Even though its analogue for $\infty $-categories has for years been a widely used tool1, a comprehensive treatment of the tensor product of presentable categories seems to be currently missing.
1.3.4 List of Gaps in the Category Theory Literature
The Clowder Project aims to address several significant gaps in the existing literature on category theory, as detailed below. See also [Emily, Gaps in the category theory literature you'd like to see filled].
An exhaustive concrete description of the various limits and colimits of categories, including 2-dimensional ones, is missing.
There seems to be no unified presentation of dinatural transformation co/classifiers in the literature. These are characterised by isomorphisms of the form
\[ \operatorname {\mathrm{Nat}}\webleft (F,G\webright )\cong \operatorname {\mathrm{DiNat}}\webleft (\Gamma \webleft (F\webright ),G\webright ), \]
and were originally studied in Dubuc–Street’s paper introducing dinatural transformations, [DS, Dinatural Transformations].
Even though these arguably form a fundamental piece of the framework of co/end calculus, it seems that all foundational treatments that followed after ended up not covering this concept.
The tensor product of symmetric monoidal categories had been a missing concept from the literature for years. Recently, [GJO, The Symmetric Monoidal 2-Category of Permutative Categories] covered the case of permutative categories. It would be nice, however, to also have a treatment of the non-strict case available.
A comprehensive and exhaustive treatment of the theory of promonoidal categories is currently missing. There are several important notions undefined, like:
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Promonoidal profunctors.
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Dualisability internal to a promonoidal category.
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Invertibility internal to a promonoidal category.
Moreover, it would be nice to record how promonoidal categories may be viewed as categorifications of “hypermonoids” (i.e. monoids in $\mathrm{Rel}$).
A comprehensive and exhaustive treatment of the theory of multicategories is currently missing. There are several important notions undefined, like:
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Co/limits internal to multicategories.
It would be nice to have an extensive collection of examples of what a given 2-categorical notion looks like in a 2-category. For instance, it would be nice to explicitly list what internal adjunctions look like in $\boldsymbol {\mathsf{Rel}}$, $\mathsf{Span}$, $\mathsf{Prof}$, etc.
See Chapter 8: Relations, Section 8.4 for a concrete example of what is meant by this gap.
The literature on centres and traces of categories is really small. There are lots of results missing1 and very few worked examples2.
- 1E.g. There’s a certain interaction between traces of categories and Leinster’s eventual image.
- 2E.g. what is the trace of Connes’s cycle category? Such a computation doesn’t seem to be available.
Natural transformations satisfy an isomorphism of the form
\[ \operatorname {\mathrm{Nat}}\webleft (F,G\webright )\cong \int _{A\in \mathcal{C}}\operatorname {\mathrm{Hom}}_{\mathcal{D}}\webleft (F_{A},G_{A}\webright ). \]
It is then exceedingly natural to define natural cotransformations via an isomorphism of the form
\[ \operatorname {\mathrm{CoNat}}\webleft (F,G\webright )\cong \int ^{A\in \mathcal{C}}\operatorname {\mathrm{Hom}}_{\mathcal{D}}\webleft (F_{A},G_{A}\webright ) \]
and study their properties. This generalises traces of categories, since we have
\[ \mathrm{Tr}\webleft (\mathcal{C}\webright )=\operatorname {\mathrm{CoNat}}\webleft (\operatorname {\mathrm{id}}_{\mathcal{C}},\operatorname {\mathrm{id}}_{\mathcal{C}}\webright ), \]
much like $\mathrm{Z}\webleft (\mathcal{C}\webright )=\operatorname {\mathrm{Nat}}\webleft (\operatorname {\mathrm{id}}_{\mathcal{C}},\operatorname {\mathrm{id}}_{\mathcal{C}}\webright )$.
There are several results, notions, and examples in the theory of Isbell duality missing from the literature, and a truly comprehensive treatment is still lacking.1
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1For instance, there appears to be no mention of the duality pairings \begin{align*} \mathsf{Spec}\webleft (F\webright )\boxtimes F & \to \mathrm{Tr}\webleft (\mathcal{C}\webright ),\\ \mathcal{F}\boxtimes \mathsf{O}\webleft (\mathcal{F}\webright ) & \to \mathrm{Tr}\webleft (\mathcal{C}\webright ) \end{align*}in the currently available literature.
The currently available treatments of 2-dimensional co/ends are unsatisfactory.1
- 1For instance, none of them define 2-dimensional co/ends via 2-dimensional dinatural transformations and then go on to develop a general theory from there.
A comprehensive treatment of factorisation systems is currently missing; see [Fosco, Answer to “Gaps in the category theory literature you'd like to see filled”].
Several proofs of coherence theorems for string diagrams currently have gaps; see [Snyder, Answer to “Gaps in the category theory literature you'd like to see filled”].
The currently available treatments of variants of category theory such as fibred category theory, enriched category theory, or internal category theory are unsatisfactory for a number of reasons.
Ideally, there should be a comprehensive and (simultaneously) approachable treatment for these topics. See also [Hu, Answer to “Gaps in the category theory literature you'd like to see filled”].
