Mitchell Riley

Mitchell Riley is postdoctoral researcher at [[CQTS]]  @ NYU Abu Dhabi.

* [institute page](https://nyuad.nyu.edu/en/research/faculty-labs-and-projects/cqts/mitchell-riley.html) at NYU Abu Dhabi

* [old institute page](https://math.uiowa.edu/people/mitchell-riley) at Iowa

## Selected writings

The [[Kenzo]]-program for [[constructive mathematics|constructive]] [[algebraic topology]] ([[computational topology]]) re-written in [[Haskell]]:

* [[Mitchell Riley]]: [github.com/mvr/at](https://github.com/mvr/at)

On [[adjoint logic]]:

* [[Daniel Licata]], [[Mike Shulman]], and [[Mitchell Riley]], _A Fibrational Framework for Substructural and Modal Logics (extended version)_, in Proceedings of 2nd International Conference on Formal Structures for Computation and Deduction (FSCD 2017) ([doi: 10.4230/LIPIcs.FSCD.2017.25](http://drops.dagstuhl.de/opus/volltexte/2017/7740/), [pdf](http://dlicata.web.wesleyan.edu/pubs/lsr17multi/lsr17multi-ex.pdf))

On [[optics (in computer science)]], such as [[lenses (in computer science)|lenses]]:

* [[Mitchell Riley]], *Categories of optics* [[arXiv:1809.00738](https://arxiv.org/abs/1809.00738), video:[YT](https://www.youtube.com/watch?v=Qy7Y4-mgwbw)]

On [[homotopy type theory|homotopy]] [[dependent linear type theory]] of dependent [[stable homotopy types]] (via a form of [[bunched logic]]) with [[categorical semantics]] in [[parametrized spectra]]:

* {#RileyFinsterLicata21} [[Mitchell Riley]], [[Eric Finster]], [[Daniel Licata]], *Synthetic Spectra via a Monadic and Comonadic Modality* [[arXiv:2102.04099](https://arxiv.org/abs/2102.04099)]

* [[Mitchell Riley]], *Extending Homotopy Type Theory with Linear Type Formers*, talk at *The ForML LAb* (2021)  [[web](https://the-au-forml-lab.github.io/colloquium_talks/Riley.html), [video](https://www.youtube.com/watch?v=sPxtdCtjSDc)]

* {#Riley22Thesis} [[Mitchell Riley]], *A Bunched Homotopy Type Theory for Synthetic Stable Homotopy Theory*, PhD Thesis (2022) [[doi:10.14418/wes01.3.139](https://doi.org/10.14418/wes01.3.139), [ir:3269](https://digitalcollections.wesleyan.edu/object/ir%3A3269),  [pdf](https://mvr.hosting.nyu.edu/pubs/thesis.pdf)]

* {#Riley22} [[Mitchell Riley]], *Linear Homotopy Type Theory*, talk at: [HoTTEST Event for Junior Researchers 2022](https://www.uwo.ca/math/faculty/kapulkin/seminars/hottest_junior_2022.html)  (Jan 2022) [slides: [pdf](https://www.uwo.ca/math/faculty/kapulkin/seminars/hottestfiles/Riley-2022-01-20-HoTTEST.pdf), [[Riley-LHoTT-talk.pdf:file]], video: [YT](https://www.youtube.com/watch?v=o2oWhHabjdM)]

  > **Abstract.** Some ∞-toposes support constructions that are inherently 'linear', such as the external smash product of parameterised spectra. These cannot be added axiomatically to ordinary HoTT, because there is no way to enforce this linearity: there are no restrictions on variable uses. This talk describes an extension of HoTT with linear tensor and hom type formers, as a kind of 'binary modality' and its right adjoint. Trying to stay compatible with existing results in HoTT naturally leads us to a novel kind of bunched dependent type theory. Our type theory is intended to be as human-usable as possible, with an eye towards synthetic stable homotopy theory.

* [[Mitchell Riley]], *Dependent Type Theories à la Carte*, talk at *[[CQTS]] Initial Researcher's Meeting* (Sep 2022) [[[CQTS-InitialResearcherMeeting-Riley-220913.pdf:file]]]

On [[cohesive homotopy type theory]] with a pair of commuting cohesive structures (such as for [[differential cohomology|differential]] [[orbifold cohomology]]):

* [[David Jaz Myers]], [[Mitchell Riley]], *Commuting Cohesions* [[arXiv:2301.13780](https://arxiv.org/abs/2301.13780)]

Proof that [[Conway's game of life]] is omniperiodic:

Proof of omniperiodicity:

* Nico Brown, Carson Cheng, Tanner Jacobi, Maia Karpovich, Matthias Merzenich, David Raucci, [[Mitchell Riley]], *Conway's Game of Life is Omniperiodic* [[arXiv:2312.02799](https://arxiv.org/abs/2312.02799)]

* [The Physics arXiv Blog](https://www.discovermagazine.com/blog/physics-arxiv), *[Mathematicians Prove the "Omniperiodicity" of Conway's Game of Life](https://www.discovermagazine.com/the-sciences/mathematicians-prove-the-omniperiodicity-of-conways-game-of-life)* (Dec 2024)

category: people


Mitchell-Bénabou language

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## Idea

The **Mitchell--Bénabou language** is a particularly simple form of the [[internal language]] of an [[elementary topos]] $E$.  It makes use of the fact that in the presence of a [[subobject classifier]] $\Omega$, there is no need to treat formulas separately from [[terms]], since a formula or [[proposition]] can be identified with a term of [[type]] $\Omega$.

Specifically, the language is a [[type theory]] $L(E)$ where:

* the (closed) **[[types]]** are the [[objects]] of $E$;

* the **[[variables]]** of type $A$ are interpreted as [[identity morphism|identity morphisms]] $\mathrm{id}_A: A \to A$ in $E$;

* the **[[terms]]** of type $B$ in variables $x_i$ of type $X_i$ are interpreted as [[morphisms]] from the product of the $X_i$ to $B$

* the **[[formulas]]** are terms of type $\Omega$, where $\Omega$ is the [[subobject classifier]];

* the [[propositional logic|propositional logical]] connectives are induced from the internal [[Heyting algebra]] structure of $\Omega$;

* the (type-bounded) [[quantifiers]] are induced from the internal completeness of $\Omega$ (i.e., the quantifiers are given by suitable morphisms from internal [[power object|powers]] of $\Omega$ to $\Omega$)

* for each type $X$ there are also two binary relations $=_X$ (defined applying the [[diagonal]] map to the [[product]] term of the arguments) and $\in_X$ (defined applying the evaluation map to the product of the term and the power term of the arguments);

* a formula is [[true]] if the arrow which interprets it factors through the arrow $true: 1 \to \Omega$.

