The Clowder Project

Set theory through a category theory lens:

1. Isbell duality for sets.
2. Density comonads and codensity monads for sets.

Relations:

1. 2-Categorical monomorphisms and epimorphisms in $\textbf{Rel}$.
2. Co/limits in $\textbf{Rel}$.
3. Apartness composition, categorical properties of $\textbf{Rel}$ with apartness, and apartness relations.
4. Apartness defines a composition for relations, but its analogue
   \[ \mathfrak {q}\mathbin {\square }\mathfrak {p}\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\int _{A\in \mathcal{C}}\mathfrak {p}^{-_{1}}_{A}\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}\mathfrak {q}^{A}_{-_{2}} \]

   fails to be unital for profunctors. Is there a less obvious analogue of apartness composition for profunctors?

5. Codensity monad $\text{Ran}_{J}\webleft (J\webright )$ of a relation (What about $\text{Rift}_{J}\webleft (J\webright )$?)
6. Relative comonads in the $2$-category of relations
7. Discrete fibrations and Street fibrations in $\textbf{Rel}$.
8. Consider adding the sections
   • The Monoidal Bicategory of Relations
   • The Monoidal Double Category of Relations
   to .

Spans:

1. Universal property of the bicategory of spans, https://ncatlab.org/nlab/show/span
2. Write about cospans.

Un/Straightening:

1. Write proper sections on straightening for lax functors from sets to Rel or Span (displayed sets)

Categories:

1. Expand and add a proof to it.
2. Sections and retractions; retracts, https://ncatlab.org/nlab/show/retract.
3. Regular categories: https://arxiv.org/pdf/2004.08964.pdf.
4. Are pseudoepic functors those functors whose restricted Yoneda embedding is pseudomonic and Yoneda preserves absolute colimits?
5. Absolutely dense functors enriched over $\mathbb {R}^{+}$ apparently reduce to topological density

Types of Morphisms in Categories:

1. Behaviour in $\mathsf{Fun}\webleft (\mathcal{C},\mathcal{D}\webright )$, e.g. pointwise sections vs. sections in $\mathsf{Fun}\webleft (\mathcal{C},\mathcal{D}\webright )$.
2. A faithful functor from balanced category is conservative

Yoneda stuff:

1. Properties of restricted Yoneda embedding, e.g. if the restricted Yoneda embedding is full, then what can we conclude? Related: https://qchu.wordpress.com/2015/05/17/generators/

Adjunctions:

1. Adjunctions, units, counits, and fully faithfulness as in https://mathoverflow.net/questions/100808/properties-of-functors-and-their-adjoints.
2. Morphisms between adjunctions and bicategory $\mathsf{Adj}\webleft (\mathcal{C}\webright )$.
3. https://ncatlab.org/nlab/show/transformation+of+adjoints

Constructions With Categories:

1. Comparison between pseudopullbacks and isocomma categories: the “evident” functor $\mathcal{C}\times ^{\mathsf{ps}}_{\mathcal{E}}\mathcal{D}\to \mathcal{C}\mathbin {\overset {\leftrightarrow }{\times }}_{\mathcal{E}}\mathcal{D}$ is essentially surjective and full, but not faithful in general.

Co/limits:

1. Add the characterisations of absolutely dense functors given in to .
2. Absolutely dense functors, https://ncatlab.org/nlab/show/absolutely+dense+functor. Also theorem 1.1 here: http://www.tac.mta.ca/tac/volumes/8/n20/n20.pdf.
3. Dense functors, codense functors, and absolutely codense functors.

Co/ends:

1. Examples of co/ends: https://mathoverflow.net/a/461814
2. Cofinality for co/ends, https://mathoverflow.net/questions/353876

Fibred category theory:

1. Internal $\mathbf{Hom}$ in categories of co/Cartesian fibrations.
2. Tensor structures on fibered categories by Luca Terenzi: https://arxiv.org/abs/2401.13491. Check also the other papers by Luca Terenzi.
3. https://ncatlab.org/nlab/show/cartesian+natural+transformation (this is a cartesian morphism in $\mathsf{Fun}\webleft (\mathcal{C},\mathcal{D}\webright )$ apparently)
4. CoCartesian fibration classifying $\mathsf{Fun}\webleft (F,G\webright )$, https://mathoverflow.net/questions/457533/cocartesian-fibration-classifying-mathrmfunf-g

