James bundles

We study cubical sets without degeneracies, which we call {square}-sets. These sets arise naturally in a number of settings and they have a beautiful intrinsic geometry; in particular a {square}-set C has an infinite family of associated {square}-sets Ji(C), for i = 1, 2, ..., which we call James complexes. There are mock bundle projections pi: |Ji(C)| -> |C| (which we call James bundles) defining classes in unstable cohomotopy which generalise the classical James–Hopf invariants of {Omega}(S2). The algebra of these classes mimics the algebra of the cohomotopy of {Omega}(S2) and the reduction to cohomology defines a sequence of natural characteristic classes for a {square}-set. An associated map to BO leads to a generalised cohomology theory with geometric interpretation similar to that for Mahowald orientation

Higher homotopy operations and cohomology

We explain how higher homotopy operations, defined topologically, may be identified under mild assumptions with (the last of) the Dwyer-Kan-Smith cohomological obstructions to rectifying homotopy-commutative diagrams.Comment: 28 page

Cubical rigidification, the cobar construction, and the based loop space

We prove the following generalization of a classical result of Adams: for any pointed and connected topological space $(X,b)$, that is not necessarily simply connected, the cobar construction of the differential graded (dg) coalgebra of normalized singular chains in $X$ with vertices at $b$ is weakly equivalent as a differential graded associative algebra (dga) to the singular chains on the Moore based loop space of $X$ at $b$. We deduce this statement from several more general categorical results of independent interest. We construct a functor $\mathfrak{C}_{\square_c}$ from simplicial sets to categories enriched over cubical sets with connections which, after triangulation of their mapping spaces, coincides with Lurie's rigidification functor $\mathfrak{C}$ from simplicial sets to simplicial categories. Taking normalized chains of the mapping spaces of $\mathfrak{C}_{\square_c}$ yields a functor $\Lambda$ from simplicial sets to dg categories which is the left adjoint to the dg nerve functor. For any simplicial set $S$ with $S_0=\{x\}$, $\Lambda(S)(x,x)$ is a dga isomorphic to $\Omega Q_{\Delta}(S)$, the cobar construction on the dg coalgebra $Q_{\Delta}(S)$ of normalized chains on $S$. We use these facts to show that $Q_{\Delta}$ sends categorical equivalences between simplicial sets to maps of connected dg coalgebras which induce quasi-isomorphisms of dga's under the cobar functor

Gamma spaces and information

We investigate the role of Segal’s Gamma-spaces in the context of classical and quantum information, based on categories of finite probabilities with stochastic maps and density matrices with quantum channels. The information loss functional extends to the setting of probabilistic Gamma-spaces considered here. The Segal construction of connective spectra from Gamma-spaces can be used in this setting to obtain spectra associated to certain categories of gapped systems

The homotopy theory of differentiable sheaves

Many important theorems in differential topology relate properties of manifolds to properties of their underlying homotopy types -- defined e.g. using the total singular complex or the \v{C}ech nerve of a good open cover. Upon embedding the category of manifolds into the $\infty$-topos $\mathbf{Diff}^\infty$ of differentiable sheaves one gains a further notion of underlying homotopy type: the shape of the corresponding differentiable sheaf. In a first series of results we prove using simple cofinality and descent arguments that the shape of any manifold coincides with many other notions of underlying homotopy types such as the ones mentioned above. Our techniques moreover allow for computations, such as the homotopy type of the Haefliger stack, following Carchedi. This leads to more refined questions, such as what it means for a mapping differential sheaf to have the correct shape. To answer these we construct model structures as well as more general homotopical calculi on the $\infty$-category $\mathbf{Diff}^\infty$ (which restrict to its full subcategory of $0$-truncated objects,$\mathbf{Diff}^\infty_{\leq 0}$) with shape equivalences as the weak equivalences. These tools are moreover developed in such a way so as to be highly customisable, with a view towards future applications, e.g. in geometric topology. Finally, working with the $\infty$-topos $\mathbf{Diff}^0$ of sheaves on topological manifolds, we give new and conceptual proofs of some classical statements in algebraic topology. These include Dugger and Isaksen's hypercovering theorem, and the fact that the Quillen adjunction between simplicial sets and topological spaces is a Quillen equivalence.Comment: 100 pages, 1 figur

Gamma spaces and information

We investigate the role of Segal’s Gamma-spaces in the context of classical and quantum information, based on categories of finite probabilities with stochastic maps and density matrices with quantum channels. The information loss functional extends to the setting of probabilistic Gamma-spaces considered here. The Segal construction of connective spectra from Gamma-spaces can be used in this setting to obtain spectra associated to certain categories of gapped systems

Associative nn-categories

We define novel fully combinatorial models of higher categories. Our definitions are based on a connection of higher categories to "directed spaces". Directed spaces are locally modelled on manifold diagrams, which are stratifications of the n-cube such that strata are transversal to the flag foliation of the n-cube. The first part of this thesis develops a combinatorial model for manifold diagrams called singular n-cubes. In the second part we apply this model to build our notions of higher categories. Singular n-cubes are "directed triangulations" of space together with a decomposition into a collection of subspaces or strata. Singular n-cubes can be naturally organised into two categories. The first, whose morphisms are bundles themselves, is used for the inductive definition of singular (n+1)-cubes. The second, whose morphisms are "open" base changes, admits an (epi,mono) factorisation system. Monomorphisms will be called embeddings of cubes. Epimorphisms will be called collapses and describe how triangulations can be coarsened. Each cube has a unique coarsest triangulation called its normal form. The existence of normal forms makes the equality relation of (combinatorially represented) manifold diagrams decidable. As the main application of the resulting combinatorial framework for manifold diagrams, we give algebraic definitions of various notions of higher categories. Namely, we define associative n-categories, presented associative n-categories and presented associative n-groupoids. All three notions will have strict units and associators; the only weak coherences are homotopies, but we develop a mechanism for recovering the usual coherence data of weak n-categories, such as associators and pentagonators and their higher analogues. This will motivate the conjecture that the theory of associative higher categories is equivalent to its fully weak counterpart.Comment: 499 page