Related papers: The James construction and $\pi_4(\mathbb{S}^3)$ i…
The goal of this thesis is to prove that $\pi_4(S^3) \simeq \mathbb{Z}/2\mathbb{Z}$ in homotopy type theory. In particular it is a constructive and purely homotopy-theoretic proof. We first recall the basic concepts of homotopy type theory,…
A recent pre-print of W\"arn gives a novel pen-and-paper construction of a type family characterizing the path spaces of an arbitrary pushout, and a natural language argument for its correctness. We present the first formalization of the…
We consider the category Grpd(Asm$(A)$) of groupoids defined internally to the category of assemblies on a partial combinatory algebra $A$. In this thesis we exhibit the structure of a $\pi$-tribe on Grpd(Asm$(A)$) showing the category to…
Real numbers in constructive mathematics have always seemed to require compromises of one form or another. Classical proofs of Cauchy completeness require countable choice, Bishop's setoid construction introduces persistent bookkeeping…
In a recent paper, the second author and Joana Cirici proved a theorem that says that given appropriate hypotheses, $n$-formality of a differential graded algebraic structure is equivalent to the existence of a chain-level lift of a…
In this paper, we determine the homotopy groups \pi_4(\Sigma K(A,1)) and \pi_5(\Sigma K(A,1)) for abelian groups A by using different facts and methods from group theory and homotopy theory: derived functors, the Carlsson simplicial…
Given a span of spaces, one can form the homotopy pushout and then take the homotopy pullback of the resulting cospan. We give a concrete description of this pullback as the colimit of a sequence of approximations, using what we call the…
Homotopy type theory is an interpretation of Martin-L\"of's constructive type theory into abstract homotopy theory. There results a link between constructive mathematics and algebraic topology, providing topological semantics for…
In homotopy type theory we can define the join of maps as a binary operation on maps with a common co-domain. This operation is commutative, associative, and the unique map from the empty type into the common codomain is a neutral element.…
We consider various $A_{\infty}$-algebras of differential (super)forms, which are related to gauge theories and demonstrate explicitly how certain reformulations of gauge theories lead to the transfer of the corresponding…
We use derived localization of the bar and nerve constructions to provide simple proofs of a number of results in algebraic topology. This includes a recent generalization of Adams' cobar-construction to the non-simply connected case, and a…
We develop formal theories of conversion for Church-style lambda-terms with Pi-types in first-order syntax using one-sorted variables names and Stoughton's multiple substitutions. We then formalize the Pure Type Systems along some…
This paper proves that the homotopy type of a pointed, simply-connected, 2-reduced simplicial set is determined by the chain-complex augmented by functorial diagonal and higher diagonal maps (a simple generalization of the ones used to…
Directed Algebraic Topology studies spaces equipped with a form of direction, to include models of non-reversible processes. In the present extension we also want to cover critical processes, indecomposable and unstoppable. The previous…
This book is an account of certain topics in general and algebraic topology. Instead of laying out a synopsis of each chapter, here is a sample of some of what is taken up: 1) Nilpotency and its role in homotopy theory. 2) Bousfield's…
Let $J(X)$ denote the James construction on a space $X$ and $J_n(X)$ be the $n$-th stage of the James filtration of $J(X)$. It is known that $[J(X),\Omega Y]\cong \lim\limits_{\leftarrow} [J_n(X),\Omega Y]$ for any space $Y$. When…
The aim of this paper is to explain how, through the work of a number of people, some algebraic structures related to groupoids have yielded algebraic descriptions of homotopy n-types. Further, these descriptions are explicit, and in some…
The purpose of this survey article is to introduce the reader to a connection between Logic, Geometry, and Algebra which has recently come to light in the form of an interpretation of the constructive type theory of Martin-L\"of into…
Groups $\Pi_k(X;\sigma)$ of "flagged homotopies" are introduced of which the usual (abelian for $k>1$) homotopy groups $\pi_k(X;p)$ is the limit case for flags $\sigma$ contracted to a point $p$. Calculus of exterior forms with values in…
In the 2-local stable homotopy category the group of left-bu-module automorphisms of bu\wedge bo which induce the identity on mod 2 homology is isomorphic to the group of infinite upper triangular matrices with entries in the 2-adic…