The Clowder Project

Set Theory:

1. https://math.stackexchange.com/questions/200389/show-that-the-set-of-all-finite-subsets-of-mathbbn-is-countable
2. https://mathoverflow.net/a/479528
3. https://www.maths.ed.ac.uk/~tl/ast/ast.pdf

Pointed sets:

1. Universal property of the smash product of pointed sets:
   1. Record the weaker version of Chapter 5: Tensor Products of Pointed Sets, saying that $\wedge $ is uniquely determined by those requirements:
      1. State as is
      2. Restate as saying that a “moduli category” of those is contractible/equivalent to $\mathsf{pt}$
   2. Study the “moduli category” of monoidal structures on $\mathsf{Sets}_{*}$ having $\wedge $ and $S^{0}$; is it contractible?
      1. Lax vs. oplax vs. etc. is a thing here.
      2. Do the same for $\webleft (\mathsf{Sets},\times ,\text{pt}\webright )$
2. Universal properties (plural!) of the left tensor product of pointed sets
3. Universal properties (plural!) of the right tensor product of pointed sets

Relations:

1. Alternative compositions for $\textbf{Rel}$:
   1. Figure which magma structures on $\{ \mathsf{t},\mathsf{f}\} $ induce associative and unital compositions on $\mathrm{Rel}$.
      1. One that does is
         \[ \webleft (S\circ R\webright )^{c}_{a}\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\int ^{b\in B}S^{c}_{b}\mathbin {\text{xor}}R^{b}_{a} \]

         with unit given by $\Delta _{\mathtt{f}}$.

      2. Another is apartness composition
   2. Figure what are the 2-categorical properties (internal monomorphisms, internal adjunctions, monads, etc.) of $\textbf{Rel}$ equipped with alternative compositions such as the apartness composition of the xor composition defined above
      1. In particular: Apartness composition, categorical properties of $\textbf{Rel}$ with apartness, and apartness relations.
2. Characterise the 2-categorical monomorphisms and epimorphisms in $\textbf{Rel}$.
3. Co/limits in $\textbf{Rel}$.
4. Codensity monad $\text{Ran}_{J}\webleft (J\webright )$ of a relation (What about $\text{Rift}_{J}\webleft (J\webright )$?)
   1. Density comonad $\text{Lan}_{J}\webleft (J\webright )$ of a relation when it exists (what about $\text{Lift}_{J}\webleft (J\webright )$?)
5. Explore relative co/monads in $\textbf{Rel}$, defined to be co/monoids in $\mathbf{Rel}\webleft (A,B\webright )$ with its left/right skew monoidal structures of Chapter 6: Relations, Section 6.4 and Section 6.5
6. Fibrations in $\textbf{Rel}$, like discrete fibrations and Street fibrations
7. Consider adding the sections
   • The Monoidal Bicategory of Relations
   • The Monoidal Double Category of Relations
   to .
8. internal relations, https://ncatlab.org/nlab/show/internal+relation

Spans:

1. Spans: study certain compositions of spans like composing $B\xleftarrow {f}A=A$ and $A=A\xleftarrow {g}B$ into a span $B\xleftarrow {f}A\xleftarrow {g}B$
2. Comparison double functor from Span to Rel and vice versa
3. Apartness composition for spans and alternate compositions for spans in general
4. non-Cartesian analogue of spans
   1. View spans as morphisms $S\to A\times B$ and consider instead morphisms $S\to A\otimes _{\mathcal{C}}B$
5. Record the universal property of the bicategory of spans of https://ncatlab.org/nlab/show/span
6. https://ncatlab.org/nlab/show/span+trace
7. Cospans.
8. Multispans.

Un/Straightening for Indexed and Fibred Sets:

1. Analogue of adjoints for Grothendieck construction for indexed and fibred sets
2. Write proper sections on straightening for lax functors from Sets to Rel or Span (displayed sets)
3. co/units for un/straightening adjunction

Categories:

1. https://en.wikipedia.org/wiki/Category_algebra
2. simple objects
3. https://mathoverflow.net/questions/442212/properties-of-categorical-zeta-function
4. Polynomial functors, https://ncatlab.org/nlab/show/polynomial+functor, https://arxiv.org/abs/2312.00990
5. https://ncatlab.org/nlab/show/simple+object
6. https://mathoverflow.net/questions/442212/properties-of-categorical-zeta-function
7. https://arxiv.org/abs/2409.17489
8. https://mathoverflow.net/a/478644
9. Posetal category associated to a poset as a right adjoint
10. “Presetal category” associated to a preordered set
11. Vopenka’s principle simplifies stuff in the theory of locally presentable categories. If we build categories using type theory or HoTT, what stuff from vopenka holds?
12. Are pseudoepic functors those functors whose restricted Yoneda embedding is pseudomonic and Yoneda preserves absolute colimits?
13. Absolutely dense functors enriched over $\mathbb {R}^{+}$ apparently reduce to topological density
14. Is there a reasonable notion of category homology? It is very common for the geometric realisation of a category to be contractible (e.g. having an initial or terminal object), but maybe some notion of directed homology could work here
15. Nerves of categories:
    1. Dihedral and symmetric nerves of categories via groupoids (define them first for groupoids and then Kan extend along $\mathsf{Grpd}\hookrightarrow \mathsf{Cats}$)
       1. Same applies to twisted nerves
    2. Cyclic nerve of a category
    3. Crossed Simplicial Group Categorical Nerves, https://arxiv.org/abs/1603.08768
16. Define contractible categories and add a discussion of universal properties as stating that certain categories are contractible. (Example of non-unique isomorphisms as e.g. being a group of order $5$ corresponds to all objects being isomorphic but the category not being contractible)
17. Expand and add a proof to it.
18. Sections and retractions; retracts, https://ncatlab.org/nlab/show/retract.
19. Groupoid cardinality
    1. https://ncatlab.org/nlab/show/groupoid+cardinality
    2. https://arxiv.org/abs/2104.11399
    3. https://terrytao.wordpress.com/2017/04/13/counting-objects-up-to-isomorphism-groupoid-cardinality/
    4. https://arxiv.org/abs/0809.2130
    5. https://qchu.wordpress.com/2012/11/08/groupoid-cardinality/
    6. https://mathoverflow.net/questions/363292/what-is-the-groupoid-cardinality-of-the-category-of-vector-spaces-over-a-finite
20. combinatorial species
    1. https://mathoverflow.net/questions/22462/what-are-some-examples-of-interesting-uses-of-the-theory-of-combinatorial-specie
    2. https://en.wikipedia.org/wiki/Combinatorial_species
21. Leinster’s the eventual image, https://arxiv.org/abs/2210.00302
22. https://ncatlab.org/nlab/show/separable+functor
23. Dagger categories:
    1. https://en.wikipedia.org/wiki/Dagger_category
    2. https://ncatlab.org/nlab/show/dagger+category
    3. Dagger compact categories, https://en.wikipedia.org/wiki/Dagger_compact_category
    4. https://mathoverflow.net/questions/220032/are-dagger-categories-truly-evil
    5. generalisation of dagger categories to categories with duality, i.e. categories $\mathcal{C}$ together with a functor $\dagger \colon \mathcal{C}^{\mathsf{op}}\to \mathcal{C}$
       1. Perhaps with the additional condition that $\dagger \circ \dagger =\text{id}$
       2. categories with involutions in general

Regular Categories:

1. https://arxiv.org/pdf/2004.08964.pdf.
2. Internal relations

Types of Morphisms in Categories:

