“I am not saying that I believe in the law of the excluded middle, I am just saying that it isn’t not true. ”

Kavin Satheeskumar

Recently, I’ve been reading Cisinski’s Higher Categories and Homotopical Algebra. I wanted to write down some of the things I’ve learned. In this post, we start from a review of presheaves of sets and simplicial sets, and then build towards the definition of an \(\infty\)-category. Let \(A\) be a category, recall a presheaf on \(A\) is a contravariant functor \(X:A^{\mathrm{op}}\rightarrow \mathbf{Set}\), where we denote \(X_a=X(a)\) and \(u^*:b\rightarrow a\) the induced morphism for each \(u:a\rightarrow b\). And the category of presheaves on \(A\) is denoted as \(\widehat{A}\). The category of elements \(\int_AX\) (or \(\int X\)) is the category where objects are \((a,s)\) where \(a\in A\) and \(s\in X_a\), and a morphisms \(u:(a,s)\rightarrow (b,t)\) is a morphism \(u:a\rightarrow b\) where \(u^*(t)=s\). It comes equipped with a faithful functor \(\varphi_X:\int_A X\rightarrow \widehat{A}\) given by \((a,s)\mapsto \mathscr{H}_a\) on objects and \(u\mapsto \mathscr{H}(u)\) on morphisms, where \(\mathscr{H}:A\rightarrow \widehat{A}\) is the Yoneda embedding. In this post, we ignore all set-theoretic size issues.

Read More

Recently, I gave another talk on the Short Attention Span Math Seminars which is a friendly introduction to category theory. Here are my slides, and here are the slides for the first five minutes. In my talk, I first talked about the idea of abstractions, then I talked about the categorical way of thing and along the way talked about many ideas like universal properties, functors, natural transformations. In the end, this culminated in the Yoneda lemma or the fundamental theorem of category theory. I also discussed a little an application of it to algebraic geometry.

“Algebraic geometry seems to have acquired the reputation of being esoteric, exclusive, and very abstract, with adherents who are secretly plotting to take over all the rest of mathematics. In one respect this last point is accurate.”

David Mumford

Continuing last week, we are finally going to define the derived category \(D(\mathcal A)\) for an abelian category \(\mathcal A\). Before we actually do that, we need to introduce localization of categories, and we will define the derived category in terms of a localization of the homotopy category of complexes. Localization of categories is very analogous to localization of rings or modules. Given any category (with no assumption of abelian, additive, triangulated, etc), we formally invert a class of morphisms. We will give one definition in terms of a localization construction and show that it is unsatisfactory in that it does not have a lot of useful and desirable properties. We will then develop a better description of derived categories in terms of roofs.

Read More

“It is impossible to be a mathematician without being a poet in soul.”

Sofia Kovalevskaya

Continuing last week where we defined the homotopy category of complexes, we will take a look at triangulated categories. This will help us build towards the definition of derived categories. In some sense, triangulated categories are approximations of abelian categories. They are not strictly speaking a weaker version of abelian categories, and they don’t really imply each other. Triangulated categories takes a different approach to abelian categories. Instead of kernels, cokernels, and strict morphisms, triangulated categories bypass these notions and start out with “distinguished triangles”, which are similar to exact sequences.

Read More