In the first part of this note an elementary proof is given of the fact that algebraic functors, that is, functors induced by morphisms of Lawvere theories, have left adjoints provided that the category K in which the models of these theories take their values is locally presentable. The main focus however lies on the special cases of the underlying functor of the category Grp(K) of internal gr...

These notes are based on lectures in algebraic topology taught by Peter May and Henry Chan at the 2016 University of Chicago Math REU. They are loosely chronological, having been reorganized for my benefit and significantly annotated by my personal exposition, plus solutions to in-class/HW exercises, plus content from readings (from May’s Finite Book), books (e.g. May’s Concise Course, Munkres’...

We exhibit examples of triangulated categories which are neither the stable category of a Frobenius category nor a full triangulated subcategory of the homotopy category of a stable model category. Even more drastically, our examples do not admit any non-trivial exact functors to or from these algebraic respectively topological triangulated categories. Introduction. Triangulated categories are ...

We show that if U∗ is a hypercover of a topological space X then the natural map hocolim U∗→X is a weak equivalence. This fact is used to construct topological realization functors for the A1-homotopy theory of schemes over real and complex fields. In an appendix, we also prove a theorem about computing homotopy colimits of spaces that are not cofibrant. Mathematics Subject Classification (2000...

The aim of this work is to lay the foundations of differential geometry and Lie theory over the general class of topological base fields and -rings for which a differential calculus has been developed in [BGN04], without any restriction on the dimension or on the characteristic. Two basic features distinguish our approach from the classical real (finite or infinite dimensional) theory, namely t...

The aim of this work is to lay the foundations of differential geometry and Lie theory over the general class of topological base fields and -rings for which a differential calculus has been developed in [BGN04], without any restriction on the dimension or on the characteristic. Two basic features distinguish our approach from the classical real (finite or infinite dimensional) theory, namely t...

The very nature of the so-called Postnikov invariants is carefully studied. Two functors, precisely defined, explain the exact nature of the connection between the category of topological spaces and the category of Postnikov towers. On one hand, these functors are in particular effective and lead to concrete machine computations through the general machine program Kenzo. On the other hand, the ...

We survey the present trends in theory of universal arrows to forgetful functors from various categories of topological algebra and functional analysis to categories of topology and topological algebra. Among them are free topological groups, free locally convex spaces, free Banach-Lie algebras, and more. An accent is put on relationship of those constructions with other areas of mathematics an...

This dissertation provides an introduction to the ideas employed in topological quantum field theory. We illustrate how the field began by considering knot invariants of three-manifolds and demonstrating the consequences of defining such a theory axiomatically. The ideas of category theory are introduced and we show that what we are actually concerned with are symmetric monoidal functors from t...

It is the purpose of this paper to go somewhat deeper into the structure of fuzzy topological spaces. In doing so we found we had to alter the definition of a fuzzy topology used up to now. We shall also introduce two functors 3 and i which will allow us to see more clearly the connection between fuzzy topological spaces and topological spaces. Finally we shall introduce the concept of fuzzy co...