* one can also construct type families and [[dependent types]], just as in any [[locally cartesian closed category]]: the types indexed by elements of some closed type $A$ are the objects of the [[slice category]] over $A$; [[dependent sum|sums]] and [[dependent product|products]] of type families (i.e., $\Sigma$- and $\Pi$-types) are given by the [[left adjoint|left]] and [[right adjoint]]s to [[base change|change-of-base functors]], respectively. As these slice categories will be topoi themselves, all the above structure can be interpreted for type families as well.

## Applications

The Mitchell--Bénabou language, like the [[internal logic]] of any [[category]], is a powerful way to describe various objects in a topos as if they were [[sets]].  It can be viewed as making the topos into a generalized [[set theory]] or a [[type theory]], so that we can write and prove statements in a topos, and properties of a topos, using [[first order logic|first order]] [[intuitionistic logic|intuitionistic predicate logic]].

As is usual for type theories, we can conversely generate a [[syntactic category|syntactic]] or [[free topos]] $E(L)$ from any suitable theory $L$ phrased in the above language.  The universal property of this topos says that [[logical functors]] $E(L)\to E$, for any other topos $E$, are equivalent to models of the theory $L$ in $E$.

## Related concepts

* [[Kripke–Joyal semantics]]

* [[internal logic of an (infinity,1)-topos]]

## References
  {#References}

* {#MacLaneMoerdijk92} [[Saunders MacLane]], [[Ieke Moerdijk]], *[[Sheaves in Geometry and Logic]]*, Springer (1992) [[doi:10.1007/978-1-4612-0927-0](https://link.springer.com/book/10.1007/978-1-4612-0927-0)]

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Mitsuhiro Takeuchi

__Mitsuhiro Takeuchi__ is a Japanese algebraist. Much of his work is dedicated to bialgebras, Hopf algebras, coalgebras, (co)module algebras, [[Hopf-Galois extension]]s and various generalizations of these concepts as well as applications of bialgebras to the theory of algebraic and formal groups and quotients.

* _A simple proof of Gabriel and Popesco’s theorem_, J. Alg. 18,  112--113 (1971) [pdf](https://core.ac.uk/download/pdf/82301148.pdf)
* H. F. Kreimer, M. Takeuchi, _Hopf algebras and Galois extensions of an algebra_, Indiana Univ. Math. J. 30 (1981), 615-692 [web](https://www.iumj.indiana.edu/IUMJ/fulltext.php?year=1981&volume=30&artid=30052) [pdf](https://www.iumj.indiana.edu/IUMJ/FTDLOAD/1981/30/30052/pdf) [djvu](https://www.iumj.indiana.edu/IUMJ/FTDLOAD/1981/30/30052/djvu)
* [[Stefaan Caenepeel]], Septimiu Crivei, Andrei Marcus, Mitsuhiro Takeuchi, _Morita equivalences induced by bimodules over Hopf–Galois extensions_, J. Algebra __314__ (2007) 267--302 [pdf](https://core.ac.uk/download/pdf/81180359.pdf)
* Y. Doi, Mitsuhiro Takeuchi, _Cleft comodule algebras for a bialgebra_, Comm. Alg. __14__ (1986) 801--818 [doi](https://doi.org/10.1080/00927878608823337)
* M. Takeuchi, _Matched pairs of groups and bismash products of Hopf algebras_, Comm. Alg. __9__:8 (1981) 841--882 [doi](https://doi.org/10.1080/00927878108822621)

He introduced left adjoint to the forgetful functor from Hopf algebras to coalgebras and (by composing) also to the forgetful functor to vector spaces ([[free Hopf algebra]] functor), together with a first example of a Hopf algebra with non-bijective antipode in

* _Free Hopf algebras generated by coalgebras_, J. Math. Soc. Japan __23__:4 (1971) 561--582

[[Takeuchi product]] of $A$-rings (where $A$ is a noncommutative ring) is named after him, on basis of his fundamental work

_Groups of algebras over $A \times \bar{A}$, J. Math. Soc. Japan __29__, 459–492, 1977, [MR0506407](http://www.ams.org/mathscinet-getitem?mr=0506407), [euclid](http://projecteuclid.org/euclid.jmsj/1240432948)

Notice that the notation of End vs Coend in the above paper is interchanged (as compared to MacLane's [[CWM]]).

category: people


Mittag-Leffler condition

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## Idea

An [[inverse sequence]] ([[sequential diagram]]) $G_\bullet$ of [[groups]] is said to satisfy the _Mittag-Leffler condition_ if the [[images]] of groups from far down the sequence do not get smaller.

This is a condition used to assure the vanishing of the first [[derived functor]] of the [[limit]]-functor, $\underset{\longleftarrow}{lim}^1 G_\bullet$. See at _[[lim^1 and Milnor sequences]]_.

This is relevant for the preservation of exactness when applying limiting processes to exact sequences.  _

## Definition

An inverse sequence of groups consists of some groups $G_n$ indexed by the [[natural numbers]]  and between them group [[homomorphisms]]: if $m \gt n$, there is a homomorphism $p^m_n : G_m \to G_n$ and if $l\gt m  \gt n$, $p^m_n p^l_m= p^l_n$, so that we really just need the $p^{n+1}_n$s to define everything.

An inverse system $G = \{G_n,p^m_n\}$ is said to _satisfy the Mittag-Leffler property_ (or _condition_) if

for any $n$, there is an $n' \gt n$ such that for any $n'' \gt n'$,

$$p^{n''}_n(G_{n''}) = p^{n'}_n(G_{n'}).$$

## Discussion

This is the 'classical' form of the condition. It can also be applied in any category where images make sense.

An inverse sequence is a special type of [[pro-object]].

Any Mittag-Leffler pro-object is known to be [[essentially epimorphic pro-object|essentially epimorphic]] in the sense that it is isomorphic to a pro-object whose connecting morphisms are epis, that is to a [[strict pro-object]].

## Literature

Related $n$Lab entries include [[movable pro-object]].

Mittag-Leffler property of pro-objects in the category of pointed sets and in the category of (nonabelian) groups is studied in Ch.II, Sec. 6.2 of

* [[S. Mardešić]], J. Segal, _Shape theory_, North Holland 1982

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Mitutaka Murayama

## Selected writings

On [[universal principal bundle|universal]] [[equivariant principal bundles]]:

* [[Mitutaka Murayama]], [[Kazuhisa Shimakawa]], _Universal equivariant bundles_, Proc. Amer. Math. Soc. 123 (1995), 1289-1295 ([doi:10.1090/S0002-9939-1995-1231040-9](https://doi.org/10.1090/S0002-9939-1995-1231040-9))

category: people


Mitya Boryachenko

* [webpage](http://www.math.lsa.umich.edu/~mityab/)

## related $n$Lab entries

* [[dualizing object]]

category: people


Mitya Boyarchenko

* [webpage](http://www.math.uchicago.edu/~mitya/)

## related $n$Lab entries

* [[symmetric spectrum]]

category: people


mix rule

# The mix rule
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## Definition

The **mix rule** is an [[inference rule]] that can be added to [[linear logic]].  The most common mix rule is the binary mix rule

$$ \frac{\Gamma \vdash \Delta \quad \Gamma' \vdash \Delta'} {\Gamma, \Gamma' \vdash \Delta, \Delta'} $$

(or $\Gamma \otimes \Gamma' \vdash \Delta \parr \Delta'$), but to be [[unbiased]] one should also consider the nullary mix rule

$$ \frac{} {\vdash} $$

(or $\mathbf{1} \vdash \bot$), which together are equivalent to $\mathbf{1} \equiv \bot$, and imply $n$-ary mix rules for all $n$.

