Van Kampen Colimits and Path Uniqueness

Fibred semantics is the foundation of the model-instance pattern of software engineering. Software models can often be formalized as objects of presheaf topoi, i.e, categories of objects that can be represented as algebras as well as coalgebras, e.g., the category of directed graphs. Multimodeling requires to construct colimits of models, decomposition is given by pullback. Compositionality requires an exact interplay of these operations, i.e., diagrams must enjoy the Van Kampen property. However, checking the validity of the Van Kampen property algorithmically based on its definition is often impossible. In this paper we state a necessary and sufficient yet efficiently checkable condition for the Van Kampen property to hold in presheaf topoi. It is based on a uniqueness property of path-like structures within the defining congruence classes that make up the colimiting cocone of the models. We thus add to the statement "Being Van Kampen is a Universal Property" by Heindel and Soboci\'{n}ski the fact that the Van Kampen property reveals a presheaf-based structural uniqueness feature

Being Van Kampen in Presheaf Topoi is a Uniqueness Property

Fibred semantics is the foundation of the model-instance pattern of software engineering. Software models can often be formalized as objects of presheaf topoi, e.g. the category of directed graphs. Multimodeling requires to construct colimits of diagrams of single models and their instances, while decomposition of instances of the multimodel is given by pullback. Compositionality requires an exact interplay of these operations, i.e., the diagrams must enjoy the Van Kampen property. However, checking the validity of the Van Kampen property algorithmically based on its definition is often impossible. In this paper we state a necessary and sufficient yet easily checkable condition for the Van Kampen property to hold for diagrams in presheaf topoi. It is based on a uniqueness property of path-like structures within the defining congruence classes that make up the colimiting cocone of the models. We thus add to the statement "Being Van Kampen is a Universal Property" by Heindel and Sobocinski presented at CALCO 2009 the fact that the Van Kampen property reveals a set-based structural uniqueness feature

Loop groups and diffeomorphism groups of the circle as colimits

We show that loop groups and the universal cover of $\mathrm{Diff}_+(S^1)$ can be expressed as colimits of groups of loops/diffeomorphisms supported in subintervals of $S^1$. Analogous results hold for based loop groups and for the based diffeomorphism group of $S^1$. These results continue to hold for the corresponding centrally extended groups. We use the above results to construct a comparison functor from the representations of a loop group conformal net to the representations of the corresponding affine Lie algebra. We also establish an equivalence of categories between solitonic representations of the loop group conformal net, and locally normal representations of the based loop group.Comment: updated to published versio

Discrete approximations for complex Kac-Moody groups

We construct a map from the classifying space of a discrete Kac-Moody group over the algebraic closure of the field with p elements to the classifying space of a complex topological Kac-Moody group and prove that it is a homology equivalence at primes q different from p. This generalises a classical result of Quillen-Friedlander-Mislin for Lie groups. As an application, we construct unstable Adams operations for general Kac-Moody groups compatible with the Frobenius homomorphism. In contrast to the Lie case, the homotopy fixed points of these unstable Adams operations cannot be realized at q as the classifying spaces of Kac-Moody groups over finite fields. Our results rely on new integral homology decompositions for certain infinite dimensional unipotent subgroups of discrete Kac-Moody groups.Comment: New title and revised introduction, references added; results and proofs unchanged, 31 pages, 1 figur

Homotopy Exact Sequence for the Pro-Étale Fundamental Group

The pro-étale fundamental group of a scheme was introduced by Bhatt and Scholze. It generalizes the fundamental groups of schemes introduced by Grothendieck in SGA1 and SGA3. The corresponding category of covers of a scheme X consists of schemes Y -> X that are étale and satisfy the valuative criterion of properness. They are called "geometric covers". As we do not assume Y -> X to be of finite type, we get more than just finite étale covers. The basic example is given by a nodal curve and a cover by an infnite chain of P^1's glued in a suitable way. This cover was already "detected" by the SGA3 fundamental group, but for more complicated schemes (e.g. an elliptic curve with two points glued) one gets more. The prominent feature of the pro-étale fundamental group is that its finite-dimensional Q_\ell-representations are able to detect all Q_\ell-local systems on X. The greater generality comes at the price of working with a more complicated class of topological groups - Noohi groups. This class includes profinite and prodiscrete groups. An important feature is that it also includes groups like GL_n(Q_\ell). In this thesis, I prove some fundamental results for the pro-étale fundamental group, that generalize the results of Grothendieck on the usual étale fundamental group to this more general context. The main results concern the homotopy exact sequence of the pro-étale fundamental groups arising from a proper morphism of connected (Nagata) noetherian schemes with geometrically connected and reduced fibers. There are two separate cases: over a general base scheme S and over a spectrum of a field k. Over the general base, one does not have exactness on the left of the sequence and the main difficulty is the exactness in the middle. Moreover, one has to use a suitable notion of exactness, involving some kind of topological closures. Over Spec(k), one can drop the properness assumption and the sequence is exact even on the level of abstract groups. In this case, we show additionally the exactness on the left (i.e. that the "geometric" fundamental group embeds into the "arithmetic" one) and this is the most difficult part. As in the classical case, the statements on exactness of the pro-étale fundamental groups translate to a statement in terms of geometric covers. We provide a detailed dictionary between the two languages. In terms of covers, the main theorems we prove are as follows. For a geometrically connected scheme X of finite type over a field k, we show that a connected geometric cover of X_\bar{k} can be dominated by a geometric cover defined over a finite extension l of the base field k. Unlike in the case of finite étale covers, this is non-trivial. Over a general base scheme S (and with assumptions about X -> S as above), we prove existence of an "infinite Stein factorization" of a geometric cover Y -> X. The resulting scheme is a geometric cover of S. If Y -> X is finite, it matches the usual Stein factorization applied to Y -> X -> S. For infinite covers, the standard definition of Stein factorization fails and we proceed differently: using a descent argument along a large pro-étale cover. Over a field, we refine an abstract version of the van Kampen theorem to provide a presentation of the pro-étale fundamental group in terms of a free product of profinite groups, discrete groups and relations (up to a certain form of completion). We use it to study the Galois actions. We also prove the Künneth formula for the pro-étale fundamental groups, generalizing the classical result of Grothendieck. We use the abstract van Kampen in the proof, again

Structured Decompositions: Structural and Algorithmic Compositionality

We introduce structured decompositions: category-theoretic generalizations of many combinatorial invariants -- including tree-width, layered tree-width, co-tree-width and graph decomposition width -- which have played a central role in the study of structural and algorithmic compositionality in both graph theory and parameterized complexity. Structured decompositions allow us to generalize combinatorial invariants to new settings (for example decompositions of matroids) in which they describe algorithmically useful structural compositionality. As an application of our theory we prove an algorithmic meta theorem for the Sub_P-composition problem which, when instantiated in the category of graphs, yields compositional algorithms for NP-hard problems such as: Maximum Bipartite Subgraph, Maximum Planar Subgraph and Longest Path