1. Characterisation of epimorphisms in the category of fields, https://math.stackexchange.com/q/4941660
2. Strong epimorphisms
3. Behaviour in $\mathsf{Fun}\webleft (\mathcal{C},\mathcal{D}\webright )$, e.g. pointwise sections vs. sections in $\mathsf{Fun}\webleft (\mathcal{C},\mathcal{D}\webright )$.
4. Faithful functors from balanced categories are conservative
5. Natural cotransformations:
   1. If there is a natural transformation between functors between categories, taking nerves gives a homotopy equivalence (or something like that). What happens for natural cotransformations?
   2. Natural transformations come with a vertical composition map
      \[ \circ \colon \coprod _{G\in \mathsf{Fun}\webleft (\mathcal{C},\mathcal{D}\webright )}\text{Nat}\webleft (G,H\webright )\times \text{Nat}\webleft (F,G\webright )\to \text{Nat}\webleft (F,H\webright ). \]

      As Morgan Rogers shows here, there’s no vertical cocomposition map of the form

      \[ \text{CoNat}\webleft (F,H\webright )\to \prod _{G\in \mathsf{Fun}\webleft (\mathcal{C},\mathcal{D}\webright )}\text{CoNat}\webleft (G,H\webright )\times \text{CoNat}\webleft (F,G\webright ) \]

      or of the form

      \[ \text{CoNat}\webleft (F,H\webright )\to \prod _{G\in \mathsf{Fun}\webleft (\mathcal{C},\mathcal{D}\webright )}\text{CoNat}\webleft (G,H\webright )\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}\text{CoNat}\webleft (F,G\webright ) \]

      for natural cotransformations.

   3. Cap product for CoNat and Nat
      1. recovers map $\mathrm{Z}\webleft (G\webright )\times \mathrm{Cl}\webleft (G\webright )\to \mathrm{Cl}\webleft (G\webright )$.
   4. What is the geometric realisation of $\mathrm{CoTrans}\webleft (F,G\webright )$?
      1. Related: https://mathoverflow.net/questions/89753/geometric-realization-of-hochschild-complex
   5. What is the totalisation of $\mathrm{Trans}\webleft (F,G\webright )$?
      1. If we view sets as discrete topological spaces, what are the homotopy/homology groups of it? The nLab says this (https://ncatlab.org/nlab/show/totalization):

         The homotopy groups of the totalization of a cosimplicial space are computed by a Bousfield-Kan spectral sequence.

         The homology groups by an Eilenberg-Moore spectral sequence.

   6. Abstract

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

Presheaves and the Yoneda Lemma:

1. Yoneda extension along ${\text{よ}}_{\mathcal{D}}\circ F\colon \mathcal{C}\to \mathsf{PSh}\webleft (\mathcal{D}\webright )$, giving a functor left adjoint to the precomposition functor $F^{*}\colon \mathsf{PSh}\webleft (\mathcal{D}\webright )\to \mathsf{PSh}\webleft (\mathcal{C}\webright )$.
2. Consider the diagram
   
3. Does the functor tensor product admit a right adjoint (“Hom”) in some sense?
4. Yoneda embedding preserves limits
5. universal objects and universal elements
6. adjoints to the Yoneda embedding and total categories
7. The co-Yoneda lemma: co/presheaves are colimits of co/representables
8. Properties of categories of copresheaves
9. Contravariant restricted Yoneda embedding
10. Contravariant Yoneda extensions
11. Make table of $\text{Lift}_{{\text{よ}}}\webleft ({\text{よ}}\webright )$, $\text{Ran}_{{\text{よ}}}\webleft ({\text{よ}}\webright )$, $\text{Ran}_{{\text{よ}}}\webleft (\style {display: inline-block; transform: rotate(180deg)}{よ}\mkern -2.5mu\webright )$, etc.
12. 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/
13. Tensor product of functors and relation to profunctors
14. rifts and rans and lifts and lans involving yoneda in $\mathsf{Cats}$ and $\mathsf{Prof}$
15. Tensor product of functors and relation to rifts and rans of profunctors

Isbell Duality:

1. Important: I should reconsider going with the notation $\mathsf{O}$ and $\mathsf{Spec}$. Although a bit common in the (somewhat scarce) literature on Isbell duality, I have doubts regarding how useful/nice of a choice $\mathsf{O}$ and $\mathsf{Spec}$ are, and whether there are better choices of notation for them.
2. Interaction with $\times $, $\textup{Hom}$, $F_{!}$, $F^{*}$, and $F_{*}$
3. Interactions between presheaves and copresheaves:
   1. Natural transformations from a presheaf to a copresheaf and vice versa
   2. Mixed Day convolution?
4. Isbell duality for monoids:
   1. Set up a dictionary between properties of $\mathsf{Sets}^{\mathrm{L}}_{A}$ or $\mathsf{Sets}^{\mathrm{R}}_{A}$ and properties of $A$
   2. Do the same for $\mathsf{O}$ given by $A\mapsto \mathsf{Sets}^{\mathrm{L}}_{A}\webleft (X,A\webright )$
   3. Do the same for $\mathsf{Spec}$ given by $A\mapsto \mathsf{Sets}^{\mathrm{R}}_{A}\webleft (X,A\webright )$
   4. Do the same for $\mathsf{O}\circ \mathsf{Spec}$
   5. Do the same for $\mathsf{Spec}\circ \mathsf{O}$
   6. Algebras for $\mathsf{Spec}\circ \mathsf{O}$
   7. Coalgebras for $\mathsf{O}\circ \mathsf{Spec}$
5. Properties of $\mathsf{Spec}$ (e.g. fully faithfulness) vs. properties of $\mathcal{C}$
6. Properties of $\mathsf{O}$ (e.g. fully faithfulness) vs. properties of $\mathcal{C}$
7. co/unit being monomorphism/epimorphism
8. reflexive completion
9. Isbell duality for simplicial sets; what’s the reflexive completion?
10. Isbell envelope
11. What does Isbell duality look like, when Cat(Aop,Set) is identified with the category of discrete opfibrations over A, using A.5.14?
12. Generalizations of Isbell duality:
    1. Monoidal Isbell duality: monoidality for Isbell adjunction with day convolution (6.3 of coend cofriend)
    2. Isbell duality with sheaves
    3. Isbell duality with Lawvere theories, product preserving functors or whatever
    4. Isbell duality for profunctors
       1. In view of of , can we just use right Kan lifts/extensions?
       2. Right Kan lift/extension of Hom functors (there’s probably a version of the Yoneda lemma here)
          1. What is $\text{Rift}_{F}\webleft (\textup{Hom}_{\mathcal{C}}\webright )$
          2. What is $\text{Ran}_{F}\webleft (\textup{Hom}_{\mathcal{C}}\webright )$
          3. What is $\text{Rift}_{\textup{Hom}_{\mathcal{C}}}\webleft (F\webright )$
          4. What is $\text{Ran}_{\textup{Hom}_{\mathcal{C}}}\webleft (F\webright )$
          5. What is $\text{Lift}_{F}\webleft (\textup{Hom}_{\mathcal{C}}\webright )$
          6. What is $\text{Lan}_{F}\webleft (\textup{Hom}_{\mathcal{C}}\webright )$
          7. What is $\text{Lift}_{\textup{Hom}_{\mathcal{C}}}\webleft (F\webright )$
          8. What is $\text{Lan}_{\textup{Hom}_{\mathcal{C}}}\webleft (F\webright )$
13. Tensor product of functors and Isbell duality
    1. What is $\mathcal{F}\boxtimes _{\mathcal{C}}\mathsf{O}\webleft (\mathcal{F}\webright )$?
    2. What is $\mathsf{Spec}\webleft (F\webright )\boxtimes _{\mathcal{C}}F$?
    3. I think there is a canonical morphism
       \[ \mathcal{F}\boxtimes _{\mathcal{C}}\mathsf{O}\webleft (\mathcal{F}\webright )\to \mathrm{Tr}\webleft (\mathcal{C}\webright ). \]

       By the way, what is $\mathrm{Tr}\webleft (\mathbb {\Delta }\webright )$? What is $\mathrm{Tr}\webleft (\mathsf{B}{A}\webright )$? What about $\text{Nat}\webleft (\text{id}_{\mathcal{C}},\text{id}_{\mathcal{C}}\webright )$ for $\mathcal{C}=\mathsf{B}{A}$ or $\mathcal{C}=\mathbb {\Delta }$

14. Isbell with coends:
    1. $\textup{Hom}\webleft (F\webleft (A\webright ),h_A\webright )$ but it’s a coend
    2. Conatural transformations and all that
15. Co/limit preservation for O/Spec
16. Isbell duality for N vs. N + N
17. What do we get if we replace $\mathsf{O}\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\text{Nat}\webleft (-,h_{X}\webright )$ by $\text{Nat}^{\webleft [W\webright ]}\webleft (-,h_{X}\webright )$, and in particular by $\text{DiNat}\webleft (-,h_{X}\webright )$?

