In addition to exploring constructions and properties of limits and colimits in categories of topological algebras, we study special subcategories of topological algebras and their properties. In particular, under certain conditions, reflective subcategories when paired with topological structures give rise to reflective subcategories and epireflective subcategories give rise to epireflective s...

In Lecture 4, we learned about an algebraic method for describing and classifying structures. In this lecture, we look at using algebra to find combinatorial descriptions of topological spaces. We begin by looking at an equivalence relation called homotopy that gives a classification of spaces that is coarser that homeomorphism, but respects the finer classification. That is, two spaces that ha...

2 Properties of Moduli Functors 22 2.1 Nilcomplete, Cohesive, and Integrable Functors . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.2 Relativized Properties of Functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 2.3 Finiteness Conditions on Functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 2.4 Moduli of Spectral Deligne-Mumford St...

1. Categories 2 1.1. First Definition and Examples 2 1.2. An Alternative Definition: The Arrows-Only Perspective 7 1.3. Some Constructions 8 1.4. The Category of Relations 9 1.5. Special Objects and Arrows 10 1.6. Exercises 14 2. Functors and Natural Transformations 16 2.1. Functors 16 2.2. Full and Faithful Functors 20 2.3. Contravariant Functors 21 2.4. Products of Categories 23 3. Natural Tr...

We propose a notion of 1-homotopy for generalized maps. This notion generalizes those of natural transformation and ordinary homotopy for functors. The 1-homotopy type of a Lie groupoid is shown to be invariant under Morita equivalence. As an application we consider orbifolds as groupoids and study the notion of orbifold 1-homotopy type induced by a 1-homotopy between presentations of the orbif...

We study the category O for a general Coxeter system using a formulation of Fiebig. The translation functors, the Zuckerman functors and the twisting functors are defined. We prove the fundamental properties of these functors, the duality of Zuckerman functor and generalization of Verma’s result about homomorphisms between Verma modules.

2 Derived Functors 2 2.1 Classical Derived Functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.2 Localisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.3 Bounded Derived Functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.4 Linear Derived Categories . . . . . . . . . . . . . . . . . . . . . . . . . . . ....

There is an ongoing massive effort by many researchers to link category theory and geometry, especially homotopy coherence and categorical coherence. This constitutes just a part of the broad undertaking known as categorification as described by Baez and Dolan. This effort has as a partial goal that of understanding the categories and functors that correspond to loop spaces and their associated...

Poincar\'e maps and suspension flows are examples of fundamental constructions in the study dynamical systems. This aimed to show that these define an adjoint pair functors if categories systems suitably set. First, we consider construction category on topological manifolds, which not necessarily smooth. We well-known results can be generalized is functorial, a with global sections adequately d...