Author Archives: Joe Moeller

Combinatorics, Lecture 2 (1 Oct 2019)

In Lecture 1, we gave the idea of what a species is and saw a few examples. Now we’ll explore the idea more and see more examples. We’ll also talk about the connection to generating functions, and some operations that let us build new species from old. Thanks to Tim Hosgood for helping me outContinueContinue reading “Combinatorics, Lecture 2 (1 Oct 2019)”

Combinatorics, Lecture 1 (26 Sep 2019)

John Baez is teaching a course on combinatorics this quarter. I’m taking detailed notes and texing them up. I’m also going to start blogging them. Credit to Tim Hosgood for the pictures. Prehistory of the course Larry Harper taught this course in the past. John is going to be talking about combinatorial species. He previouslyContinueContinue reading “Combinatorics, Lecture 1 (26 Sep 2019)”

What is the Grothendieck construction like?

This is my best attempt at an intuitive introduction to the Grothendieck construction. I’ll give you the definition, but not before warming up to the idea. I’ll start with the earliest conceptual ancestor I could come up with: addition. Numbers, Addition What am I going to tell you about addition that you don’t already know?ContinueContinue reading “What is the Grothendieck construction like?”

Reference List: Operads and Combinatorial Species

(1972) The Geometry of Iterated Loop Spaces – May [pdf](1973) Homotopy Invariant Algebraic Structures on Topological Spaces – Boardman, Vogt(1981) Une théorie combinatoire des séries formelles – Joyal [link](1989) The Relation between Burnside Rings and Combinatorial Species – Labelle, Yeh [link](1990) Generatingfunctionology – Wilf [link](1997) Combinatorial Species and Tree-like Structures – Bergeron, Leroux, Labelle [firstContinueContinue reading “Reference List: Operads and Combinatorial Species”

Algebraic Analysis notes Lecture 11 (4 Feb 2019)

Notes for lecture 10 Last time: for an abelian category A, C(A) is the category of complexes in A. Say $latex f, g \in \mathrm{Hom}_{C(A)} (X, Y)$ are homotopic, f~g, if there are maps $latex s^i : X^i \to Y^i$ such that $latex f^i -g^i = d_Y s + sd_X$. Definition The homotopy category K(A)ContinueContinue reading “Algebraic Analysis notes Lecture 11 (4 Feb 2019)”

Algebraic Analysis notes Lecture 10 (1 Feb 2019)

Notes for lecture 9 Last time: $latex \Gamma : Sh(X; k) \to k-mod$ global sections functor is left exact. We’ll leave sheaves for now to look at derived categories. What do sheaves have to do with cohomology? Poincare Lemma: Let M be a manifold. Consider the following complex of sheaves: where d is the deContinueContinue reading “Algebraic Analysis notes Lecture 10 (1 Feb 2019)”

Algebraic Analysis notes Lecture 9 (30 Jan 2019)

Notes for lecture 8 Last time we showed that Sh(X;k) is an abelian category. So we’ll get: a notion of simple objectscomplexes, exactness, cohomology of complexes5-lemmasnake lemmaJordan-Holder theorem for abelian categories of finite length For $latex \phi \colon F \to G$, we saw $latex ker (\phi)_x \cong ker (\phi_x)$ and $latex cok (\phi)_x \cong cokContinueContinue reading “Algebraic Analysis notes Lecture 9 (30 Jan 2019)”

Algebraic Analysis notes Lecture 8 (28 Jan 2019)

Notes for lecture 7 Last time: we defined additive categories, and kernels for morphisms in additive categories. Definition: The cokernel of a morphism $latex \phi$ (if it exists) is the universal object $latex cok \phi$ with the dual universal property: Definition: an additive category C is abelian if (A4) for any $latex \phi \colon XContinueContinue reading “Algebraic Analysis notes Lecture 8 (28 Jan 2019)”