## Semantics

Ordinary linear logic has semantics in [[polycategories]], [[linearly distributive categories]], and [[star-autonomous categories]], depending on how many connectives are assumed.  A linearly distributive category satisfies the binary mix rule if there is a morphism $\bot \to \mathbf{1}$ satisfying a certain coherence axiom.  It additionally satisfies the nullary mix rule if this morphism is an [[isomorphism]].  In [CS97](#CS97) these are called **mix categories** and **isomix categories** respectively.

## References

* [[Robin Cockett]] and [[R.A.G. Seely]], 1997. _Proof theory for full intuitionistic linear logic, bilinear logic, and mix categories_. 1997. [[TAC]] 5:3. [Web](http://tac.mta.ca/tac/volumes/1997/n5/3-05abs.html).
  {#CS97}

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mixed characteristic

A commutative ring $R$ of [[characteristic]] $0$ is called a *ring of mixed characteristic* if $R$ has an Ideal $I$ such that the [[quotient]] $R/I$ is of prime characteristic.


mixed complex

## Idea

A **mixed complex** is both a [[chain complex]] and a [[cochain complex]] in a compatible way.  Equivalently, it is a [[dg-module]] over the [[dg-algebra]] $k[B]/(B^2)$, where $k$ is the base [[ring]], $B$ is of degree $-1$, and $dB = 0$.  Mixed complexes were introduced in the study of [[cyclic homology]].

## References

* [[Christian Kassel]], _Cyclic homology, comodules and mixed complexes_, J. Alg. 107 (1987), 195–216.

[[!redirects mixed complexes]]


Mixed Hodge Structures

This page compiles seome links concerning the book:

* [[Chris Peters]], [[Jozef Steenbrink]]:

  \linebreak

  **Mixed Hodge Structures**

  \linebreak

  Ergebisse der Mathematik (2008)

  [doi:10.1007/978-3-540-77017-6](https://doi.org/10.1007/978-3-540-77017-6)

  [pdf](http://www.arithgeo.ethz.ch/alpbach2012/Peters_Steenbrinck)

on [[Hodge theory]] in general and [[mixed Hodge structures]] in particular.

#Contents#
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## 7 Applications to algebraic cycles and to Singularities

### 7.1 The Hodge conjectures

### 7.2 Deligne cohomology

* [[Deligne cohomology]]

* [[intermediate Jacobian]]

### 7.2 The filtered de Rham complex and applications

category: reference


mixed model structure

## Definition

Let $M$ be a [[category]] with two closed [[model category|model structures]] $(C_q,W_q,F_q)$ and $(C_h,W_h,F_h)$, and assume that $F_h\subseteq F_q$ and $W_h \subseteq W_q$.

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###### Theorem
There is a (necessarily unique closed) **mixed model structure** $(C_m,W_q,F_h)$ on $M$ in which the fibrations are the $h$-fibrations, but the weak equivalences are the $q$-equivalences.
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## Properties

\begin{proposition}
Suppose $i$ and $j$ are _m_-cofibrations in the commutative diagram

\begin{tikzcd}
 & A \dlar[swap]{i} \drar{j} & \\
 X \arrow{rr}{f} & & Y
\end{tikzcd}

1. If $f$ is a _q_-equivalence, then it is an _h_-equivalence. In particular, a _q_-equivalence between _m_-cofibrant objects is an _h_-equivalence.

2. If $f$ is an _h_-cofibration, then it is an _m_-cofibration. In particular, an _h_-cofibration between _m_-cofibrant objects is an _m_-cofibration.

\end{proposition}
([May–Ponto 2012, proposition 17.3.4](#MayPonto2012))

\begin{theorem}
A map $i \colon A \to X$ is an _m_-cofibration iff it is an _h_-cofibration which is a composition

$$
  A \stackrel{j}{\to} K \stackrel{w}{\to} X
$$

of a _q_-cofibration $j$ and an _h_-equivalence $w$. In particular, an object $X$ is _m_-cofibrant iff it is _h_-equivalent to a _q_-cofibrant object.

\end{theorem}
([May–Ponto 2012, theorem 17.3.5](#MayPonto2012))

## Examples

* From the Quillen and Strøm [[model structures on topological spaces]], we obtain a mixed model structure in which the weak equivalences are the [[weak homotopy equivalences]], the fibrations are the [[Hurewicz fibrations]], and the cofibrant objects are those of the [[m-cofibrant space|homotopy type of a CW complex]].  (This example influences the notation above; the $q$-model structure is the Quillen one, the $h$-model structure is the "Hurewicz" (or "homotopy") one.)

* Similarly, we can mix the $q$- and $h$- [[model structures on chain complexes]].

* Let $T$ be an [[accessible functor|accessible]] [[strict 2-monad]] on a [[locally finitely presentable category|locally finitely presentable]] [[strict 2-category]] $K$.  Then by a theorem of Lack, the category $T Alg_s$ of strict $T$-[[algebra over a monad|algebras]] admits a [[transferred model structure]] from the [[2-trivial model structure]] on $K$, where the weak equivalences and fibrations are the morphisms which become internal equivalences and internal isofibrations in $K$.

  On the other hand, $T Alg_s$ has its own 2-trivial model structure.  Since the forgetful functor $T Alg_s \to K$ preserves equivalences and isofibrations (the latter since it has a strict left 2-adjoint), we can mix these two model structures to obtain one whose weak equivalences are the equivalences in $K$ (which are also the equivalences in the category $T Alg$ of $T$-algebras and [[pseudo morphisms]]), but whose fibrations are the isofibrations in $T Alg_s$.

## See also

* [[intermediate model structure]]

## References

The original paper is

* [[Michael Cole]], _Mixing model structures_, Top. Appl. 153 (2006) 1016–1032

There is also an exposition in

* {#MayPonto2012} [[Peter May]] and [[Kate Ponto]], _More Concise Algebraic Topology_ (2012), section 17.3.


mixed motive

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## Idea

Where the [[category of pure motives]] has [[smooth variety|smooth]] [[projective varieties]] as its [[objects]], the _category of mixed motives_ is supposed to be constructed from all [[smooth varieties]].