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.
2. Quotients of categories by actions of monoidal categories
   1. Quotients of categories by actions of monoids $\mathsf{B}{A}$
   2. Quotients of categories by actions of monoids $A_{\mathsf{disc}}$
   3. Lax, oplax, pseudo, strict, etc. quotients of categories
   4. lax Kan extensions along $\mathsf{B}{\mathcal{C}}\to \mathsf{B}{\mathcal{D}}$ for $\mathcal{C}\to \mathcal{D}$ a monoidal functor
3. Quotient of $\mathsf{Fun}\webleft (\mathsf{B}{A},\mathcal{C}\webright )$ by the $A$-action.
   1. This is used to build the cycle and $p$-cycle categories from the paracycle category.
   2. The quotient of $\mathsf{Fun}\webleft (\mathsf{B}\mathbb {N},\mathcal{C}\webright )$ by the $\mathbb {N}$-action should act as a kind of cyclic directed loop space of $\mathcal{C}$
4. $\mathsf{Fun}\webleft (\mathsf{B}\mathbb {N},\mathcal{C}\webright )$ as a homotopy pullback in $\mathsf{Cats}_{\mathsf{2}}$
   1. $\mathsf{Fun}\webleft (\mathsf{B}\mathbb {Z},\mathcal{C}\webright )$ as a homotopy pullback in $\mathsf{Grpd}_{\mathsf{2}}$
   2. Free loop space objects

Limits and colimits:

1. Initial/terminal objects as left/right adjoints to $!_{\mathcal{C}}\colon \mathcal{C}\to \mathsf{pt}$.
2. A small cocomplete category is a poset, https://mathoverflow.net/questions/108737/small-categories-and-completeness
3. Co/limits in $\mathsf{B}{A}$, including e.g. co/equalisers in $\mathsf{B}{A}$
4. Add the characterisations of absolutely dense functors given in to .
5. 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.
6. Dense functors, codense functors, and absolutely codense functors.
7. van kampen colimits

Completions and cocompletions:

1. https://mathoverflow.net/questions/429003/manifolds-as-cauchy-completed-objects
2. what is the conservative cocompletion of smooth manifolds? Is it related to diffeological spaces?
3. what is the conservative completion of smooth manifolds? Is it related to diffeological spaces?
4. what is the conservative bicompletion of smooth manifolds? Is it related to diffeological spaces?
5. completion of a category under exponentials
6. https://mathoverflow.net/questions/468897/cocompletion-without-cocontinuous-functors
7. The free cocompletion of a category;
8. The free completion of a category;
9. The free completion under finite products;
10. The free cocompletion under finite coproducts;
11. The free bicompletion of a category;
12. The free bicompletion of a category under nonempty products and nonempty coproducts (https://ncatlab.org/nlab/show/free+bicompletion);
13. Cauchy completions
14. Dedekind–MacNeille completions
15. Isbell completion (https://ncatlab.org/nlab/show/reflexive+completion)
16. Isbell envelope

Ends and Coends:

1. Ask Fosco about whether composition of dinatural transformations into higher dinaturals could be useful for https://arxiv.org/abs/2409.10237
2. Cyclic co/ends
   1. Try to mimic the construction given in Haugseng for the cycle, paracycle, cube, etc. categories
   2. cyclotomic stuff for cyclic co/ends
      1. Check out Ayala–Mazel-Gee–Rozenblyum’s Symmetries of the cyclic nerve
      2. isogenetic $\mathbb {N}^{\times }$-action (what the fuck does this mean?)
3. After stating the co/ends
   \[ \begin{aligned} \int ^{A\in \mathcal{C}}h_{A}\odot \mathcal{F}^{A},\\ \int ^{A\in \mathcal{C}}h^{A}\odot F_{A}, \end{aligned} \qquad \begin{aligned} \int _{A\in \mathcal{C}}\mathsf{Sets}\webleft (h_{A},\mathcal{F}^{A}\webright ),\\ \int _{A\in \mathcal{C}}\mathsf{Sets}\webleft (h^{A},F_{A}\webright ) \end{aligned} \]

   in the co/end version of the Yoneda lemma, add a remark explaining what the co/ends

   \[ \begin{aligned} \int _{A\in \mathcal{C}}h_{A}\odot \mathcal{F}^{A},\\ \int _{A\in \mathcal{C}}h^{A}\odot F_{A}, \end{aligned} \qquad \begin{aligned} \int ^{A\in \mathcal{C}}\mathsf{Sets}\webleft (h_{A},\mathcal{F}^{A}\webright ),\\ \int ^{A\in \mathcal{C}}\mathsf{Sets}\webleft (h^{A},F_{A}\webright ) \end{aligned} \]

   and the co/ends

   \[ \begin{aligned} \int ^{A\in \mathcal{C}}\mathcal{F}^{A}\odot h_{A},\\ \int ^{A\in \mathcal{C}}F_{A}\odot h^{A},\\ \int _{A\in \mathcal{C}}\mathcal{F}^{A}\odot h_{A},\\ \int _{A\in \mathcal{C}}F_{A}\odot h^{A},\\ \end{aligned} \qquad \begin{aligned} \int _{A\in \mathcal{C}}\mathsf{Sets}\webleft (\mathcal{F}^{A},h_{A}\webright ),\\ \int _{A\in \mathcal{C}}\mathsf{Sets}\webleft (F_{A},h^{A}\webright ),\\ \int ^{A\in \mathcal{C}}\mathsf{Sets}\webleft (\mathcal{F}^{A},h_{A}\webright ),\\ \int ^{A\in \mathcal{C}}\mathsf{Sets}\webleft (F_{A},h^{A}\webright ) \end{aligned} \]

   are.

4. ends $\mathcal{C}\to \mathcal{D}$ with $\odot $ is a special case of ends for a certain enrichment over $\mathcal{D}$
5. try to figure out what the end/coend
   \[ \int ^{X\in \mathcal{C}}h^{A}_{X}\times h^{X}_{B},\qquad \int _{X\in \mathcal{C}}h^{A}_{X}\times h^{X}_{B} \]

   are for $\mathcal{C}=\mathsf{B}{A}$. (I think the coend is like tensor product of $A$ as a left $A$-set with it as a right $A$-set)

6. Cyclic ends
7. Dihedral ends
8. Does Haugseng’s constructions give a way to define cyclic co/homology with coefficients in a bimodule?
9. Category of elements of dinatural transformation classifier
10. Examples of co/ends: https://mathoverflow.net/a/461814
11. Cofinality for co/ends, https://mathoverflow.net/questions/353876
12. “Fourier transforms” as in https://arxiv.org/pdf/1501.02503#page=168 or https://tetrapharmakon.github.io/stuff/itaca.pdf

Weighted/diagonal category theory:

1. co/ends as centre/trace-infused co/limits: compare the co/end of $\textup{Hom}_{\mathcal{C}}$ with the co/limit of $\textup{Hom}_{\mathcal{C}}$
2. Codensity $W$-weighted monads, $\text{Ran}^{\webleft [W\webright ]}_{F}\webleft (F\webright )$;
3. Codensity diagonal monads, $\mathrm{DiRan}_{F}\webleft (F\webright )$;

Profunctors:

1. 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 with the unit $h^{A}_{-}$. Is it unital for some other unit? Is there a less obvious analogue of apartness composition for profunctors? Or maybe does $\mathsf{Prof}$ equipped with $\square $ and units $h^{A}_{-}$ form a skew bicategory?

