• John Baez, Two Miracles of Algebraic Geometry is a really nice post (what else did you expect from John!) by John Baez where he explains the categorical properties of the Albanese variety as a functor. It really highlights why algebraic geometry is special, and it does so by asking this seemingly naive MathOverflow question, in which he does something that was never envisioned by the Italian school of algebraic geometry, namely turning the Albanese variety into an endofunctor.

    A funny quote from the blogpost is

    But forgetful functors often go unspoken in ordinary mathematical English: they’re not just forgetful, they're forgotten.

  • Shane Kelly, Some observations about motivic tensor triangulated geometry over a finite field is a set of notes for a summer school from last year, which was my first real introduction to the stable homotopy category and why an algebraic geometer could be interesting in all this. It applies all the (proven) machinery to reduce very abstract questions about the tensor triangulated geometry of the stable homotopy category to the (for me at least) little less abstract category of Voevodsky motives, and then appeals to certain conjectures in the special case of finite fields to compute that the Balmer spectrum of this topological gadget is actually $latex \mathrm{Spec}\,\mathbb{Q}$!