-
1.
Explicit descriptions of the major types of co/limits in $\mathsf{Sets}$, including in particular explicit descriptions of pushouts and coequalisers (see Definition 4.2.4.1.1, Remark 4.2.4.1.3, Definition 4.2.5.1.1, and Remark 4.2.5.1.3).
-
2.
A discussion of powersets as decategorifications of categories of presheaves, including in particular results such as:
-
(a)
A discussion of the internal Hom of a powerset (Section 4.4.7).
-
(b)
A 0-categorical version of the Yoneda lemma (
, ), which we term the Yoneda lemma for sets (Proposition 4.5.5.1.1).
-
(c)
A characterisation of powersets as free cocompletions (Section 4.4.5), mimicking the corresponding statement for categories of presheaves (
).
-
(d)
A characterisation of powersets as free completions (Section 4.4.6), mimicking the corresponding statement for categories of copresheaves (
).
-
(e)
A $\webleft (-1\webright )$-categorical version of un/straightening (Item 2 of Proposition 4.5.1.1.4 and Remark 4.5.1.1.5).
-
(f)
A 0-categorical form of Isbell duality internal to powersets (Section 4.4.8).
-
(a)
-
3.
A lengthy discussion of the adjoint triple
\[ f_{!}\dashv f^{-1}\dashv f_{*}\colon \mathcal{P}\webleft (A\webright )\overset {\rightleftarrows }{\to }\mathcal{P}\webleft (B\webright ) \]of functors (i.e. morphisms of posets) between $\mathcal{P}\webleft (A\webright )$ and $\mathcal{P}\webleft (B\webright )$ induced by a map of sets $f\colon A\to B$, including in particular:
-
(a)
How $f^{-1}$ can be described as a precomposition while $f_{!}$ and $f_{*}$ can be described as Kan extensions (Remark 4.6.1.1.4, Remark 4.6.2.1.2, and Remark 4.6.3.1.4).
-
(b)
An extensive list of the properties of $f_{!}$, $f^{-1}$, and $f_{*}$ (Proposition 4.6.1.1.5, Proposition 4.6.1.1.6, Proposition 4.6.2.1.3, Proposition 4.6.2.1.4, Proposition 4.6.3.1.7, and Proposition 4.6.3.1.8).
-
(c)
How the functors $f_{!}$, $f^{-1}$, $f_{*}$, along with the functors
\begin{align*} -_{1}\cap -_{2} & \colon \mathcal{P}\webleft (X\webright )\times \mathcal{P}\webleft (X\webright ) \to \mathcal{P}\webleft (X\webright ),\\ \webleft [-_{1},-_{2}\webright ]_{X} & \colon \mathcal{P}\webleft (X\webright )^{\mathsf{op}}\times \mathcal{P}\webleft (X\webright ) \to \mathcal{P}\webleft (X\webright ) \end{align*}may be viewed as a six-functor formalism with the empty set $\text{Ø}$ as the dualising object (Section 4.6.4).
4 Constructions With Sets
This chapter develops some material relating to constructions with sets with an eye towards its categorical and higher-categorical counterparts to be introduced later in this work. Of particular interest are perhaps the following:
