A c-direct category is a small category equipped with an ordinal degree function such that every morphism is level or degree-raising. Every c-direct category is c-Reedy. The c-Reedy model structure on any functor category from a c-direct category to a model category coincides with the projective model structure. In this framework, a realization functor is a colimit-preserving functor satisfying some mild homotopical conditions from the category of presheaves on a c-direct category with cofibrant representables to a model category. We prove that any two such realization functors are weakly equivalent on cofibrant presheaves. For categories of cubes, we prove that thick categories have cofibrant representables. As an application, we introduce the $\varepsilon$-branching space of an $\mathcal A$-set for any thick category of cubes $\mathcal A$. It is obtained as a coend over a c-direct category with cofibrant representables constructed from $\mathcal A$. We prove that, on free $\mathcal A$-sets generated by precubical sets, this new definition coincides with the earlier one. We prove that, for cofibrant $\mathcal A$-sets, the resulting space is independent of $\varepsilon$ up to homotopy.

The theory of fiber bundles over small categories is developed, viewing them as locally constant functors to the category of small categories. The Grothendieck construction yields a total category equipped with a projection that is a bifibration. We show that, up to natural isomorphism, every such bundle admits a constant fiber, and that the monodromy gives a representation of the fundamental groupoid in the automorphism group of the fiber, which allows the classification of fiber bundles up to isomorphism. The gauge group of the bundle is proved to be isomorphic to the centralizer of the monodromy subgroup. We then give a precise analysis of sections and (lax) fixed points of the fiber bundle. Beat points for functors are introduced, and it is proved that every fiber bundle with some finiteness and acyclic conditions admits a minimal core, using a rigidity lemma for finite acyclic categories. These concepts are illustrated with explicit examples.

In this paper, we first introduce Nijenhuis Lie 2-algebras as the categorification of Nijenhuis Lie algebras. We prove that the category of Nijenhuis Lie 2-algebras is equivalent to the category of 2-term Nijenhuis $L_\infty$-algebras. Next, given a Nijenhuis Lie algebra, we introduce the notion of a 2-representation and show that the corresponding semidirect product inherits a Nijenhuis Lie 2-algebra structure. On the other hand, we consider a $2$-term representation up to homotopy of a Nijenhuis Lie algebra and obtain a $2$-term Nijenhuis $L_\infty$-algebra as the semidirect product. Finally, we show that the category of $2$-representations and the category of $2$-term representations up to homotopy of a Nijenhuis Lie algebra are equivalent.

We propose an $\infty$-categorical approach to Khovanov--Qi's Hopfological algebra that, in particular, refines several foundational aspects of the theory by recasting the previous constructions in terms of $\infty$-categories of modules in monoidal $\infty$-categories. This perspective leads to a more general variant of Hopfological algebra that takes place over an arbitrary rigidly-compactly generated symmetric monoidal stable $\infty$-category, which we also outline in the article. In the appendix, we compare the construction of Hopfological derived categories to that of Holm--Jørgensen's $Q$-shaped derived categories.

In 2018, Kalck and Yang showed that the singularity categories associated with $3$-dimensional Gorenstein quotient singularities are triangle equivalent (up to direct summands) to small cluster categories associated with McKay quivers with potential. We introduce higher McKay quivers with potential and generalize Kalck and Yang's theorem to arbitrary dimensions. The singularity categories we consider occur as the stable categories of categories of Cohen-Macaulay modules. We refine our description of the singularity categories by showing that these categories of Cohen-Macaulay modules are equivalent to Higgs categories in the sense of Wu. Moreover, we describe the singularity categories in the non-Gorenstein case.

In this paper, we prove tilting equivalence for the finite almost derived algebraic cobordism spectrum $\mathrm{dMGL}^{a,\rm fin}$ of perfectoid algebras. More precisely, if $V$ is an integral perfectoid valuation ring and $A$ is an integral perfectoid $V$-algebra, then the tilting functor induces a weak equivalence \[ \mathrm{dMGL}^{a,\rm fin}(A) \simeq \mathrm{dMGL}^{a,\rm fin}(A^\flat). \] This invariant is a finite syntomic, derived, and non-$\mathbb{A}^1$-local version of algebraic cobordism, designed to retain infinitesimal deformation data over mixed characteristic bases. To prove the result, we first establish the corresponding finite non-unital statement and isolate a form of excisive approximation for pointed $\infty$-categories, including non-presentable ones. In the locally finitely presentable case, this agrees with the framework of Heuts. We also define approximation functors along natural transformations and apply them to the comparison between periodic algebraic cobordism and homotopy $K$-theory, obtaining Bott periodicity and Gabber rigidity.

We develop a nilpotent approximation theory for Smith ideals, extending adic completion for commutative rings to monoid objects in locally presentable symmetric monoidal abelian categories and to $\mathbb{E}_\infty$-algebra objects in stable symmetric monoidal model categories. The main result is a formal completeness theorem: finite generation of a Smith ideal forces completeness of its nilpotent approximation. This gives a categorical analogue of the finite generation completeness phenomenon in classical adic completion, while remaining distinct from ordinary adic completion of quotient rings. As applications, we construct an almost mathematics version of nilpotent approximation and prove a homotopical completeness theorem for weakly compact Smith ideals. We then apply the general theory to motivic spectra. For the canonical morphism from algebraic cobordism to algebraic K-theory, we construct the corresponding K-theoretic nilpotent approximation of algebraic cobordism, prove its homotopical completeness and Bott periodicity, and establish a mod-$\ell$ Gabber rigidity theorem for the analogous approximation of $\mathbf{MGL}/\ell$ by $\mathbb{K}/l$.

There are many metrics available to compare phylogenetic trees since this is a fundamental task in computational biology. In this paper, we focus on one such metric, the $\ell^\infty$-cophenetic metric introduced by Cardona et al. This metric works by representing a phylogenetic tree with $n$ labeled leaves as a point in $\mathbb{R}^{n(n+1)/2}$ known as the cophenetic vector, then comparing the two resulting Euclidean points using the $\ell^\infty$ distance. Meanwhile, the interleaving distance is a formal categorical construction generalized from the definition of Chazal et al., originally introduced to compare persistence modules arising from the field of topological data analysis. We show that the $\ell^\infty$-cophenetic metric is an example of an interleaving distance. To do this, we define phylogenetic trees as a category of merge trees with some additional structure; namely labelings on the leaves plus a requirement that morphisms respect these labels. Then we can use the definition of a flow on this category to give an interleaving distance. Finally, we show that, because of the additional structure given by the categories defined, the map sending a labeled merge tree to the cophenetic vector is, in fact, an isometric embedding, thus proving that the $\ell^\infty$-cophenetic metric is, in fact, an interleaving distance.