   Is $\Delta _{\text{Ø}}$ a unit?

2. Figure what monoidal category structures on $\mathsf{Sets}$ induce associative and unital compositions on $\mathsf{Prof}$.
3. https://mathoverflow.net/questions/470213/a-distributor-between-categories-induces-a-distributor-between-their-categories
4. Different compositions for profunctors from monoidal structures on the category of sets (e.g. https://mathoverflow.net/questions/155939/what-other-monoidal-structures-exist-on-the-category-of-sets)
5. Nucleus of a profunctor;
6. Isbell duality for profunctors:
   1. https://mathoverflow.net/questions/259525/isbell-duality-for-profunctors
   2. https://mathoverflow.net/questions/260322/the-mathfrak-l-functor-on-textsfprof
   3. https://mathoverflow.net/questions/262462/again-on-the-mathfrak-l-functor-on-mathsfprof

Centres and Traces of Categories:

1. Explicitly work out the trace and $\pi _{0}\mathsf{Fun}\webleft (\mathsf{B}\mathbb {N},-\webright )$ for monoids with few elements.
2. $\webleft [1_{A}\webright ]$ can contain more than one element. An example is $\mathsf{Sets}\webleft (\mathbb {N},\mathbb {N}\webright )$ and the maps given by
   \begin{align*} \webleft\{ 0,1,2,3,\ldots \webright\} & \mapsto \webleft\{ 0,0,1,2,\ldots \webright\} ,\\ \webleft\{ 0,1,2,3,\ldots \webright\} & \mapsto \webleft\{ 2,3,4,5,\ldots \webright\} . \end{align*}

   Show also that if $c\in \webleft [1_{A}\webright ]$, then $c$ is idempotent.

3. Drinfeld centre
4. trace of the symmetric simplex category; it’s probably different from that of $\mathsf{FinSets}$
5. Trace of $\mathsf{Rep}_{G}$ and interaction with induction, restriction, etc.
6. $\pi _{0}\webleft (\mathsf{B}\mathbb {N},\mathsf{B}{A}\webright )$, $K\webleft (\mathsf{B}\mathbb {N},\mathsf{B}{A}\webright )$, and $\mathrm{Tr}\webleft (\mathsf{B}\mathbb {N},\mathsf{B}{A}\webright )$ as concepts of conjugacy for monoids, their equivalents for categories, and comparison with traces
7. Comparison between $\pi _{0}\webleft (\mathsf{Fun}\webleft (\mathsf{B}\mathbb {N},\mathcal{C}\webright )\webright )$ and $K\webleft (\mathsf{Fun}\webleft (\mathsf{B}\mathbb {N},\mathcal{C}\webright )\webright )$
8. Lax, oplax, pseudo, and strict trace of simplex 2-category
9. duality over $\Gamma $ might give a map from product of a monoid with a set to $\mathrm{Tr}\webleft (\Gamma \webright )$
10. Studying the set $\text{Nat}\webleft (\text{id}_{\mathcal{C}},F\webright )$ as a notion of categorical trace:
    1. Ganter–Kapranov define the trace of a $1$-endomorphism $f\colon A\to A$ in a $2$-category $\mathcal{C}$ to be the set $\textup{Hom}_{\mathcal{C}}\webleft (\text{id}_{A},f\webright )$;
       1. https://arxiv.org/abs/math/0602510
       2. https://golem.ph.utexas.edu/string/archives/000757.html
       3. https://ncatlab.org/nlab/show/categorical+trace

       We should study this notion in detail, and also study $\text{Nat}\webleft (F,\text{id}_{\mathcal{C}}\webright )$ as well as $\text{CoNat}\webleft (\text{id}_{\mathcal{C}},F\webright )$ and $\text{CoNat}\webleft (F,\text{id}_{\mathcal{C}}\webright )$.

11. Centre of bicategories
12. Lax centres and lax traces
13. Examples of traces:
    1. Discrete categories
    2. Posets
       1. $\mathsf{Open}\webleft (X\webright )$
    3. Trace of small but non-finite categories:
       1. $\mathsf{Sets}$
       2. $\mathsf{Rep}\webleft (G\webright )$
       3. category of finite groups
       4. category of finite abelian groups
       5. category of finite $p$-groups for fixed $p$
       6. category of finite $p$-groups for all $p$
       7. category of finite fields
       8. category of finite topological spaces
       9. category of finite [insert a mathematical object here]
14. When is the trace of a groupoid just the disjoint sum of sets of conjugacy classes?
15. Set-theoretical issues when defining traces
    1. Sets is a large category, and yet we can speak of its centre
       \begin{align*} \mathrm{Z}\webleft (\mathsf{Sets}\webright ) & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\int _{A\in \mathsf{Sets}}\mathsf{Sets}\webleft (X,X\webright )\\ & \cong \text{Nat}\webleft (\text{id}_{\mathsf{Sets}},\text{id}_{\mathsf{Sets}}\webright )\\ & \cong \text{pt}. \end{align*}

       Is there a way to do the same for the trace of sets, or otherwise work with traces of large categories?

16. Understand how traces are defined via universal properties in Xinwen Zhu’s Geometric Satake, categorical traces, and arithmetic of Shimura varieties.
17. trace as an $\text{Obj}\webleft (\mathcal{C}\webright )$-indexed set
    1. properties, functoriality, etc.
18. Maybe actually call $\mathsf{Fun}\webleft (\mathsf{B}\mathbb {N},\mathcal{C}\webright )$ the categorical directed loop space of $\mathcal{C}$?
19. Cyclic version of $\mathsf{Fun}\webleft (\mathsf{B}\mathbb {N},\mathcal{C}\webright )$
20. Traces of categories, nerves of categories, and the cycle category

Categorical Hochschild Homology:

1. To any functor we have an associated natural transformation (). Do we have sharp transformations associated to natural transformation?
2. build Hochschild co/simplicial set and study its homotopy groups
3. $\mathsf{Fun}\webleft (\mathsf{B}\mathbb {N},X_{\bullet }\webright )$ vs. $\mathsf{Fun}\webleft (\Delta ^{1}/\partial \Delta ^{1},X_{\bullet }\webright )$
   1. Their $\pi _{0}$’s vs. the $\pi _{0}$’s of $\textup{Hom}_{X_{\bullet }}\webleft (x,x\webright )$, of $\textup{Hom}^{\mathrm{L}}_{X_{\bullet }}\webleft (x,x\webright )$, and $\textup{Hom}^{\mathrm{R}}_{X_{\bullet }}\webleft (x,x\webright )$.

