaliser functors with homology decompositions of $p$-local finite groups. We prove that the cohomology ring of Robinson’s construction is in some precise sense very close to the cohomology ring of the fusion system it realizes.

In this paper we construct binary self-dual codes using the étale cohomology of $\mu_2$ on the spectra of rings of $S$-integers of global fields. We will show that up to equivalence, all selfdual codes of length at least 4 arise from Hilbert pairings on rings of $S$-integers of $\mathbb{Q}$. This is an arithmetic counterpart of a result of Kreck and Puppe, who used cobordism theory to show that all self-dual codes arise from Poincaré duality on real three manifolds.

We prove that the protocol complex of the immediate snapshot read/write complex for $n + 1$ processors is a simplicial subdivision of the input complex. Our proof is purely geometric, using the Schlegel diagram construction.

A kind of motivic algebra of spectral categories and modules over them is developed to introduce $K$-motives of algebraic varieties. As an application, bivariant algebraic $K$-theory $K(X; Y)$ as well as bivariant motivic cohomology groups $H^{p;q}(X; Y; \mathbb{Z})$ are defined and studied. We use Grayson’s machinery to produce the Grayson motivic spectral sequence connecting bivariant $K$-theory to bivariant motivic cohomology. It is shown that the spectral sequence is naturally realized in the triangulated category of $K$-motives constructed in the paper. It is also shown that ordinary algebraic $K$-theory is represented by the $K$-motive of the point.

We present different ways of endowing a particular category of graphs with Quillen model structures. We show, among other things, that the core of a graph can be seen as its homotopy type in an appropriate Quillen model structure, and that an infinity of Quillen model structures exist for our particular category of graphs.