• An explicit description of major types of co/limits in $\mathsf{Sets}$, including in particular pushouts and coequalisers (Section 1);
• A discussion of basic operations with sets and their properties (Section 2);
• A discussion of powersets as decategorifications of categories of presheaves (Section 3.2);
• A lengthy discussion of the adjoint triple $$f_*\dashv f^{-1}\dashv f_!\colon\mathcal{P}(A)\mathrel{\raise{0.5em}{\underset{\leftrightarrows}{\to}}}\mathcal{P}(B)$$ of functors (morphisms of posets) between $\mathcal{P}(A)$ and $\mathcal{P}(B)$ induced by a map of sets $f\colon A\to B$, along with a discussion of the properties of $f_*$, $f^{-1}$, and $f_!$ (Sections 3.3–3.5);
• A lengthy discussion on pointed sets and constructions with them (Section 4), including in particular a discussion of the various tensor products involving htem, like smash products (Section 4.6), tensors and cotensors by sets (Section 4.7), and the “left” and “right” skew tensor products of pointed sets (Section 4.8).

• A discussion of indexed sets (i.e. functors $K_\mathrm{disc}\to\mathsf{Sets}$ with $K$ a set), constructions with them like dependent sums and dependent products, and their properties (Section 1);
• A discussion of fibred sets (i.e. maps of sets $X\to K$), constructions with them like dependent sums and dependent products, and their properties (Section 2);
• A discussion of the un/straightening construction for indexed and fibred sets. (Section 3)

• A basic discussion and definition of relations (Section 1.1);
• How relations may be viewed as decategorification of profunctors (Remarks 1.1.5 and 1.1.6)
• A discussion of the various kind of categories (a category, a monoidal category, a 2-category, a double category) that relations form (Sections 1.2 to 1.5);
• The various categorical properties of the 2-category of relations, including self-duality, identifications of adjunctions in $\mathsf{Rel}$ with functions, of monads in $\mathsf{Rel}$ with preorders, of comonads in $\mathsf{Rel}$ with subsets, the partial co/completeness of $\mathsf{Rel}$, and its closedness, including how right Kan extensions and right Kan lifts exist in $\mathsf{Rel}$ (Section 1.6);
• A discussion of operations with relations, including graphs, domains, ranges, unions, intersections, products, inverse relations, composition of relations, and collages (Section 2);
• A discussion of equivalence relations (Section 3) and quotient sets (Section 3.5);
• A lengthy discussion of the adjoint pairs \begin{align*} R_{*} \dashv R_{-1} &\colon\mathcal{P}(A)\rightleftarrows\mathcal{P}(B),\\ R^{-1} \dashv R_{!} &\colon\mathcal{P}(B)\rightleftarrows\mathcal{P}(A) \end{align*} of functors (morphisms of posets) between $\mathcal{P}(A)$ and $\mathcal{P}(B)$ induced by a relation $R\colon A\mathrel{\to\mkern-18mu|\mkern7mu}B$, along with a discussion of the properties of $R_*$, $R_{-1}$, $R^{-1}$, and $R_!$ (Section 4).

  These two pairs of adjoint functors are the counterpart for relations of the adjoint triple $f_*\dashv f^{-1}\dashv f_!$ induced by a function $f\colon A\to B$ studied in Constructions With Sets, and indeed we have $R_{-1}=R^{-1}$ if and only if $R$ is total and functional. Thus when $R$ comes from a function this pair of adjunctions reduces to the triple adjunction $f_*\dashv f^{-1}\dashv f_!$.

  The pairs $R_*\dashv R_{-1}$ and $R^{-1}\dashv R_!$ later make an appearance in the context of continuous, open, and closed relations between topological spaces (see the Relations Between Topological Spaces section in the Topological Spaces chapter).