Monoidal Categories:

1. Notion of braided monoidal categories in which the braiding is not an isomorphism. Relation to https://arxiv.org/abs/1307.5969
2. Proving a certain diagram between free monoidal categories commutes involves Fermat’s little theorem. Can we reverse this and prove Fermat’s little theorem from the commutativty of that diagram?
3. https://nilesjohnson.net/notes/grPic-P2S.pdf
4. Proof that monoidal equivalences $F$ of monoidal categories automatically admit monoidal natural isomorphisms $\text{id}_{\mathcal{C}}\cong F^{-1}\circ F$ and $\text{id}_{\mathcal{D}}\cong F\circ F^{-1}$.
5. Proof that category with products is monoidal under the Cartesian monoidal structure, [MO 382264].
6. Explore 2-categorical algebra:
   1. Find a construction of the free 2-group on a monoidal category. Apply it to the multiplicative structure on the category of finite sets and permutations, as well as to the multiplicative structure on the 1-truncation of the sphere spectrum, and try to figure out whether this looks like a categorification of $\mathbb {Q}$.
   2. What is the free 2-group on $\webleft (\mathbb {\Delta },\oplus ,\webleft [0\webright ]\webright )$?
7. Categorify the preorder $\leq $ on $\mathbb {N}$ to a promonad $\mathfrak {p}$ on the groupoid of finite sets and permutations $\mathbb {F}$:
   1. A preorder is a monad in $\mathrm{Rel}$
   2. A promonad is a monad in $\mathsf{Prof}$.
   3. There’s a promonad $\mathfrak {p}$ in $\mathbb {F}$ defined by
      \[ \mathfrak {p}\webleft (m,n\webright )\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft\{ \text{surjections from $\webleft\{ 1,\ldots ,m\webright\} $ to $\webleft\{ 1,\ldots ,n\webright\} $}\webright\} \]

      This promonad categorifies $\leq $ in that its values are the witnesses to the fact that $m$ is bigger than $n$ (i.e. surjections).

   4. Figure out whether this promonad extends to the 1-truncation of the sphere spectrum, and perhaps to other categorified analogues of monoids/groups/rings.
8. https://arxiv.org/abs/1307.5969
9. https://arxiv.org/abs/1306.3215
10. https://mathoverflow.net/questions/477219/reference-for-the-monoidal-category-structure-x-otimes-y-x-y-x-times-y
11. Include an explicit proof of
12. Include an explicit proof of
13. 
14. obstruction theory for braided enhancements of monoidal categories, using the “moduli category of braided enhancements”
15. Define symmetric and exterior algebras internal to braided monoidal categories
    1. https://mathoverflow.net/questions/471372/is-there-an-alternating-power-functor-on-braided-monoidal-categories
    2. https://arxiv.org/abs/math/0504155
16. https://mathoverflow.net/q/382364
17. https://mathoverflow.net/q/471490
18. Concepts of bicategories applied to monoidal categories (e.g. internal adjunctions lead to dualisable objects)
19. Involutive Category Theory
20. https://mathoverflow.net/questions/474662/the-analogy-between-dualizable-categories-and-compact-hausdorff-spaces

Bimonoidal Categories:

1. Include an explicit proof of

Six Functor Formalisms:

1. https://mathoverflow.net/questions/258159/yoga-of-six-functors-for-group-representations
2. Is the 1-categorical analogue of six functor formalisms given by Mann interesting?
   1. Mann defines:

      A six functor formalism is an $\infty $-functor $f\colon \mathsf{Corr}\webleft (C,E\webright )\to \mathsf{Cats}_{\infty }$ such that $-\otimes A$, $f^{*}$, and $f_{!}$ admit right adjoints

   2. Is the notion

      A 1-categorical six functor formalism is a (lax?) $2$-functor $f\colon \mathsf{Corr}\webleft (C,E\webright )\to \mathsf{Cats}_{2}$ (or should $\mathsf{Cats}$ be the target?) such that $-\otimes A$, $f^{*}$, and $f_{!}$ admit right adjoints

      interesting?

3. Interaction of the six functors with Kan extensions (e.g. how the left Kan extension of $-\otimes A$ may interact with the other functors)
4. Contexts like Wirthmuller Grothendieck etc
5. formalisation by cisinski and deglise
6. How do the following examples fit?
   1. base change between $\mathcal{C}_{/X}$ and $\mathcal{C}_{/Y}$
   2. $f_{!}\dashv f_{*}\dashv f^{*}$ adjunction between powersets
   3. $f_{!}\dashv f_{*}\dashv f^{*}$ adjunction between $\mathsf{Span}\webleft (\text{pt},A\webright )$ and $\mathsf{Span}\webleft (\text{pt},B\webright )$
   4. quadruple adjunction between powersets induced by a relation
   5. adjunctions between categories of presheaves induced by a functor or a profunctor
   6. Adjunction between left $A$-sets and left $B$-sets

   Do they have exceptional $f^{!}$? Is there a notion of Fourier–Mukai transform for them? What kind of compatibility conditions (proper base change, etc.) do we have?

Skew Monoidal Categories:

1. Free monoidal category on a skew monoidal category
2. Skew monoidal structures associated to a locally Cartesian closed category
3. Does the $\mathbb {E}_{1}$ tensor product of monoids admit a skew monoidal category structure?
4. 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?
5. 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?
6. Add example: $\mathcal{C}$ with coproducts, take $\mathcal{C}_{X/}$ and define
   \[ \webleft (X\xrightarrow {f}A\webright )\oplus \webleft (X\xrightarrow {g}B\webright )\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft [X\to X\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}X\xrightarrow {f\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}g}A\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}B\webright ] \]
7. Duals:
   1. Dualisable objects in monoidal categories and traces of endomorphisms of them, including also examples for monoidal categories which are not autonomous/rigid, such as $\webleft (\mathsf{Fun}\webleft (\mathcal{C},\mathcal{C}\webright ),\circ ,\text{id}_{\mathcal{C}}\webright )$.
   2. compact closed categories
   3. star autonomous categories
   4. Chu construction
   5. Balanced monoidal categories, https://ncatlab.org/nlab/show/balanced+monoidal+category
   6. Traced monoidal categories, https://ncatlab.org/nlab/show/traced+monoidal+category
8. Invertible objects and Picard groupoids
9. https://mathoverflow.net/questions/155939/what-other-monoidal-structures-exist-on-the-category-of-sets
10. Free braided monoidal category with a braided monoid: https://ncatlab.org/nlab/show/vine
11. https://golem.ph.utexas.edu/category/2024/08/skew_monoidal_categories_throu.html

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

Operads and Multicategories:

1. Simplicial lists in operad theory I

Monads:

1. Kantorovich monad (https://ncatlab.org/nlab/show/Kantorovich+monad) and probability monads in general, https://ncatlab.org/nlab/show/monads+of+probability%2C+measures%2C+and+valuations.

Enriched Categories:

1. $\mathcal{V}$-matrices

Bicategories:

1. https://mathoverflow.net/questions/478867/2-category-structure-on-modr
2. Bicategories of matrices, as in Street’s Variation through enrichment, also https://arxiv.org/abs/2410.18877
3. https://mathoverflow.net/a/86933
4. What are the internal 2-adjunctions in the fundamental $2$-groupoid of a space?
5. 2-category structure on $\mathsf{Mod}_{R}$, where a $2$-morphism is a commutative square. Characterisation of adjuntions therein (but be careful if this ends up being in an NDA)
6. Cook up a very large list of examples of bicategories, like the ones I made for the AI problems. In particular, find an interesting bicategory of representations qualitatively different from the one I described in the Epoch AI problem
7. 2-category structure on category of $R$-algebras as enriched $\mathsf{Mod}_{R}$-categories
8. Let $\mathcal{C}$ be a bicategory, let $A,B\in \text{Obj}\webleft (\mathcal{C}\webright )$, and let $F,G\in \text{Obj}\webleft (\mathsf{Hom}_{\mathcal{C}}\webleft (A,B\webright )\webright )$.
   1. Does precomposition with $\lambda ^{\mathcal{C}}_{A|F}\colon \text{id}_{A}\circ F\Rightarrow F$ induce an isomorphism of sets
      \[ \textup{Hom}_{\mathsf{Hom}_{\mathcal{C}}\webleft (A,B\webright )}\webleft (F,G\webright )\cong \textup{Hom}_{\mathsf{Hom}_{\mathcal{C}}\webleft (A,B\webright )}\webleft (F\circ \text{id}_{A},G\webright ) \]

      for each $F,G\in \text{Obj}\webleft (\mathsf{Hom}_{\mathcal{C}}\webleft (A,B\webright )\webright )$?

