Lecture notes from M275 at SJSU, a graduate-level course at SJSU in Algebraic Topology taught by Professor Richard Kulbelka. These notes were taken using Vim and/or GVim equipped with L A T E X-Suite, which sped up typesetting significantly; latexmk with the flag-pvc was useful for compiling these notes in real-time. 1.1 The Question Question. Given two topological spaces, can we determine if t...

1 Topological Field Theories and Higher Categories 2 1.1 Classical Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 Extending Down: Lower Dimensional Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.3 Higher Category Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 1.4 Extending Up: Diffeo...

1 Topological Field Theories and Higher Categories 2 1.1 Classical Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 Extending Down: Lower Dimensional Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.3 Higher Category Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 1.4 Extending Up: Diffeo...

Engaging in explanation, even to oneself, can enhance learning. What underlies this effect? Williams & Lombrozo (in press) propose that explanation exerts subsumptive constraints on processing, driving learners to discover underlying patterns. A category-learning experiment demonstrates that explanation can enhance or impair learning depending on whether these constraints match the structure of...

A 2-categorical generalisation of the notion of elementary topos is provided, and some of the properties of the yoneda structure [SW78] it generates are explored. Results enabling one to exhibit objects as cocomplete in the sense definable within a yoneda structure are presented. Examples relevant to the globular approach to higher dimensional category theory are discussed. This paper also cont...

This development proves Yoneda's lemma and aims to be readable by humans. It only defines what is needed for the lemma: categories, func-tors and natural transformations. Limits, adjunctions and other important concepts are not included. There is no explanation or discussion in this document. See [O'K04] for this and a survey of category theory formalisations.

Homotopy Theory: The Interaction of Category Theory and Homotopy theory A revised version of the 2001 article

We extend the theory of Quillen adjunctions by combining ideas of homotopical algebra and of enriched category theory. Our results describe how the formulas for homotopy colimits of Bousfield and Kan arise from general formulas describing the derived functor of the weighted colimit functor.

One of the open problems in higher category theory is the systematic construction of the higher dimensional analogues of the Gray tensor product. In this paper we begin to adapt the machinery of globular operads [1] to this task. We present a general construction of a tensor product on the category of n-globular sets from any normalised (n + 1)-operad A, in such a way that the algebras for A ma...