• A discussion of spans (Section 5) and their relation to functions (Proposition 5.2.1) and relations (Propositions 5.3.1 and 5.3.3 and Remark 5.3.5);
• A discussion of “hyperpointed sets” (Section 6). I don’t know why I wrote this...

• A discussion of posets, constructions with them, and co/limits inside posets (Sections 1–4);
• A discussion of so-called relative preorders from a set $X$ to a set $Y$. These are supposed to be an extension of the notion of a preorder $\preceq_X\colon X\mathrel{\to\mkern-18mu|\mkern7mu}X$ on a set $X$ but where we allow the source and target of $\preceq_X$ to be entirely different sets.

  The basic idea is that we may view preorders as precisely the monads in the bicategory $\mathsf{Rel}$ of relations, so relative preorders are to be defined as relative monads in $\mathsf{Rel}$ in the sense of nLab, relative monad.

  Thus, if you’re interested in relative monads, you might like reading Appendix A.

• A discussion of categories, functors, natural transformations, and profunctors (Sections 1–3);
• A discussion of monomorphisms and epimorphisms (Sections 4–5);

  (I’ve been meaning to revise these for a while now...)

• A discussion of adjunctions (Section 6).

  (P.S.: I wrote Section 6.3 a while ago just for fun; it is completely useless.)

• A discussion of the Yoneda lemma (Sections 7–8).
• Some miscellaneous material on a bunch of random things I have to either revise or push to somewhere else in the notes (Appendices A and B).

(Please ignore them!)

• A discussion of co/limits, 2-co/limits, some weighted 2-co/limits, pseudo co/limits, lax co/limits, and oplax co/limits of categories, all with very explicit descriptions (Sections 1–6);
• A discussion of deloopings of monoids, classifying spaces of categories, opposite categories, categories of pointed objects (i.e. $\mathbb{E}_{0}$-monoids), joins, arrow categories, the funny tensor product, and the category of simplices of a category (Section 7);
• A discussion of endomorphisms, automorphisms, involutions, idempotent morphisms, and the categories they form (Section 8);
• A discussion of slice categories (Section 9);
• A discussion of coslice categories (Section 10);
• A discussion of quotients of categories (Section 11), where:
• In Section 11.1 we discuss a notion (I made up) of the quotient of a category by a profunctor (to be thought of as a categorified relation);
• In Section 11.2 we discuss the usual notion of a quotient of a category by a congruence relation on morphisms;
• In Section 11.3 we discuss the notion of a quotient of a category by a generalised congruence relation, introduced in [Bednarczyk–Borzyszkowski–Pawlowski];
• In Section 11.4 we define generalised congruence relations in a two-step process, first defining the quotient $\mathcal{C}/\mathord{\simeq}$ of a category $\mathcal{C}$ by a congruence relation $\mathord{\simeq}$ on objects, and then defining a generalised congruence relation to be a congruence relation on objects $\mathord{\simeq}$ together with a (classical) congruence relation $\mathord{\sim}$ on $\mathcal{C}/\mathord{\simeq}$.
• A discussion of Gabriel–Zisman localisations (Section 12);
• A discussion of Karoubi envelopes (Section 13).

Although probably way too large, only this file contains indices of definitions and theorems for the above chapters, as well as a comprehensive table of contents for the notes. The indices are:

• Index of Notation
• Index of Algebra
• Index of Algebraic Geometry
• Index of Analysis
• Index of Category Theory
• Index of Cellular Stuff
• Index of Cubical Stuff
• Index of Cyclic Stuff
• Index of Differential Geometry
• Index of Functional Analysis
• Index of Globular Stuff
• Index of Higher Category Theory
• Index of Homological Algebra
• Index of Homotopical Algebra
• Index of Homotopy Theory
• Index of $\infty$-Categories
• Index of Measure Theory
• Index of Monoids
• Index of Number Theory
• Index of $p$-Adic Geometry
• Index of Set Theory
• Index of Simplicial Stuff
• Index of Supersymmetry
• Index of Topology