   2. Similarly, do we have an induced isomorphism of the form
      \[ \textup{Hom}_{\mathsf{Hom}_{\mathcal{C}}\webleft (A,B\webright )}\webleft (F,G\webright )\cong \textup{Hom}_{\mathsf{Hom}_{\mathcal{C}}\webleft (A,B\webright )}\webleft (F,\text{id}_{B}\circ G\webright ) \]

      and so on?

9. Are there two Duskin nerve functors? (lax/oplax/etc.?)
10. Interaction with cotransformations:
    1. Can we abstract the structure provided to $\mathsf{Cats}_{\mathsf{2}}$ by natural cotransformations?
    2. Are there analogues of cotransformations for $\textbf{Rel}$, $\mathsf{Span}$, $\mathsf{BiMod}$, $\mathsf{MonAct}$, etc.?
    3. Perhaps this might also make sense as a 1-categorical definition, e.g. comorphisms of groups from $A$ to $B$ as $\mathsf{Sets}\webleft (A,B\webright )$ quotiented by $f\webleft (ab\webright )\sim f\webleft (a\webright )f\webleft (b\webright )$.
11. Consider developing the analogue of traces for endomorphisms of dualisable objects in monoidal categories to the setting of bicategories, including e.g. the trace of a category as a trace internal to $\mathsf{Prof}$.
12. Centres of bicategories (lax, strict, etc.)
13. Concepts of monoidal categories applied to bicategories (e.g. traces)
14. 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].
15. Comonads in the bicategory of profunctors.
16. 2-limit of $\text{id},\text{id}\colon \mathsf{Sets}\rightrightarrows \mathsf{Sets}$ is $\mathsf{B}\mathbb {Z}$, https://mathoverflow.net/questions/209904/van-kampen-colimits?rq=1#comment520288_209904
17. https://mathoverflow.net/questions/473527/universal-property-of-2-presheaves-and-pseudo-lax-colax-natural-transformations
18. https://mathoverflow.net/questions/473526/free-cocompletion-of-a-2-category-under-pseudo-colimits-lax-colimits-and-colax

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

Internal adjunctions:

1. https://www.google.com/search?q=mate+of+an+adjunction
2. Moreover, by uniqueness of adjoints (, of ), this implies also that $S=f^{-1}$.
3. define bicategory $\mathsf{Adj}\webleft (\mathcal{C}\webright )$
4. walking monad
5. proposition: 2-functors preserve unitors and associators
6. https://ncatlab.org/nlab/show/2-category+of+adjunctions. Is there a 3-category too?
7. https://ncatlab.org/nlab/show/free+monad
8. https://ncatlab.org/nlab/show/CatAdj
9. https://ncatlab.org/nlab/show/Adj
10. $\mathsf{Adj}\webleft (\mathsf{Adj}\webleft (\mathcal{C}\webright )\webright )$
11. Examples of internal adjunctions
    1. Internal adjunctions in $\mathsf{Mod}$.
    2. Internal adjunctions in $\mathsf{PseudoFun}\webleft (\mathcal{C},\mathcal{D}\webright )$.
    3. Internal adjunctions in $\mathsf{LaxFun}\webleft (\mathcal{C},\mathcal{D}\webright )$.
    4. Internal adjunctions in 2-categories related to fibrations.

2-Categorical Limits:

1. https://sorilee.github.io/posts/strict-bilimit-and-its-proper-examples

Topos theory:

1. https://mathoverflow.net/q/479496
2. Grothendieck topologies on $\mathsf{B}{A}$
3. Enriched Grothendieck topologies
   1. Borceux–Quintero, https://www.numdam.org/item/CTGDC_1996__37_2_145_0/
   2. https://arxiv.org/abs/2405.19529
4. Cotopos theory:
   1. Copresheaves and copresheaf cotopoi
   2. Elementary cotopoi
      1. https://mathoverflow.net/questions/474287/intuition-for-the-internal-logic-of-a-cotopos
      2. https://mathoverflow.net/questions/394098/what-is-a-cotopos

         In case you haven’t seen it yet, Grothendieck studies (pseudo) cotopos in pursuing stacks

Formal category theory:

1. Yosegi boxes https://arxiv.org/abs/1901.01594

Homotopical Algebra:

1. https://arxiv.org/abs/2109.07803

Simplicial stuff:

1. https://arxiv.org/abs/2302.02484 and https://arxiv.org/abs/2411.19751
2. Internal adjunctions in $\mathbb {\Delta }$ are the same as Galois connections between $\webleft [n\webright ]$ and $\webleft [m\webright ]$. (NDA)
3. https://mathoverflow.net/q/478461
4. draw coherence for lax functors using the diagram for $\Delta ^{2}$
5. characterisation of simplicial sets such that left, right, and two-sided homotopies agree
6. every continuous simplicial set arises as the nerve of a poset.
7. Functor $\mathrm{sd}$ is convolution of ${\text{よ}}_{\mathbb {\Delta }}$ with itself; see https://arxiv.org/pdf/1501.02503.pdf#page=109
8. Extra degeneracies
   1. https://www.google.com/search?client=firefox-b-d&q=augmented+simplicial+objects+with+extra+degeneracies
   2. https://leanprover-community.github.io/mathlib_docs/algebraic_topology/extra_degeneracy.html
9. Comparison between $\Delta ^{1}/\partial \Delta ^{1}$ and $\mathsf{B}\mathbb {N}$

$\infty $-Categories:

1. https://arxiv.org/abs/2410.17102
2. https://arxiv.org/abs/2410.02578, https://scholar.colorado.edu/concern/graduate_thesis_or_dissertations/st74cr650, https://arxiv.org/abs/2206.00849
3. https://mathoverflow.net/questions/479716/non-strictly-unital-functors-of-infinity-categories
4. https://mathoverflow.net/questions/472253/whats-the-localization-of-the-infty-category-of-categories-under-inverting-f

Monoids:

1. https://math.stackexchange.com/a/4996051/603207, https://arxiv.org/abs/1006.5687
2. Six functor formalism for monoids, following Chapter 2: Constructions With Sets, Section 2.6.4, but in which $\cap $ and $\webleft [-,-\webright ]$ are replaced with Day convolution.
3. Monoid $\webleft (\webleft\{ 1,\ldots ,n\webright\} \cup \infty ,\operatorname*{\text{gcd}}\webright )$. The element $\infty $ can be replaced by $p^{\operatorname*{\text{min}}\webleft (e^{1}_{1},\ldots ,e^{m}_{1}\webright )}_{1}\cdots p^{\operatorname*{\text{min}}\webleft (e^{1}_{k},\ldots ,e^{m}_{k}\webright )}_{k}$.
4. Universal property of localisation of monoids as a left adjoint to the forgetful functor $\mathcal{C}\to \mathcal{D}$, where:
   • $\mathcal{C}$ is the category whose objects are pairs $\webleft (A,S\webright )$ with $A$ a monoid and $S$ a submonoid of $A$.
   • $\mathcal{D}$ is the category whose objects are pairs $\webleft (A,S\webright )$ with $A$ a monoid and $S$ a submonoid of $A$ which is also a group.
   Explore this also for localisations of rings

   Explore if we can define field spectra with an approach like this

5. Adjunction between monoids and monoids with zero corresponding to $\webleft (-\webright )^{-}\dashv \webleft (-\webright )^{+}$
6. Rock paper scissors as an example of a non-associative operation
7. https://mathoverflow.net/questions/438305/grothendieck-group-of-the-fibonacci-monoid
8. Witt monoid, https://www.google.com/search?q=Witt+monoid
9. semi-direct product of monoids, https://ncatlab.org/nlab/show/semidirect+product+group
10. morphisms of monoids as natural transformation between left $A$-sets over $A$ and $B_{A}$.
11. Figure out if 2-morphisms of monoids coming from $\mathsf{Fun}^{\otimes }\webleft (A_{\mathsf{disc}},B_{\mathsf{disc}}\webright )$, $\mathsf{PseudoFun}\webleft (\mathsf{B}{A},\mathsf{B}{B}\webright )$, etc. are interesting
12. Write sections on the quotient and set of fixed points of a set by a monoid action
13. 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:

14. Representation theory of monoids
    1. https://mathoverflow.net/questions/37115/why-arent-representations-of-monoids-studied-so-much
    2. Representation theory of groups associated to monoids (groups of units, group completions, etc.)

