Related papers: Nested cobordisms, Cyl-objects and Temperley-Lieb …
We develop a graphical calculus for monoidal categories equipped with twisted pivotal structures, which are a generalization of pivotal structures originating from the study of orientation structures in the context of the Cobordism…
We give a finite presentation of the cobordism symmetric monoidal bicategory of (smooth, oriented) closed manifolds, cobordisms and cobordisms with corners as an extension of the bicategory of closed manifolds, cobordisms and…
We give an overview of recent developments in silting theory. After an introduction on torsion pairs in triangulated categories, we discuss and compare different notions of silting and explain the interplay with t-structures and…
We present a general framework for TQFT and related constructions using the language of monoidal categories. We construct a topological category C and an algebraic category D, both monoidal, and a TQFT functor is then defined as a certain…
In this paper we present the construction of a skeletal diagram category, which we call the square with bands category. This category extends the Temperley-Lieb (TL) category, where morphisms now include diagrams of embedded curves on…
We introduce the notion of noncompact (partial) silting and (partial) tilting sets and objects in any triangulated category D with arbitrary (set-indexed) coproducts. We show that equivalence classes of partial silting sets are in bijection…
We study cobordisms of nested manifolds, which are manifolds together with embedded submanifolds, which can themselves have embedded submanifolds, etc. We identify a nested analog of the Pontryagin-Thom construction. Moreover, when the…
We construct a certain `cobordism category' ${\cal D}$ whose morphisms are suitably decorated cobordism classes between similarly decorated closed oriented 1-manifolds, and show that there is essentially a bijection between…
Every smooth manifold contains particles which propagate. These form objects and morphisms of a category equipped with a functor to the category of Abelian groups, turning this into a 0+1 topological field theory. We investigate the…
Graham and Lehrer (1998) introduced a Temperley-Lieb category $\mathsf{\widetilde{TL}}$ whose objects are the non-negative integers and the morphisms in $\mathsf{Hom}(n,m)$ are the link diagrams from $n$ to $m$ nodes. The Temperley-Lieb…
We introduce a formalism based on a combinatorial notion of cell complex subject to an inclusion-reversing duality operation. Our main goal is to open the way for a functorial definition of field theories in a context where no manifold or…
(Co)bordisms of manifolds and maps are fundamental and important objects in algebraic and differential topology of manifolds and related studies were started by Thom etc.. Cobordisms of Morse functions were introduced and have been studied…
We study a special sort of 2-dimensional extended Topological Quantum Field Theories (TQFTs) which we call open-closed TQFTs. These are defined on open-closed cobordisms by which we mean smooth compact oriented 2-manifolds with corners that…
In this paper, we introduce a category of graded commutative rings with certain algebraic morphisms, to investigate the cobordism category of plumbed 3-manifolds. In particular, we define a non-associative distributive algebra that gives…
The Clifford algebra over the three-dimensional real linear space includes its linear structure and its exterior algebra, the subspaces spanned by multivectors of the same degree determine a gradation of the Clifford algebra. Through these…
For each braided category $\mathcal{C}$ we show that, under mild hypotheses, there is an associated category of "half braided algebras" and their bimodules internal to $\mathcal{C}$ which is not only monoidal but even braided and balanced.…
We define a simplicial category called the category of derived manifolds. It contains the category of smooth manifolds as a full discrete subcategory, and it is closed under taking arbitrary intersections in a manifold. A derived manifold…
Inspired by Morse theory, we introduce a topological stack Broken, which we refer to as the moduli stack of broken lines. We show that Broken can be presented as a Lie groupoid with corners and provide a combinatorial description of sheaves…
We associate a bivariant theory to any suitable oriented Borel-Moore homology theory on the category of algebraic schemes or the category of algebraic G-schemes. Applying this to the theory of algebraic cobordism yields operational…
Together with F. Morel, we have constructed in \cite{CR, Cobord1, Cobord2} a theory of {\em algebraic cobordism}, an algebro-geometric version of the topological theory of complex cobordism. In this paper, we give a survey of the…