Related papers: A constructive approach to higher homotopy operati…
Invertibility is an important concept in category theory. In higher category theory, it becomes less obvious what the correct notion of invertibility is, as extra coherence conditions can become necessary for invertible structures to have…
Finite-order invariants of knots in arbitrary 3-manifolds (including non-orientable ones) are constructed and studied by methods of the topology of discriminant sets. Obstructions to the integrability of admissible weight systems to…
We construct combinatorial model category structures on the categories of (marked) categories and (marked) pre-additive categories, and we characterize (marked) additive categories as fibrant objects in a Bousfield localization of…
Examples of non-trivial higher string topology operations have been regrettably rare in the literature. In this paper, working in the context of string topology of classifying spaces, we provide explicit calculations of a wealth of…
Multimodal normal incestual systems are investigated in terms of multiple categories. The different sorted composition of operators are exhibited as 2-cells in multiple categories built up from 2-categories giving rise to different axioms.…
This paper describes an issue that arises when inverting elements of the homotopy groups of an equivariant commutative ring. Equivariant commutative rings possess an enhanced multiplicative structure arising from the presence of "indexed…
We show that the free construction from multicategories to permutative categories is a categorically-enriched non-symmetric multifunctor. Our main result then shows that the induced functor between categories of algebras is an equivalence…
We decribe the correspondence between normalised $\omega$-operads and certain lax monoidal structures on the category of globular sets. As with ordinary monoidal categories, one has a notion of category enriched in a lax monoidal category.…
This paper studies the homotopy theory of the Grothendieck construction using model categories and semi-model categories, provides a unifying framework for the homotopy theory of operads and their algebras and modules, and uses this…
It is well known that strict $\omega$-categories, strict $\omega$-functors, strict natural $\omega$-transformations, and so on, form a strict $\omega$-category. A similar property for weak $\omega$-categories is one of the main hypotheses…
In this work we analyze the behavior of Massey products of closed manifolds under the blow-up construction. The results obtained in the article are applied to the problem of constructing closed symplectic non-formal manifolds. The proofs…
We construct a smash product operation on secondary homotopy groups yielding the structure of a lax symmetric monoidal functor. Applications on cup-one products, Toda brackets and Whitehead products are considered. In particular we prove a…
Opetopes are algebraic descriptions of shapes corresponding to compositions in higher dimensions. As such, they offer an approach to higher-dimensional algebraic structures, and in particular, to the definition of weak $\omega$-categories,…
We show that a system of unstable higher Toda brackets can be defined inductively.
We give sufficient conditions for the existence of a model structure on operads in an arbitrary symmetric monoidal model category. General invariance properties for homotopy algebras over operads are deduced.
One of the prime motivation for topology was Homotopy theory, which captures the general idea of a continuous transformation between two entities, which may be spaces or maps. In later decades, an algebraic formulation of topology was…
Rota-Baxter operators, $\mathcal{O}$-operators on Lie algebras and their interconnected pre-Lie and post-Lie algebras are important algebraic structures with applications in mathematical physics. This paper introduces the notions of a…
Homotopy optimization is a traditional method to deal with a complicated optimization problem by solving a sequence of easy-to-hard surrogate subproblems. However, this method can be very sensitive to the continuation schedule design and…
We initiate the study of model structures on (categories induced by) lattice posets, a subject we dub homotopical combinatorics. In the case of a finite total order $[n]$, we enumerate all model structures, exhibiting a rich combinatorial…
The transfer of the generating operations of an algebra to a homotopy equivalent chain complex produces higher operations. The first goal of this paper is to describe precisely the higher structure obtained when the unary operations commute…