Monoid Actions:

1. $f_{!}\dashv f^{*}\dashv f_{*}$ adjunction
   1. Is it related to the Kan extensions adjunction for $f\colon \mathsf{B}{A}\to \mathsf{B}{B}$ and the categories $\mathsf{Sets}^{\mathrm{L}}_{A}\cong \mathsf{PSh}\webleft (\mathsf{B}{A}^{\mathsf{op}},\mathsf{Sets}\webright )$ and $\mathsf{Sets}^{\mathrm{L}}_{B}\cong \mathsf{PSh}\webleft (\mathsf{B}{B}^{\mathsf{op}},\mathsf{Sets}\webright )$?
   2. Is it related to the cobase change adjunction of https://ncatlab.org/nlab/show/base+change? Maybe we can take a morphism of monoids $f\colon A\to B$ and consider $B^{\mathrm{L}}_{A}$ as a left $A$-set, and then $\webleft (\mathsf{Sets}^{\mathrm{L}}_{A}\webright )_{A/}$ and $\webleft (\mathsf{Sets}^{\mathrm{L}}_{A}\webright )_{B^{\mathrm{L}}_{A}/}$
2. https://arxiv.org/abs/2112.10198
3. double category of monoid actions
4. Analogue of Brauer groups for $A$-sets
5. Hochschild homology for $A$-sets

Group Theory:

1. Is $\sum _{\webleft [g\webright ]\in \mathrm{Cl}\webleft (G\webright )}\frac{1}{\left\lvert g\right\rvert }$ an interesting invariant of $G$?
2. Idempotent endomorphism $f\colon A\to A$ is the same as a decomposition $A\cong B\oplus C$ via $B\cong \mathrm{Im}\webleft (f\webright )$ and $C\cong \mathrm{Ker}\webleft (f\webright )$.
   1. https://mathstrek.blog/2015/03/02/idempotents-and-decomposition/
3. https://math.stackexchange.com/questions/34271/order-of-general-and-special-linear-groups-over-finite-fields

Linear Algebra:

1. https://math.stackexchange.com/questions/1715249/the-number-of-subspaces-over-a-finite-field
2. https://math.stackexchange.com/questions/70801/how-many-k-dimensional-subspaces-there-are-in-n-dimensional-vector-space-over
3. https://en.wikipedia.org/wiki/Gaussian_binomial_coefficient
4. https://en.wikipedia.org/wiki/List_of_q-analogs

Commutative Algebra:

1. https://math.stackexchange.com/questions/637918/
2. https://categorytheory.zulipchat.com/#narrow/stream/411257-theory.3A-mathematics/topic/Big.20Witt.20ring
3. https://math.stackexchange.com/questions/535623/how-many-irreducible-factors-does-xn-1-have-over-finite-field
4. Derivations between morphisms of $R$-algebras, after https://mathoverflow.net/questions/434488
   1. Namely, a derivation from a morphism $f\colon A\to B$ of $R$-algebras to a morphism $g\colon A\to B$ of $R$-algebras is a map $D\colon B\to B$ such that we have
      \[ D\webleft (ab\webright )=g\webleft (a\webright )D\webleft (b\webright )+D\webleft (a\webright )f\webleft (b\webright ) \]

      for each $a,b\in B$.

Topological Algebra:

1. https://mathoverflow.net/q/477757
2. https://math.stackexchange.com/questions/2593556/galois-theory-for-topological-fields

Differential Graded Algebras:

1. https://mathoverflow.net/questions/476150/constructing-an-adjunction-between-algebras-and-differential-graded-algebras

Topology:

1. This paper has some cool references on convergence spaces: https://arxiv.org/abs/2410.18245
2. https://arxiv.org/abs/2402.12316
3. Write about the 6-functor formalism for sheaves on topological spaces and for topological stacks, with lots of examples.
   1. MO question titled 6-functor formalism for topological stacks: https://mathoverflow.net/q/471758

Measure Theory:

1. https://en.wikipedia.org/wiki/Valuation_%28measure_theory%29
2. There’s a theorem saying that there does not exist an infinite-dimensional “Lebesgue” measure, i.e. (from https://en.wikipedia.org/wiki/Infinite-dimensional_Lebesgue_measure):

   Let $X$ be an infinite-dimensional, separable Banach space. Then, the only locally finite and translation invariant Borel measure $\mu $ on $X$ is a trivial measure. Equivalently, there is no locally finite, strictly positive, and translation invariant measure on $X$.

   What kind of measures exist/not exist that satisfy all conditions above except being locally finite?

3. https://ncatlab.org/nlab/show/categories+of+measure+theory
4. Functions $f_{!}$, $f^{*}$, and $f_{*}$ between spaces of (probability) measures on probability/measurable spaces, mimicking how a map of sets $f\colon X\to Y$ induces morphisms of sets $f_{!}$, $f^{*}$, and $f_{*}$ between $\mathcal{P}\webleft (X\webright )$ and $\mathcal{P}\webleft (Y\webright )$.
5. Analogies between representable presheaves and the Yoneda lemma on the one hand and Dirac probability measures on the other hand
   1. Universal property of the embedding of a space $X$ into the space of probability measures on $X$
   2. Same question but for distributions
   3. non-symmetric metric on space of probability measures where we define $\mathrm{d}\webleft (\mu ,\nu \webright )$ to be the measure given by
      \[ U\mapsto \int _{U}\rho _{\mu }\, \mathrm{d}\nu , \]

      where $\rho _{\mu }$ is the probability density of $\mu $. Can we make this idea work?

6. https://arxiv.org/abs/0801.2250
7. https://mathoverflow.net/questions/325861

   In particular, I came across a PhD thesis by Martial Agueh. I thought it was interesting because it explicitly investigated the geodesics of Wasserstein space to produce solutions to a type of parabolic PDE.

Probability Theory:

1. https://www.epatters.org/wiki/stats-ml/categorical-probability-theory
2. https://ncatlab.org/nlab/show/category-theoretic+approaches+to+probability+theory
3. Categorical probability theory
4. https://golem.ph.utexas.edu/category/2024/08/introduction_to_categorical_pr.html
5. https://arxiv.org/abs/1109.1880
6. Connection between fractional differential operators and stochastic processes with jumps

Statistics:

1. https://towardsdatascience.com/t-test-from-application-to-theory-5e5051b0f9dc

Metric Spaces:

1. Lawvere metric spaces: object of $\mathcal{V}$-natural transformations corresponds to $\inf \webleft (\mathrm{d}\webleft (f\webleft (x\webright ),g\webleft (x\webright )\webright )\webright )$.
2. Does the assignment $d\webleft (x,y\webright )\mapsto d\webleft (x,y\webright )/\webleft (1+d\webleft (x,y\webright )\webright )$ constructing a bounded metric from a metric be given a universal property?
3. Explore Lawvere metric spaces in a comprehensive manner
4. metric $\mathrm{lcm}\webleft (x,y\webright )/\operatorname*{\text{gcd}}\webleft (x,y\webright )$ on $\mathbb {N}$, https://mathoverflow.net/questions/461588/. What shape do balls on $\mathbb {N}\times \mathbb {N}$ have with respect to this metric?
5. https://golem.ph.utexas.edu/category/2023/05/metric_spaces_as_enriched_categories_ii.html
6. Simon Willerton’s work on the Legendre–Fenchel transform:
   1. https://golem.ph.utexas.edu/category/2014/04/enrichment_and_the_legendrefen.html
   2. https://golem.ph.utexas.edu/category/2014/05/enrichment_and_the_legendrefen_1.html
   3. https://arxiv.org/abs/1501.03791

