1.2.1 Content and Scope
In this section, we outline what content is expected to be covered in the Clowder Project.
1.2.1.1 Elementary Category Theory
First and foremost, the Clowder Project aims to cover the foundations of category theory. This comprises all the usual topics treated in basic textbooks in category theory, such as [Mac Lane, Categories for the Working Mathematician] or [Riehl, Category Theory in Context], like adjunctions, co/limits, Kan extensions, co/ends, monoidal categories, etc.
1.2.1.2 Variants of Category Theory
Second, the Clowder Project aims to cover variants of category theory such as internal, fibred, or enriched category theory. The literature on these topics is often quite scattered and scarce, and so having a comprehensive discussion of them in Clowder aims to fill a large gap in the literature. See also Gap 1.3.4.1.14.
1.2.1.3 Higher Category Theory
Third, a detailed presentation of the theories of bicategories and double categories is planned, along with some material on tricategories.
Bicategories are another topic for which the literature is rather scattered, and, for some topics, scarce. As mentioned in the introduction, only recently has a proper textbook on bicategories appeared, [JY, 2-Dimensional Categories]. Moreover, one finds several gaps in the literature, with a number of important results missing. As one particular example, one could look at the theory of 2-dimensional co/ends, in which case a comprehensive treatment based upon lax/oplax/pseudo dinatural transformations seems to be missing.
All of the elementary and not-so-elementary topics in the theory of bicategories are planned to appear in Clowder, and the same holds true for the theory of double categories.
1.2.1.4 $\infty $-Categories
Lastly, some material on $\infty $-categories is planned, although the precise scope of this remains to be defined. Ideally, this would include both model categories as well as synthetic and concrete models for $\infty $-categories (e.g. quasicategories, complete Segal spaces, cubical quasicategories, etc.).
In this way, we view Clowder as a good complement to [Lurie, Kerodon].
1.2.1.5 Other Topics
Occasionally, material on topics not a-priori related to category theory will be included. This may be done for a variety of reasons, including:
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Illustrating general theory.
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Comparison with classical concepts, such as e.g. ionads vs.topological spaces.
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Providing a more consistent and unified treatment of a particular topic, with hyperlinks to relevant concepts or examples.
