Journal of Homotopy and Related Structures最新文献

In this paper, we develop the (A_infty )-analog of the Maurer-Cartan simplicial set associated to an (L_infty )-algebra and show how we can use this to study the deformation theory of (infty )-morphisms of algebras over non-symmetric operads. More precisely, we first recall and prove some of the main properties of (A_infty )-algebras like the Maurer-Cartan equation and twist. One of our main innovations here is the emphasis on the importance of the shuffle product. Then, we define a functor from the category of complete (curved) (A_infty )-algebras to simplicial sets, which sends a complete curved (A_infty )-algebra to the associated simplicial set of Maurer-Cartan elements. This functor has the property that it gives a Kan complex. In all of this, we do not require any assumptions on the field we are working over. We also show that this functor can be used to study deformation problems over a field of characteristic greater than or equal to 0. As a specific example of such a deformation problem, we study the deformation theory of (infty )-morphisms of algebras over non-symmetric operads.

We prove that four different ways of defining Cartesian fibrations and the Cartesian model structure are all Quillen equivalent:

1. 1.

   On marked simplicial sets (due to Lurie [31]),

2. 2.

   On bisimplicial spaces (due to deBrito [12]),

3. 3.

   On bisimplicial sets,

4. 4.

   On marked simplicial spaces.

The main way to prove these equivalences is by using the Quillen equivalences between quasi-categories and complete Segal spaces as defined by Joyal–Tierney and the straightening construction due to Lurie.

In 1947, N.E. Steenrod defined the Steenrod Squares, which are mod 2 cohomology operations, using explicit cochain formulae for cup-i products of cocycles. He later recast the construction in more general homological terms, using group homology and acyclic model methods, rather than explicit cochain formulae, to define mod p operations for all primes p. Steenrod’s student J. Adem applied the homological point of view to prove fundamental relations, known as the Adem relations, in the algebra of cohomology operations generated by the Steenrod operations. In this paper we give a proof of the mod 2 Adem relations at the cochain level. Specifically, given a mod 2 cocycle, we produce explicit cochain formulae whose coboundaries are the Adem relations among compositions of Steenrod Squares applied to the cocycle, using Steenrod’s original cochain definition of the Square operations.

Let R be a right noetherian ring. We introduce the concept of relative singularity category (Delta _{mathcal {X} }(R)) of R with respect to a contravariantly finite subcategory (mathcal {X} ) of ({text {{mod{-}}}}R.) Along with some finiteness conditions on (mathcal {X} ), we prove that (Delta _{mathcal {X} }(R)) is triangle equivalent to a subcategory of the homotopy category (mathbb {K} _mathrm{{ac}}(mathcal {X} )) of exact complexes over (mathcal {X} ). As an application, a new description of the classical singularity category (mathbb {D} _mathrm{{sg}}(R)) is given. The relative singularity categories are applied to lift a stable equivalence between two suitable subcategories of the module categories of two given right noetherian rings to get a singular equivalence between the rings. In different types of rings, including path rings, triangular matrix rings, trivial extension rings and tensor rings, we provide some consequences for their singularity categories.

The equivalence between dg duality and Verdier duality has been established for cyclic operads earlier. We propose a generalization of this correspondence from cyclic operads and dg duality to twisted modular operads and the Feynman transform. Specifically, for each twisted modular operad (mathcal {P}) (taking values in dg-vector spaces over a field k of characteristic 0), there is a certain sheaf (mathcal {F}) associated with it on the moduli space of stable metric graphs such that the Verdier dual sheaf (Dmathcal {F}) is associated with the Feynman transform (Fmathcal {P}) of (mathcal {P}). In the course of the proof, we also prove a relation between cyclic operads and modular operads originally proposed in the pioneering work of Getzler and Kapranov; however, to the best knowledge of the author, no proof has appeared. This geometric interpretation in operad theory is of fundamental importance. We believe this result will illuminate many aspects of the theory of modular operads and find many applications in the future. We illustrate an application of this result, giving another proof on the homotopy properties of the Feynman transform, which is quite intuitive and simpler than the original proof.

As a step towards understanding the (mathrm {tmf})-based Adams spectral sequence, we compute the K(1)-local homotopy of (mathrm {tmf}wedge mathrm {tmf}), using a small presentation of (L_{K(1)}mathrm {tmf}) due to Hopkins. We also describe the K(1)-local (mathrm {tmf})-based Adams spectral sequence.