The category of mixed motives is supposed to be an [[abelian category|abelian]] [[tensor category]] which contains the [[pure motives]] as the [[full subcategory]] of [[semisimple objects]].

So far there is no realisation of such a category, but there are proposals by [[Vladimir Voevodsky]] and [[Marc Levine]] of [[triangulated categories]] that behave as its [[derived category]] is expected to.

## Voevodsky's mixed motives

Here we construct [[Voevodsky]]'s [[triangulated category]] of mixed motives following [Cisinski-Deglise](#CD).

1.  Let $S$ be a [[regular scheme|regular]] and [[noetherian scheme|noetherian]] base [[scheme]].   Let $Sm_S$ be the category of [[schemes]] [[smooth]] and of [[finite type]] over $S$.  Let $N_S^{tr}$ denote the [[closed symmetric monoidal category]] of [[Nisnevich sheaves with transfer]].  We will write $L_S[X]$ for the [[sheaf]] [[representable|represented]] by $X \in Sm_S$.

2.  Let $G_S$ denote the set of representable sheaves and $H_S$ as the set of [[complexes]] in  $Cpx(N_S^{tr})$ which are the [[cones]] of morphisms $L_S[Y] \to L_S[X]$  induced by [[hypercovers]] $Y\to X$.  The pair $(G_S, H_S)$ then defines a weakly flat [[descent structure]] (in the sense of [Cisinski-Deglise](#CD), see discussion at [[model structure on chain complexes]]) and therefore induces a [[symmetric monoidal model structure]] on $Cpx(N_S^{tr})$ where the [[weak equivalences]] are the [[quasi-isomorphisms]].

3.  Let $\mathcal{T}_S$ denote the set of [[complexes]] $Cone(L_S[A^1_X] \to L_S[X])$ where $A^1_X = A^1_S \times_S X$ is the [[affine line]] over $X$.  Call a [[complex]] $K$ **$A^1$-local** if $Hom_{D(A)}(T, K[n]) = 0$ for all $T \in \mathcal{T}$ and integers $n$.   Equivalently, the [[Nisnevich]] [[hypercohomology sheaves]] are [[homotopy invariant]].  In particular, for $F$ [[fibrant]] with respect to the above [[model structure]], $F$ is $A^1$-local iff the morphism $F(X) \to F(A^1_X)$ induced by the projection is a [[quasi-isomorphism]] for all $X \in Sm_S$.  This is again equivalent to the [[cohomology]] [[presheaves]] of $F$ being [[homotopy invariant]].

4.  Consider the [[left Bousfield localization]] of the above [[model structure]] at the class of morphisms $0 \to T[n]$ for $T \in \mathcal{T}_S$ and $n \in \mathbf{Z}$.  The [[fibrant objects]] are $G_S$-local and $A^1$-local complexes.  One can prove that this [[model structure]] is still [[symmetric monoidal model structure|symmetric monoidal]].

4.  The [[homotopy category]] of this [[model category]] is called the **triangulated category of effective motives** over $S$, denoted $DM^{eff}(S)$.  It is canonically equivalent to the [[full subcategory]] of the [[derived category]] $D(N_S^{tr})$ spanned by Nisnevich fibrant and $A^1$-local complexes.

5.  (introduce symmetric Tate spectra and DM(S))

## Related concepts

* [[motive]]

  * mixed motive,

    * [[mixed Tate motive]]

  * [[pure motive]],

    * [[Chow motive]], [[numerical motive]]

  * [[Voevodsky motive]]

  * [[noncommutative motive]]

* [[motivic cohomology]]

## References

* [[Marc Levine]], _Mixed Motives_, Handbook of K-theory ([pdf](http://www.uni-due.de/~bm0032/publ/MixMotKHB.pdf))

* {#CisinskiDeglise09} [[Denis-Charles Cisinski]], [[Frédéric Déglise]], *Triangulated categories of mixed motives*, Springer Monographs in Mathematics, Springer (2019) [[arXiv:0912.2110](http://arxiv.org/abs/0912.2110), [doi:10.1007/978-3-030-33242-6](https://doi.org/10.1007/978-3-030-33242-6)]

* {#CD} [[Denis-Charles Cisinski]], [[Frédéric Déglise]], *Local and stable homological algebra in Grothendieck abelian categories*, Homology, Homotopy and Applications **11** 1 (2009) 219–260 [[arXiv:0712.3296](http://arxiv.org/abs/0712.3296), [hha:1251832567](https://projecteuclid.org/journals/homology-homotopy-and-applications/volume-11/issue-1/Local-and-stable-homological-algebra-in-Grothendieck-abelian-categories/hha/1251832567.full)]

Section 8.3 of

* [[Alain Connes]], [[Matilde Marcolli]], _[[Noncommutative Geometry, Quantum Fields and Motives]]_  ([pdf](http://www.alainconnes.org/docs/bookwebfinal.pdf))

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mixed Tate motive

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## Idea

The [[stable (infinity,1)-category]] of _mixed Tate motives_ $DTM(S)$ is the [[localizing subcategory]] of the [[stable (infinity,1)-category]] of [[mixed motives]] $DM(S)$ generated by the twists of the [[Tate motive]] $\mathbf{Z}(n)$.

## Properties

* $DTM(S)$ can be described as a [[coreflective sub-(infinity,1)-category]] of $DM(S)$, since it follows from a theorem of [[Amnon Neeman]] that the inclusion admits a [[right adjoint]].

* The existence of a [[left adjoint]] is false for [[algebraically closed fields]] that are not the [[algebraic closure]] of a [[finite field]], while for fields [[algebraic extension|algebraic]] over a [[finite field]], the [[Tate-Beilinson conjecture]] would imply that the inclusion is a [[Frobenius functor]], i.e. that the [[right adjoint]] is also a [[left adjoint]].  See ([Totaro 15](#Totaro15)).

## Related concepts

* [[mixed motives]]
* [[mixed elliptic motives]]
* [[multiple zeta values]]

## References

* [[Marc Levine]], _Mixed Motives_, [[K-theory Handbook]], [pdf](http://www.math.illinois.edu/K-theory/handbook/1-429-522.pdf).

* [[Hélène Esnault]], [[Marc Levine]], _Tate motives and the fundamental group_, [arXiv:0708.4034](http://arxiv.org/abs/0708.4034).

Section 7 in

* [[Burt Totaro]], _The motive of a classifying space_, [arXiv:1407.1366](http://arxiv.org/abs/1407.1366).

* {#Totaro15} [[Burt Totaro]], _Adjoint functors on the derived category of motives_, [arXiv:1502.05079](http://arxiv.org/abs/1502.05079).

