Related papers: The augmentation category map induced by exact Lag…

We give a presentation of the $n$-dimensional oriented cobordism category $\text{Cob}_n$ with generators corresponding to diffeomorphisms and surgeries along framed spheres, and a complete set of relations. Hence, given a functor $F$ from…

We present a categorical model for intuitionistic linear logic where objects are polynomial diagrams and morphisms are simulation diagrams. The multiplicative structure (tensor product and its adjoint) can be defined in any locally…

We introduce a functor from cochain complexes to bicomplexes, called inflation functor, which sends quasi-isomorphisms to the class of pluripotential weak equivalences. We show this functor is part of a Quillen adjunction. Its right adjoint…

Expansions of abelian categories are introduced. These are certain functors between abelian categories and provide a tool for induction/reduction arguments. Expansions arise naturally in the study of coherent sheaves on weighted projective…

The main purpose of this paper is to introduce a new category, which we call a resonance category, whose combinatorics reflect that of canonical stratifications of $n$-fold symmetric smash products. The study of the stratifications can then…

Any link that is the closure of a positive braid has a natural Legendrian representative. These were introduced in an earlier paper, where their Chekanov--Eliashberg contact homology was also evaluated. In this paper we re-phrase and…

We study the relationship between cartesian bicategories and a specialisation of Lawvere's hyperdoctrines, namely elementary existential doctrines. Both provide different ways of abstracting the structural properties of logical systems: the…

Let $\mathcal C$ be a category with finite colimits, and let $(\mathcal E,\mathcal M)$ be a factorisation system on $\mathcal C$ with $\mathcal M$ stable under pushouts. Writing $\mathcal C;\mathcal M^{\mathrm{op}}$ for the symmetric…

We prove that there are precisely two embedded exact Lagrangian fillings of the standard Legendrian Hopf link, up to compactly supported Hamiltonian isotopy. It was known that the standard Legendrian Hopf link admitted at least two such…

We construct an enhanced version of knot contact homology, and show that we can deduce from it the group ring of the knot group together with the peripheral subgroup. In particular, it completely determines a knot up to smooth isotopy. The…

Let $M$ be a closed manifold. We introduce a family of Legendrian isotopy invariants for Legendrians in $J^1M$, which we collectively call Legendrian higher torsion. Given a choice of a class $\mathcal{F}$ of fibre bundles over $M$,…

The paper establishes new relationship between cohomology, extensions and automorphisms of quandles. We derive a four term exact sequence relating quandle 1-cocycles, second quandle cohomology and certain group of automorphisms of an…

Let $T$ be a right exact functor from an abelian category $\mathscr{B}$ into another abelian category $\mathscr{A}$. Then there exists a functor ${\bf p}$ from the product category $\mathscr{A}\times\mathscr{B}$ to the comma category…

We prove that a lax $\mathbb{E}_{n+1}$-monoidal functor from $\mathcal V$ to $\mathcal W$ induces a lax $\mathbb{E}_n$-monoidal functor from $\mathcal V$-enriched $\infty$-categories to $\mathcal W$-enriched $\infty$-categories in the sense…

We present two different constructions of invariants for Legendrian knots in the standard contact space $\R^3$. These invariants are defined combinatorially, in terms of certain planar projections, and are useful in distinguishing…

A tangent category is a category with an endofunctor, called the tangent bundle functor, which is equipped with various natural transformations that capture essential properties of the classical tangent bundle of smooth manifolds. In this…

The relative cell complexes with respect to a generating set of cofibrations are an important class of morphisms in any model structure. In the particular case of the standard (algebraic) model structure on $\textbf{Top}$, we give a new…

We prove that each overtwisted contact structure has knot types that are represented by infinitely many distinct transverse knots all with the same self-linking number. In some cases, we can even classify all such knots. We also show…

We prove that Lagrangian cocores and Lagrangian linking disks of a stopped Weinstein manifold generate the Lagrangian cobordism infinity-category. As a geometric consequence, we see that any brane (after stabilization) admits a Lagrangian…

For a given family $\{(\mathrm{q}_i, \mathrm{t}_i, \mathrm{p_i} )\}_{i \in I}$ of adjoint triples between exact categories $\mathcal{C}$ or $\mathcal{D}$, we show that any cotorsion pair in $\mathcal{C}$ and $\mathcal{D}$ yield two…