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The goal of this short note is to show how to compute the limit or colimit of a diagram of presheaves, and prove that every presheaf can be written as the colimit of representable...

The notion of an adjunction between categories is a fundamental idea in category theory whose utility is so great that it appears in virtually every field of mathematics. Roughly speaking, an adjunction is...

The $\mathrm{Tor}$ functors, which are ubiquitous in algebraic topology and more generally homological algebra, encode the failure of the tensor product to be an exact functor. As a derived functor, one typically defines...

In this post, we discuss equivalent characterizations of the homotopy extension property of a topological pair $(X, A)$, which states that a homotopy...

In this post, we discuss an invariant of a complex representation of a finite group, and show how this yields a connection between irreducible real and complex representations...

This post builds off of Roger Penrose's article On the Cohomology of Impossible Figures, in which he shows a remarkable connection between impossible drawings, such as the Penrose triangle, and nontrivial cohomology groups...

The Yoneda lemma, according to Ravil Vakil in Foundations of Algebraic Geometry, is an "important exercise you should do once in your life." In this post, we give...

Here are two puzzles of my own invention concerning finite arithmetic progressions in cyclic (or monothetic) groups. An example this is a sequence of regularly scheduled meetings spread throughout multiple years (e.g. 10 meetings occuring once every 60 days)...

If you press ctrl + K on the blog page of this site, you'll be greeted by a search dialog modeled after the one on the Vue.js website. When a user searches for a post, the matching text is marked by being either highlighted or underlined...