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. In particular, it contains:
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Explicit descriptions of the major types of co/limits in $\mathsf{Sets}$, including in particular explicit descriptions of pushouts and coequalisers (see Definition 2.2.4.1.1, Remark 2.2.4.1.2, Definition 2.2.5.1.1, and Remark 2.2.5.1.2).
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A discussion of powersets as decategorifications of categories of presheaves (Remark 2.4.1.1.2 and Remark 2.4.3.1.2), including a $\webleft (-1\webright )$-categorical analogue of un/straightening, described in Item 1 and Item 2 of Proposition 2.4.3.1.6 and Remark 2.4.3.1.7.
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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 (morphisms of posets) between $\mathcal{P}\webleft (A\webright )$ and $\mathcal{P}\webleft (B\webright )$ induced by a map of sets $f\colon A\to B$, along with a discussion of the properties of $f_{*}$, $f^{-1}$, and $f_{!}$.
In line with the categorical viewpoint developed here, this adjoint triple may be described in terms of Kan extensions, and, as it turns out, it also shows up in some definitions and results in point-set topology, such as in e.g. notions of continuity for functions (
).