[[!redirects mixed Tate motives]]


mixed unitary quantum channel

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#Contents#
* table of contents
{:toc}

## Idea

In [[quantum information theory]], a *mixed unitary quantum channel* is a [[quantum channel]]

$$
  chan \,\colon\,
  \mathscr{H} \otimes \mathscr{H}^\ast
  \longrightarrow
  \mathscr{H} \otimes \mathscr{H}^\ast
$$

which admits an [operator-sum decomosition](quantum+channel#InTermsOfKrausDecompositions) all whose Kraus operators are multiples of [[unitary operators]]

$$
  s \,\colon\, S
  \;\;\;\;
    \vdash
  \;\;\;\;
  U \,\colon\, \mathscr{H} \to \mathscr{H}
$$

as

$$
  chan
  \;\colon\;
  \rho
  \;\mapsto\;
  \underset{s}{\sum}
  \,
  p_s \, U_s \cdot \rho \cdot U^\dagger
  \,,
$$

where the [[coefficients]]

$$
  p_{(-)} \,\colon\, S \to [0,1]
$$

necessarily form a [[probability distribution]] over the [[finite set|finite]] index [[set]] $S$.

## Examples

\begin{example}
  Of course, all [[unitary quantum channels]] are examples of mixed unitary quantum channels, this being the case (in particular) where the index set $S = \ast$ is the [[singleton set]].
\end{example}

\begin{example}
  A [[bit-flip quantum channel]]
  $$
    \array{
      QBit \otimes QBit^\ast
      &\xrightarrow{\phantom{-------}}&
      QBit \otimes QBit^\ast
      \\
      \rho
        &\mapsto&
      (1-p)\,\rho
       +
      p \, X \cdot \rho \cdot X
    }
  $$
  is a mixed unitary quantum channel, where the two unitary operators are the [[identity matrix]] on [[qbits]] and the $X$-[[Pauli gate]]-operator, respectively.
\end{example}

In fact:
\begin{example}\label{UnitalChannelOnSingleQBitIsMixedUnitary}
  Every [[unital quantum channel|unital]] quantum channel on a single [[qbit]] is a mixed unitary quantum channel.
\end{example}
This is attributed by [Müller-Hermes & Perry 2019](#Müller-HermesPerry19) (inside the proof of Cor. 1.4) to [Ruskai, Szarek & Werner 2002](#RuskaiSzarekWerner02), where it may be gleaned from combining Thm. 14 (every channel on qbits is the convex combination of two channels in the closure of extremal channels) there and the remark on p. 163 (that the extremal unital channels are the unitary channels).

## References

See most references on *[[quantum channels]]*.

Discussion of mixed unitary quantum channels specifically on single [[qbits]] (cf. *[[DQC1]]*):

* {#RuskaiSzarekWerner02} Mary Beth Ruskai, Stanislaw Szarek, Elizabeth Werner, *An analysis of completely-positive trace-preserving maps on $M_2$*, Linear Algebra and its Applications **347** 1–3 (2002) 159-187 [doi:10.1016/S0024-3795(01)00547-X]

* {#Müller-HermesPerry19} Alexander Müller-Hermes, Christopher Perry, *All unital qubit channels are 4-noisy operations*, Letters in Mathematical Physics **109** (2019) 1–9 [[doi:10.1007/s11005-018-1104-x](https://doi.org/10.1007/s11005-018-1104-x), [arXiv:1802.01337](https://arxiv.org/abs/1802.01337)]

[[!redirects mixed unitary quantum channels]]


Mizar

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#### Constructivism, Realizability, Computability
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# Contents
* table of contents
{: toc}

## Idea

Mizar is a [[proof assistant]] system. It is based on [[Tarski–Grothendieck set theory]] (rather than [[type theory]] as systems such as [[Coq]] or [[Agda]]). Presently Mizar stands out among other proof assistants by the large size of its library of formalized mathematics. On the other hand, the formal proof of research-level theorems in Mizar remains a challenge :

> {#ProblemOfPertinence} One of the biggest problems that worry the developers of automated deduction systems is that their systems are not sufficiently recognized and exploited by working mathematicians. Unlike the computer algebra systems, the use of which has indeed become ubiquitous in the work of mathematicians these days, deduction systems are still seldom used. They are mostly used to formalize proofs of well-established and widely known classical theorems, the Fundamental Theorem of Algebra formalized in the systems Coq and Mizar may serve as a perfect example here. Such work, however, is not always considered to be really challenging from the viewpoint of mathematicians who are concerned with obtaining their own new results. Therefore it has been recognized as a big challenge for the deduction systems community to prove that some of the state-of-the-art systems are developed enough to cope with formalizing recent mathematics. ([Naumowicz 06](#Naumowicz06))

## Related systems

[[!include proof assistants and formalization projects -- list]]

## References

* Wikipedia, _[Mizar system](http://en.wikipedia.org/wiki/Mizar_system)_

* [[Freek Wiedijk]], _[Mizar](http://www.cs.ru.nl/~freek/mizar/)_

* {#Naumowicz06} Adam Naumowicz, _An example of formalizing recent mathematical results in Mizar_, Journal of Applied Logic Volume 4, Issue 4, December 2006, Pages 396–413 Towards Computer Aided Mathematics ([publisher](http://www.sciencedirect.com/science/article/pii/S1570868305000686))

category: software

[[!redirects Mizar]]
[[!redirects Mizar system]]


Mladen Bestvina

* [webpage](https://www.math.utah.edu/~bestvina/)

* [Wikipedia entry](https://en.wikipedia.org/wiki/Mladen_Bestvina)

## Selected writings

On [[differential topology]]:

* [[Mladen Bestvina]] (notes by [[Adam Keenan]]), _Differentiable Topology and Geometry_, 2002 ([pdf](http://www.math.utah.edu/~keenan/manifoldsnotes.pdf), [[BestvinaKeenanDifferentialTopology.pdf:file]])

category: people


Mladen Pavičić

* [personal page](https://www2.irb.hr/korisnici/mpavicic/)

## Selected writings

On [[quantum logic]]:

* [[Mladen Pavičić]], *Bibliography on quantum logics and related structures*,  Int J Theor Phys **31** (1992) 373–455 [[doi:10.1007/BF00739999](https://doi.org/10.1007/BF00739999), [pdf](http://www.irb.hr/users/mpavicic/papers-ps-pdf/quantum-logic/1992-int-j-theor-phys-1.pdf)]

category: people

[[!redirects Mladen Pavicic]]


MO

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* table of contents
{:toc}

## Idea

The universal [[Thom spectrum]] (see there for more) of the [[orthogonal group]]. (...) Abstractly, this is the [[homotopy colimit]] of the [[J-homomorphism]] in [[Spectra]]:

$$
  MO = \underset{\rightarrow}{\lim}(B O \stackrel{J}{\to} B GL_1(\mathbb{S}) \to Spectra)
$$

## Properties

### Thom's theorem on $M O$
 {#ThomTheoremOnMO}

By [[Thom's theorem]] the [[stable homotopy groups]] of $M O$ form the [[bordism ring]] of unoriented manifolds

$$
  \pi_\bullet(M O) \simeq \Omega^O_\bullet
  \,.
$$

Moreover, this is the [[polynomial algebra]]

$$
  \pi_\bullet(M O)
  \simeq
  (\mathbb{Z}/2\mathbb{Z})[ x_n \;|\; n \in \mathbb{N}, \,n \geq 2, \, n \neq 2^t-1]
  \,.
$$

Due to ([Thom 54](#Thom54)). See for instance ([Kochman 96, theorem 3.7.6](#Kochman96))

The corresponding statement for [[MU]] is considerably more subtle, see _[[Milnor-Quillen theorem on MU]]_.

