Finiteness for 2-dimensional contractible complexes

While thinking about graph-complex and related operadic stuff, I found a quite interesting (at least for me) question. However, I’m a novice in the algebraic topology, so I’m unable to resolve it by myself. Definition Let us call a (pure) n-dimensional polyhedral complex the topological space glued from a finite number of n-dimensional (convex) polyhedra … Read more

Invert quasi-isomorphisms of symmetric cooperads

The theory of symmetric operads in chain complexes (say over a good enough field) is in some sense nice, because we have a well defined homotopy theory. In particular we have a notion of infinity-morphisms of operads (maybe called homotopy morphisms instead), which can be defined as a cooperad map between the appropriate bar constructions … Read more

Moduli spaces for the TCFT map HH(L)→GW(X)HH(L) \to GW(X)

Let L be a Lagrangian submanifold of a closed symplectic manifold X. What I gather from Costello (see specifically §2.5 there), is that one expects to have a morphism of closed TCFT’s Φ→Ψ where Φ is a closed TCFT derived from the unital CY A∞ algebra A of L. It sends the generator object to … Read more

Dyer–Lashof operations for more than 2 inputs

Let O be a topological operad and X an algebra over it. Let the base ring be Z2. If C∗ denotes the singular chain complex over Z2, the action of O gives us morphisms μ:C∗(O(r))⊗rC∗(X)⊗r→C∗(X) where ⊗r means the tensor product over the group algebra Z2Sr of the rth symmetric group. Now consider the covering … Read more

Planar dendroidal sets?

The meta picture is: (non-planar) dendroidal sets are to symmetric colored operads as simplicial sets are to categories. This suggests that one should have the notion of planar dendroidal sets (with a straightforwards definition?) and they should be in the same relation as above with non-symmetric colored operad. Is the theory of planar dendroidal sets … Read more

Is there a 1-categorical treatment of operadic left Kan extensions in the literature?

Lurie develops in Section 3.1.2 of Higher Algebra a notion of operadic left Kan extension used to compute free algebras, giving a left adjoint $\mathrm{Alg}_{\mathcal{O}}(\mathcal{C})\to\mathrm{Alg}_{\mathcal{O}’}(\mathcal{C})$ to the forgetful functor $\mathrm{Alg}_{\mathcal{O}}(\mathcal{C})\to\mathrm{Alg}_{\mathcal{O}’}(\mathcal{C})$ associated to any map of $\infty$-operads $\mathcal{O}\to\mathcal{O}’$. There’s a number of constructions in the $1$-categorical setting that resemble this notion a bit, including monoidal Kan … Read more

An example for a construction on monads/operads?

Suppose that C is a bicategory. (I only need a monoidal category, i.e. one object bicategory, but I will stick with bicategories, since theory of monads is more commonly stated in that setting). A monad (X, A) in C is defined as a monoid object (A : X \rightarrow X, p^A : AA \rightarrow A, … Read more

Comparing cobar constructions for different types of (co)operads (e.g. cyclic vs. non-cyclic)

(I have already asked this on Math.SE and was told that it may be a better idea to post it here.) I am trying to understand a remark made by Clément Dupont and Bruno Vallete in their preprint “Brown’s moduli spaces of curves and the gravity operad” in the proof of Proposition 1.16: They say … Read more

3-Gerstenhaber algebra structure on the cohomology of deformation complexes?

In a seminal paper “On the Deformation of Rings and Algebras“, M. Gerstenhaber showed that the deformation complex of any associative algebra (known as the Hochschild complex) is naturally endowed with a structure of (later called) Gerstenhaber algebra (consisting of a degree 0 commutative associative product together with a compatible degree -1 Lie bracket). The … Read more