$p$-Adic Analysis:

1. Analysis of functions $\mathbb {Z}_{p}\to \mathbb {Q}_{q}$, $\mathbb {Q}_{p}\to \mathbb {Q}_{q}$, $\mathbb {Z}_{p}\to \mathbb {C}_{q}$, etc.
   1. https://siegelmaxwellc.wordpress.com/publications-pre-prints/

Partial Differential Equations:

1. Geometry of PDEs:
   1. https://mathoverflow.net/questions/457268/pdes-and-algebraic-varieties
   2. Can we build a kind of algebraic geometry of PDEs starting with the notion of the zero locus of a differential operator?
      1. https://ncatlab.org/nlab/show/diffiety

Functional Analysis:

1. In the Hilbert space $\ell ^{2}\webleft (\mathbb {N};\mathbb {C}\webright )$, the operator $\webleft (x_{n}\webright )_{n\in \mathbb {N}}\mapsto \webleft (x_{n+1}\webright )_{n\in \mathbb {N}}$ admits $\webleft (x_{n}\webright )_{n\in \mathbb {N}}\mapsto \webleft (0,x_{0},x_{1},\ldots \webright )$ as its adjoint.
2. https://arxiv.org/abs/2110.06300

Lie algebras:

1. Pre-Lie algebras
2. Post-Lie algebras

Modular Representation Theory:

1. https://math.stackexchange.com/questions/167979/representation-of-cyclic-group-over-finite-field
2. https://math.stackexchange.com/questions/153429/irreducible-representations-of-a-cyclic-group-over-a-field-of-prime-order

Homotopy theory:

1. https://ncatlab.org/nlab/show/group+actions+on+spheres, https://www.maths.ed.ac.uk/~v1ranick/papers/wall7.pdf, https://math.stackexchange.com/questions/1575798/which-groups-act-freely-on-sn, https://arxiv.org/abs/math/0212280.
2. Pascal’s triangle via homology of $n$-tori, https://topospaces.subwiki.org/wiki/Homology_of_torus
3. Conditions on morphisms of spaces $f\colon X\to Y$ such that $f^{*}\colon \webleft [Y,K\webright ]\to \webleft [X,K\webright ]$ or $f_{*}\colon \webleft [K,X\webright ]\to \webleft [K,Y\webright ]$ are injective/surjective (so, epi/monomorphisms in $\mathsf{Ho}\webleft (\mathsf{Top}\webright )$) or other conditions.

Differential Geometry:

1. functor of points approach to differential geometry

Number Theory:

1. https://math.stackexchange.com/questions/10233/uses-of-quadratic-reciprocity-theorem/10719#10719
2. https://mathoverflow.net/questions/120067/what-do-theta-functions-have-to-do-with-quadratic-reciprocity

Quantum Mechanics:

1. https://ncatlab.org/nlab/show/geometrical+formulation+of+quantum+mechanics

Quantum Field Theory:

1. https://arxiv.org/abs/2309.15913 and https://arxiv.org/abs/2311.09284
2. The current ongoing work on higher gauge theory, specially Christian Saemann’s
3. The recent work about determining the value of the strong coupling constant in the long-distance range, some pointers and keywords for this are available at this scientific american article.

Combinatorics:

1. Catalan numbers, https://mathstrek.blog/2012/02/19/power-series-and-generating-functions-ii-formal-power-series/

Other:

1. Are sedenions and higher useful for anything?
2. https://mathstodon.xyz/@pschwahn/113388126188923908
3. Tambara functors, https://arxiv.org/abs/2410.23052
4. 2-vector spaces
5. 2-term chain complexes. They form a 2-category and middle-four exchange holds, the proof using the fact that we have
   \[ h_{1}\circ \alpha +\beta \circ g_{2}=k_{1}\circ \alpha +\beta \circ f_{2}, \]

   which uses the chain homotopy identities

   \begin{align*} d_{V}\circ \alpha & = g_{2}-f_{2},\\ -\beta \circ d_{V} & = h_{1}-k_{1}. \end{align*}

   Can we modify this to work for usual chain complexes, seeking an answer to https://mathoverflow.net/questions/424268? What seems to make things go wrong in that case is that the chain homotopy identities are replaced with

   \begin{align*} \alpha _{n+1}\circ d^{V}_{n}+d^{W}_{n-1}\circ \alpha _{n} & = g_{n}-f_{n},\\ \beta _{n+1}\circ d^{V}_{n}+d^{W}_{n-1}\circ \beta _{n} & = k_{n}-h_{n}. \end{align*}
6. https://arxiv.org/abs/1402.2600
7. https://grossack.site/blog
8. Classifying space of $\mathbb {Q}_{p}$
9. https://www.valth.eu/proc.htm
10. Construction of $\mathbb {R}$ via slopes:
    1. http://maths.mq.edu.au/~street/EffR.pdf
    2. https://arxiv.org/abs/math/0301015
    3. Pierre Colmez’s comment “Et si on remplace $\mathbb {Z}$ par $\mathbb {Q}$, on obtient les adèles.”
    4. I wonder if one could apply an analogue of this construction to the sphere spectrum and obtain a kind of spectral version of the real numbers, as in e.g. following the spirit of https://mathoverflow.net/questions/443018.
11. https://arxiv.org/abs/2406.04936
12. https://mathoverflow.net/a/471510
13. https://mathoverflow.net/questions/279478/the-category-theory-of-span-enriched-categories-2-segal-spaces/448523#448523
14. The works of David Kern, https://dskern.github.io/writings
15. https://qchu.wordpress.com/
16. https://aroundtoposes.com/
17. https://ncatlab.org/nlab/show/essentially+surjective+and+full+functor
18. https://mathoverflow.net/questions/415363/objects-whose-representable-presheaf-is-a-fibration
19. https://mathoverflow.net/questions/460146/universal-property-of-isbell-duality
20. http://www.tac.mta.ca/tac/volumes/36/12/36-12abs.html ( Isbell conjugacy and the reflexive completion )
21. https://ncatlab.org/nlab/show/enrichment+versus+internalisation
22. The works of Philip Saville, https://philipsaville.co.uk/
23. https://golem.ph.utexas.edu/category/2024/02/from_cartesian_to_symmetric_mo.html
24. https://mathoverflow.net/q/463855 (One-object lax transformations)
25. https://ncatlab.org/nlab/show/analytic+completion+of+a+ring
26. https://en.wikipedia.org/wiki/Quaternionic_analysis
27. https://arxiv.org/abs/2401.15051 (The Norm Functor over Schemes)
28. https://mathoverflow.net/questions/407291/ (Adjunctions with respect to profunctors)
29. https://mathoverflow.net/a/462726 ($\mathsf{Prof}$ is free completion of $\mathsf{Cats}$ under right extensions)
30. there’s some cool stuff in https://arxiv.org/abs/2312.00990 (Polynomial Functors: A Mathematical Theory of Interaction), e.g. on cofunctors.
31. https://ncatlab.org/nlab/show/adjoint+lifting+theorem
32. https://ncatlab.org/nlab/show/Gabriel%E2%80%93Ulmer+duality

General TODO:

1. Browse MO questions/answers for interesting ideas/topics
2. Change Longrightarrow to Rightarrow where appropriate
3. Try to minimize the amount of footnotes throughout the project. There should be no long footnotes.