1.1.6 Goodies
In this section we list a few sample nice results and things from the Clowder Project.
1.1.6.1 General Utility
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Chapter 14: Notes, Section 14.1 contains several tikz-cd snippets producing somewhat-hard-to-draw diagrams. Examples include cube, pentagon, and hexagon diagrams, as well as e.g. co/product diagrams with perfectly circular arrows.
1.1.6.2 Set Theory Through a Categorical Lens
Sets:
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Chapter 4: Constructions With Sets, Section 4.4.7 contains a discussion of internal Homs in powersets viewed as categories.
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More generally, Chapter 4: Constructions With Sets, Section 4.4 discusses several properties of powersets that are analogous to those of presheaf categories.
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Chapter 4: Constructions With Sets,
discusses the adjoint triple $f_{*}\dashv f^{-1}\dashv f_{!}$ between $\mathcal{P}\webleft (X\webright )$ and $\mathcal{P}\webleft (Y\webright )$ induced by a function $f\colon X\to Y$.
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Chapter 4: Constructions With Sets, Section 4.6.4 constructs a kind of “six functor formalism for (power)sets”.
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Chapter 5: Monoidal Structures on the Category of Sets contains explicit proofs that product/coproduct of sets form a monoidal structure.
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Chapter 5: Monoidal Structures on the Category of Sets, Section 5.1.10 gives a completely 1-categorical proof of the universal property of $\webleft (\mathsf{Sets},\times ,\mathrm{pt}\webright )$.
Pointed Sets:
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Chapter 7: Tensor Products of Pointed Sets constructs several tensor products of pointed sets, including a few unusual ones giving rise to skew monoidal structures on $\mathsf{Sets}_{*}$.
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Chapter 7: Tensor Products of Pointed Sets, Section 7.5.10 gives a completely 1-categorical proof of the universal property of $\webleft (\mathsf{Sets}_{*},\wedge ,S^{0}\webright )$.
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Chapter 7: Tensor Products of Pointed Sets, Proposition 7.5.12.1.1 contains a description of comonoids in $\mathsf{Sets}_{*}$ with respect to $\wedge $.
Relations:
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Chapter 8: Relations, Section 8.4 contains a discussion of several properties of the 2-category of relations like descriptions of internal adjunctions and internal monads.
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Chapter 8: Relations, Section 8.6 and Section 8.7 contains a discussion of two skew monoidal structures on the category $\mathbf{Rel}\webleft (A,B\webright )$ of relations from a set $A$ to a set $B$.
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Chapter 9: Constructions With Relations, Section 9.2 contains a description of left/right Kan extensions and lifts internal to the 2-category of relations.
1.1.6.3 Category Theory
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Chapter 11: Categories contains a description of several properties of functors, including somewhat lesser known ones such as dominant functors or pseudoepic functors.
