We introduce crossed interval groups and construct a crossed interval analog IS of the Fiedorowicz-Loday symmetric category ∆S. We prove that the functor FS(−) of the free IS-extension of an I-object does not change homotopy type. We then observe that the operad B of natural operations on the Hochschild cohomology equals FS(T ), where T is an operad whose homotopy type is known. We conclude fro...

Since the classical iterative methods for solving nonlinear ill-posed problems are locally convergent, this paper constructs a robust and widely convergent method for identifying parameter based on homotopy algorithm, and investigates this method’s convergence in the light of Lyapunov theory. Furthermore, we consider 1-D elliptic type equation to testify that the homotopy regularization can ide...

Link homotopy has been an active area of research for knot theorists since its introduction by Milnor in the 1950s. We introduce a new equivalence relation on spatial graphs called component homotopy, which reduces to link homotopy in the classical case. Unlike previous attempts at generalizing link homotopy to spatial graphs, our new relation allows analogues of some standard link homotopy res...

Homotopy type theory is an extension of Martin-Löf type theory, based on a correspondence with homotopy theory and higher category theory. In homotopy type theory, the propositional equality type becomes proof-relevant, and corresponds to paths in a space. This allows for a new class of datatypes, called higher inductive types, which are specified by constructors not only for points but also fo...

With a view toward studying the homotopy type of spaces of Boolean formulae, we introduce a simplicial complex, called the theta complex, associated to any hypergraph. In particular, the set of satisfiable formulae in k-conjunctive normal form with ≤ n variables has the homotopy type of Θ(Cube(n, n− k)), where Cube(n, n− k) is a hypergraph associated to the (n− k)-skeleton of an n-cube. We make...

We introduce a new way of formalizing the intensional identity type based on the notion of computational paths which will be taken to be proofs of propositional equality, and thus terms of the identity type. Our approach results in an elimination rule different than the one given by Martin-Löf in his intensional identity type. In order to show the validity and power of our approach, we formulat...

Univalent homotopy type theory (HoTT) may be seen as a language for the category of ∞-groupoids. It is being developed as a new foundation for mathematics and as an internal language for (elementary) higher toposes. We develop the theory of factorization systems, reflective subuniverses, and modalities in homotopy type theory, including their construction using a “localization” higher inductive...

One of the most interesting entities of homotopy type theory is the identity type. It gives rise to an interesting interpretation of the equality, since one can semantically interpret the equality between two terms of the same type as a collection of homotopical paths between points of the same space. Since this is only a semantical interpretation, the addition of paths to the syntax of homotop...

In a type-theoretic fibration category in the sense of Shulman (representing a dependent type theory with at least 1, Σ, Π, and identity types), we define the type of coherently constant functions A ω −→ B. This involves an infinite tower of coherence conditions, and we therefore need the category to have Reedy limits of diagrams over ωop. Our main result is that, if the category further has pr...

Spaces of maps into classifying spaces for equivariant crossed complexes, II: The general topological group case. Abstract The results of a previous paper [3] on the equivariant homotopy theory of crossed complexes are generalised from the case of a discrete group to general topological groups. The principal new ingredient necessary for this is an analysis of homotopy coherence theory for cross...