## Related concepts

* [[homology of MO]]

* [[orientation in generalized cohomology]]

\linebreak

[[!include flavours of cobordism cohomology theories -- table]]

## References

* {#Thom54} [[René Thom]], _Quelques propriétés globales des variétés différentiables_ Comment. Math. Helv. 28, (1954). 17-86

* {#Novikov62} [[Sergei Novikov]], _Homotopy properties of Thom complexes_, Mat. Sbornik 57 (1962), no. 4, 407–442, 407–442 ([pdf](http://www.mi-ras.ru/~snovikov/6.pdf), [[NovikovThomComplexes.pdf:file]])

Textbook accounts:

* {#Stong68} [[Robert Stong]], Chapter VI of: _Notes on Cobordism theory_, Princeton University Press, 1968 ([toc pdf](http://pi.math.virginia.edu/StongConf/Stongbookcontents.pdf), [ISBN:9780691649016](http://press.princeton.edu/titles/6465.html), [pdf](https://www.maths.ed.ac.uk/~v1ranick/papers/stongcob.pdf))

* {#Kochman96} [[Stanley Kochman]], section 1.5 and section 3.7 of: _[[Bordism, Stable Homotopy and Adams Spectral Sequences]]_, AMS 1996

Review:

* {#Malkiewich11} [[Cary Malkiewich]], section 2 of: _Unoriented cobordism and $M O$_, 2011 ([pdf](http://math.uiuc.edu/~cmalkiew/cobordism.pdf))

* [[Branko Juran]], _Thom spaces and the Oriented Cobordism Ring_, 2020 ([pdf](http://www.math.uni-bonn.de/people/daniel/2020/CharClasses/Vortrag4.pdf), [[JuranMO.pdf:file]])

Discussion of [[MO]]-bordism with [[MSO]]-[[manifold with boundary|boundaries]]:

* G. E. Mitchell, _Bordism of Manifolds with Oriented Boundaries_, Proceedings of the American Mathematical Society Vol. 47, No. 1 (Jan., 1975), pp. 208-214 ([doi:10.2307/2040234](https://doi.org/10.2307/2040234))

In the incarnation of $MO$ as a [[symmetric spectrum]]:

* {#Schwede12} [[Stefan Schwede]], Example I.2.8 in _Symmetric spectra_, 2012 ([pdf](http://www.math.uni-bonn.de/~schwede/SymSpec.pdf))

In the incarnation as an [[orthogonal spectrum]] (in fact as an [[equivariant spectrum]] in [[global equivariant stable homotopy theory]]):

* [[Stefan Schwede]], chapter V.4 of _[[Global homotopy theory]]_, 2015


MO5

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{:toc}

## Idea

The _MO5_  ([Dasgupta-Mukhil 95, Sec. 2FFurther ](#DasguptaMukhil95), [Witten 95, Sec. 3.3](#Witten95)) is one of the types of $\tfrac{1}{2}$ [[BPS state|BPS]] [[orientifold]] [[fixed loci]] in [[M-theory]]/on [[D=11 N=1 supergravity|D=11 N=1]] [[super-spacetime]]. Locally on [[super Minkowski spacetime]] it is the [[super-geometry|super]]-[[orientifold]]  $\mathbb{R}^{10,1\vert \mathbf{32}} \sslash \mathbb{Z}^{\Gamma^{56789}}_2$ given by the global [[orbifold quotient]] by the [[group of order 2]] which is generated by the [[Clifford algebra]]-element $\Gamma_5 \Gamma_6 \Gamma_7 \Gamma_8 \Gamma_9 \in Pin^+(10,1)$ in the positive [[pin group]] (for precise details see [HSS 18, Lemma 4.12](#HSS18)).

## Properties
 {#Properties}

### Charge, anomaly cancellation and duality to O4-planes
 {#ChargeAndAnomalyCancellation}

Under [[duality between M-theory and type IIA string theory]], for [[KK-compactification]] along a [[circle]]-[[fiber]] parallel to the MO5, a plain MO5 becomes the [[O4-plane]], specifically the $O^- 4$-plane, while an MO5 with an [[M5-brane]] on top of it becomes the $O^+ 4$-plane ([Gimon 9, Sec. III8](#Gimon98),[Hanany-Kol 00, Sec. 3.1](#HananyKol00)).

Hence the spacetime [[dimensions]] of the MO5 is the same that of the [[black brane|black]] [[M5-brane]], but its charge as an [[O-plane]] is $- \tfrac{1}{2}$ that of a single [[M5-brane]] ([Dasgupta-Mukhil 95, Sec. 2](#DasguptaMukhil95), [Witten 95, 3.3](#Witten95), [Hori 98, 2.1](#Hori98)), so that am [[M-theory|M-theoretic]] lift of [[RR-field tadpole cancellation]] requires the presence of 16 [[M5-branes]] in a [[toroidal orbifold|toroidal]] MO5-type [[orientifold]] $\mathbb{T}^{\mathbf{5}_{sgn}} \!\sslash\! \mathbb{Z}_2$.

### As a sector of heterotic M-theory on ADE-singularities

By the classification of [MFF 12, Sec. 8.3](M5-brane#MFF12), the MO5-plane is _not_ in fact a $\tfrac{1}{2}$ [[BPS state|BPS]] solution of [[D=11 N=1 supergravity]]. But the [[brane intersection|intersection]] of an [[MK6]] with an [[MO9-plane]] is a $\tfrac{1}{4}$ [[BPS state|BPS]] solution to [[D=11 N=1 supergravity]].

> graphics grabbed from [HSS18, Example 2.2.7](#HSS18)

Thus, [[heterotic M-theory on ADE-orbifolds]] the MO5-plane becomes the [[fixed locus]] of the [[diagonal]] $\mathbb{Z}_2 \overset{ diag }{\hookrightarrow } \mathbb{Z}_2 \times G^{DE}$, hence the [[brane intersection|intersection]] of an [[MO9]] with an [[MK6]]${}_{DE}$ ([FLO 99, Sec. 4](#FLO99)), also called the $-\tfrac{1}{2}M5$, since under [[duality between M-theory and type IIA string theory]] this becomes the [[half NS5-brane]] of [[type I' string theory]].