Monoidal categories:

1. Free braided monoidal category with a braided monoid: https://ncatlab.org/nlab/show/vine

Skew monoidal categories:

1. Does the $\mathbb {E}_{1}$ tensor product of monoids admit a skew monoidal category structure?
2. Is there a (right?) skew monoidal category structure on $\mathsf{Fun}\webleft (\mathcal{C},\mathcal{D}\webright )$ using right Kan extensions instead of left Kan extensions?
3. Similarly, are there skew monoidal category structures on the subcategory of $\mathbf{Rel}\webleft (A,B\webright )$ spanned by the functions using left Kan extensions and left Kan lifts?

Higher categories:

1. Internal adjunctions in $\mathsf{Mod}$ as in Section 6.3 of [Johnson–Yau, 2-Dimensional Categories]; see Example 6.2.6 of [Johnson–Yau, 2-Dimensional Categories].
2. Comonads in the bicategory of profunctors.

Monoids:

1. Isbell’s zigzag theorem for semigroups: the following conditions are equivalent:
   1. A morphism $f\colon A\to B$ of semigroups is an epimorphism.
   2. For each $b\in B$, one of the following conditions is satisfied:
      • We have $f\webleft (a\webright )=b$.
      • There exist some $m\in \mathbb {N}_{\geq 1}$ and two factorisations
        \begin{align*} b & = a_{0}y_{1},\\ b & = x_{m}a_{2m} \end{align*}

        connected by relations

        \begin{align*} a_{0} = x_{1}a_{1},\\ a_{1}y_{1} = a_{2}y_{2},\\ x_{1}a_{2} = x_{2}a_{3},\\ a_{2m-1}y_{m} = a_{2m} \end{align*}

        such that, for each $1\leq i\leq m$, we have $a_{i}\in \mathrm{Im}\webleft (f\webright )$.

   Wikipedia says in https://en.wikipedia.org/wiki/Isbell%27s_zigzag_theorem:

   For monoids, this theorem can be written more concisely:

Types of morphisms in bicategories:

1. Behaviour in 2-categories of pseudofunctors (or lax functors, etc.), e.g. pointwise pseudoepic morphisms in vs. pseudoepic morphisms in 2-categories of pseudofunctors.
2. Statements like “coequifiers are lax epimorphisms”, Item 2 of Examples 2.4 of https://arxiv.org/abs/2109.09836, along with most of the other statements/examples there.
3. Dense, absolutely dense, etc. morphisms in bicategories

Other:

1. https://qchu.wordpress.com/
2. https://aroundtoposes.com/
3. https://ncatlab.org/nlab/show/essentially+surjective+and+full+functor
4. https://mathoverflow.net/questions/415363/objects-whose-representable-presheaf-is-a-fibration
5. https://mathoverflow.net/questions/460146/universal-property-of-isbell-duality
6. http://www.tac.mta.ca/tac/volumes/36/12/36-12abs.html ( Isbell conjugacy and the reflexive completion )
7. https://ncatlab.org/nlab/show/enrichment+versus+internalisation
8. The works of Philip Saville, https://philipsaville.co.uk/
9. https://golem.ph.utexas.edu/category/2024/02/from_cartesian_to_symmetric_mo.html
10. https://mathoverflow.net/q/463855 (One-object lax transformations)
11. https://ncatlab.org/nlab/show/analytic+completion+of+a+ring
12. https://en.wikipedia.org/wiki/Quaternionic_analysis
13. https://arxiv.org/abs/2401.15051 (The Norm Functor over Schemes)
14. https://mathoverflow.net/questions/407291/ (Adjunctions with respect to profunctors)
15. https://mathoverflow.net/a/462726 ($\mathsf{Prof}$ is free completion of $\mathsf{Cats}$ under right extensions)
16. there’s some cool stuff in https://arxiv.org/abs/2312.00990 (Polynomial Functors: A Mathematical Theory of Interaction), e.g. on cofunctors.
17. https://ncatlab.org/nlab/show/adjoint+lifting+theorem
18. https://ncatlab.org/nlab/show/Gabriel%E2%80%93Ulmer+duality