### As geometric engineering of $D=6$ $\mathcal{N} = (2,0)$ SCFTs

Coincident [[M5-branes]] on an [[MO5-plane]] are supposed to [[geometric engineering of QFTs|geometrically engineer]] [[D=6 N=(2,0) SCFTs]] with [[classification of simple Lie groups|D-series]] [[gauge groups]] ([AKY 98, Sec. IIB](#AKY98)).

But if these are really $-\tfrac{1}{2} M5 = MK6 \cap MO9$-branes of [[heterotic M-theory on ADE-orbifolds]], then they will [[geometric engineering of QFTs|geometrically engineer]] [[D=6 N=(1,0) SCFTs]], instead, see [there](heterotic+M-theory+on+ADE-orbifolds#GeometricEngineeringOfDIs6NIs1SCFTs).

## References

### General

The original articles are

* {#DasguptaMukhil95} Keshav Dasgupta, [[Sunil Mukhi]], _Orbifolds of M-theory_, Nucl. Phys. B465 (1996) 399-412 ([arXiv:hep-th/9512196](https://arxiv.org/abs/hep-th/9512196))

* {#Witten95} [[Edward Witten]], _Five-branes And M-Theory On An Orbifold_, Nucl. Phys. B463:383-397, 1996 ([arXiv:hep-th/9512219](https://arxiv.org/abs/hep-th/9512219))

* {#Hori98} [[Kentaro Hori]], _Consistency Conditions for Fivebrane in M Theory on $\mathbb{R}^5/\mathbb{Z}_2$ Orbifold_, Nucl.Phys.B539:35-78, 1999 ([arXiv:hep-th/9805141](https://arxiv.org/abs/hep-th/9805141))

Classification view of [[rational equivariant homotopy theory|rational]] [[equivariant Cohomotopy]] of [[super-spacetime]]:

* {#HSS18} [[John Huerta]], [[Hisham Sati]], [[Urs Schreiber]], Example 2.2.2 of: _[[schreiber:Equivariant homotopy and super M-branes|Real ADE-equivariant (co)homotopy and Super M-branes]]_, CMP (2019) ([arXiv:1805.05987](https://arxiv.org/abs/1805.05987), [doi:10.1007/s00220-019-03442-3](http://link.springer.com/article/10.1007/s00220-019-03442-3))

### As a sector of the $-\tfrac{1}{2}M5$ in heterotic M-theory at ADE-singularities

Discussion in the more general context of [[heterotic M-theory on ADE-orbifolds]] is in

* {#FLO99} Michael Faux, [[Dieter Lüst]], [[Burt Ovrut]], Section 4 of _Intersecting Orbifold Planes and Local Anomaly Cancellation in M-Theory_, Nucl. Phys. B554: 437-483, 1999 ([arXiv:hep-th/9903028](https://arxiv.org/abs/hep-th/9903028))

and in view of [[super-embedding formalism|super-embeddings]] of [[M5-branes]] in

* {#FSS19c} [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]], Section 4 of: _[[schreiber:Super-exceptional embedding construction of the M5-brane|Super-exceptional geometry: origin of heterotic M-theory and super-exceptional embedding construction of M5]]_ ([arXiv:1908.00042](https://arxiv.org/abs/1908.00042))

### Relation to O4-planes

Relation of the MO5 to the [[O4-plane]] under [[duality between M-theory and type IIA string theory]]:

* {#Hori98} [[Kentaro Hori]], Section 3 of: _Consistency Conditions for Fivebrane in M Theory on $\mathbb{R}^5/\mathbb{Z}_2$ Orbifold_, Nucl. Phys. B539:35-78, 1999 ([arXiv:hep-th/9805141](https://arxiv.org/abs/hep-th/9805141))

* {#Gimon98} [[Eric Gimon]], _On the M-theory Interpretation of Orientifold Planes_ ([arXiv:hep-th/9806226](https://arxiv.org/abs/hep-th/9806226), [spire:472499](http://inspirehep.net/record/472499))

* {#HananyKol00} [[Amihay Hanany]], [[Barak Kol]], Section 3.1 in: _On Orientifolds, Discrete Torsion, Branes and M Theory_, JHEP 0006 (2000) 013 ([arXiv:hep-th/0003025](https://arxiv.org/abs/hep-th/0003025))

* Yoonseok Hwang, Joonho Kim, Seok Kim, _M5-branes, orientifolds, and S-duality_,  J. High Energ. Phys. (2016) 2016: 148 ([arXiv:1607.08557](https://arxiv.org/abs/1607.08557))

### Geometric engineering of orthogonal $D=6$, $\mathcal{N}=(2,0)$ SCFTs

As [[geometric engineering of QFTs|geometric engineering]] of [[D=6 N=(2,0) SCFTs]] with [[classification of simple Lie groups|D-series]] [[gauge groups]]:

* {#AKY98} Changhyun Ahn, Hoil Kim, Hyun Seok Yang, _$SO(2N)$ $(0,2)$ SCFT and M Theory on $AdS_7 \times \mathbb{R}P^4$_, Phys.Rev. D59 (1999) 106002 ([arXiv:hep-th/9808182](https://arxiv.org/abs/hep-th/9808182))

[[!redirects MO5-plane]]


Mochizuki's corollary 3.12

[[!redirects Mochizuki's corollary 3.1.2]]
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#Contents#
* table of contents
{:toc}

## Idea

_Mochizuki's corollary 3.12_ or _Mochizuki's inequality_ is a conjecture by [[Shinichi Mochizuki]] in [[number theory]] and [[algebraic geometry]] that first appeared in [[Mochizuki's proof of abc|his attemped proof]] of the [[abc conjecture]] from [[inter-universal Teichmüller theory]]. Mochizuki's proof of Mochizuki's corollary 3.12 from [[inter-universal Teichmüller theory]] was shown to be incorrect by [[Peter Scholze]] and [[Jakob Stix]].

## Statement

Let $E$ be an [[elliptic curve]] over a [[number field]] $F$ with [[initial theta data]] $(\overline{F}/F, E_F, l, \underline{M}, \underline{V}, V_{mod}^{bad}, \underline{\epsilon})$. Then the following inequality holds:

$$
  -\underline{\widehat{deg}}(P_q)\leq\underline{\widehat{deg}}_{lgp}(P_{hull(U_\Theta)})
$$

TODO: details

## Consequences

If Mochizuki's corollary 3.12 happens to be true, then [[Szpiro's conjecture]] and the [[abc conjecture]] follow as a consequence.

## Related concepts

* [[Szpiro's conjecture]]

* [[initial theta data]]

* [[inter-universal Teichmüller theory]]

## References

* [[Shinichi Mochizuki]], _Inter-universal Teichmüller theory III, Canonical splittings of the Log-theta-lattice_ (2012) ([pdf](http://www.kurims.kyoto-u.ac.jp/~motizuki/Inter-universal%20Teichmuller%20Theory%20III.pdf))

* [[Peter Scholze]], [[Jakob Stix]], _Why abc is still a conjecture_, ([pdf](https://ncatlab.org/nlab/files/why_abc_is_still_a_conjecture.pdf))

* [[Taylor Dupuy]], [[Anton Hilado]], _The Statement of Mochizuki's Corollary 3.12, Initial Theta Data, and the First Two Indeterminacies_, ([arxiv:2004.13228](https://arxiv.org/abs/2004.13228))

* Taylor Dupuy, Anton Hilado, _Probabilistic Szpiro, Baby Szpiro, and Explicit Szpiro from Mochizuki's Corollary 3.12_, ([arxiv:2004.13108](https://arxiv.org/abs/2004.13108))

Lectures on Mochizuki's inequality:

* Taylor Dupuy, "A User's Guide to Mochizuki's Inequality", Spring 2019, University of Tennesse Knoxville, Barrett Lectures ([slides](https://www.dropbox.com/s/b63figtzrg86u5t/IUTFAQ.pdf?))

Claim of proof of similar statements:

* [[Kirti Joshi]], *Construction of Arithmetic Teichmuller Spaces III: A Rosetta Stone and a proof of Mochizuki's Corollary 3.12* [[arXiv:2401.13508](https://arxiv.org/abs/2401.13508)]

[[!redirects Mochizuki's Corollary 3.12]]
[[!redirects Mochizuki's inequality]]


Mochizuki's proof of abc

#Contents#
* table of contents
{:toc}

## Idea

In the preprints listed at _[[inter-universal Teichmüller theory]]_, [[Shinichi Mochizuki]] has claimed, in particular, a proof of the _[[abc conjecture]]_. Over the years, the proposed proof has been met with a fair amount of scepticism from various authors, some of which is referenced below.

## Reactions

### Dimitrov's commentary

> The following is verbatim quoted from [Dimitrov 2012](#Dimitrov12).

***

**Completely rewritten. (9/26)**

It seems indeed that nothing like Theorem 1.10 from Mochizuki's IUTT-IV could hold.

Here is an infinite set of counterexamples, assuming for convenience two standard conjectures (the first being in fact a consequence of ABC), that contradict Thm. 1.10 *very* badly.

*Assumptions:*

- A (Consequence of ABC) *For all but finitely many elliptic curves over $\mathbb{Q}$, the conductor $N$ and the minimal discriminant $\Delta$ satisfy $\log{|\Delta|} \lt (\log{N})^2$.*

- B (Uniform Serre Open Image conjecture) *For each* $d \in \mathbb{N}$, *there is a constant* $c(d) \lt \infty$ *such that for every number field* $F/\mathbb{Q}$ with $[F:\mathbb{Q}] \leq d$, *and every
     non-CM elliptic curve* $E$ *over* $F$, *and every prime* $\ell \geq c(d)$, *the Galois representation of* $G_F$ *on* $E[\ell]$ *has full image* $\mathrm{GL}_2(\mathbb{Z}/{\ell})$. (In fact, it is sufficient to take the weaker version in which $F$ is held fixed. )

Further, as far as I can tell from the proof of Theorem 1.10 of IUTTIV, the only reason for taking $F \coloneqq F_{\mathrm{tpd}}\big( \sqrt{-1}, E_{F_{\mathrm{tpd}}}[3\cdot 5]
 \big)$ --- rather than simply $F \coloneqq F_{\mathrm{tpd}}(\sqrt{-1})$ --- was to ensure that $E$ has semistable reduction over $F$. *Since I will only work in what follows with semistable elliptic curves over* $\mathbb{Q}$, *I will assume, for a mild technical convenience in the examples below, that for elliptic curves already semistable over* $F_{\mathrm{tpd}}$, *we may actually take* $F \coloneqq F_{\mathrm{tpd}}(\sqrt{-1})$ *in Theorem 1.10.*

*The infinite set of counterexamples.* They come from Masser's paper *Masser: Note on a conjecture of Szpiro, *Asterisque* 1990*, as follows. Masser has produced an infinite set of Frey-Hellougarch (i.e., semistable and with rational 2-torsion) elliptic curves over $\mathbb{Q}$ whose conductor $N$ and minimal discriminant $\Delta$ satisfy
\[
\label{eqone}
\frac{1}{6}\log{|\Delta|} \geq \log{N} + \frac{\sqrt{\log{N}}}{\log{\log{N}}}.
\]
(Thus, $N$ in these examples may be taken arbitrarily large. ) By (A) above, taking $N$ big enough will ensure that
\[
\label{eqtwo}
\log{|\Delta|} \lt (\log{N})^2.
\]
Next, the sum of the logarithms of the primes in the interval $\big( (\log{N})^2, 3(\log{N})^2 \big)$ is $2(\log{N})^2 + o((\log{N})^2)$, so it is certainly $\gt (\log{N})^2$ for $N \gg 0$ big enough.  Thus, by \eqref{eqtwo}, it is easy to see that the interval $\big( (\log{N})^2, 3(\log{N})^2 \big)$ contains a prime $\ell$ which divides neither $|\Delta|$ nor any of the exponents $\alpha = \mathrm{ord}_p(\Delta)$ in the prime factorization $|\Delta| = \prod p^{\alpha}$ of $|\Delta|$.

Consider now the pair $(E,\ell)$: it has $F_{\mathrm{mod}} = \mathbb{Q}$, and since $E$ has rational $2$-torsion, $F_{\mathrm{tpd}} = \mathbb{Q}$ as well. Let $F \coloneqq \mathbb{Q} \big(
\sqrt{-1}\big)$. I claim that, upon taking $N$ big enough, the pair $(E_F,\ell)$ arises from an **initial $\Theta$-datum** as  in IUTT-I, Definition 3.1. Indeed:

- Certainly (a), (e), (f) of IUTT-I, Def. 3.1 are satisfied (with appropriate $\underline{\mathbb{V}}, \, \underline{\epsilon}$);
- (b) of IUTT-I, Def. 3.1 is satisfied since by construction $E$ is semistable over $\mathbb{Q}$;
- (c) of IUTT-I, Def. 3.1 is satisfied, in view of (B) above and the choice of $\ell$, as soon as $N \gg 0$ is big enough (recall that $\ell \gt (\log{N})^2$ by construction!), and by the observation that, for $v$ a place of $F = \mathbb{Q}(\sqrt{-1})$, the order of the $v$-adic $q$-parameter of $E$ equals $\mathrm{ord}_v (\Delta)$, which equals $\mathrm{ord}_p(\Delta)$ for $v \mid p \gt 2$, and $2\cdot\mathrm{ord}_2(\Delta)$ for $v \mid 2$;

while $\mathbb{V}_{\mathrm{mod}}^{\mathrm{bad}}$ consists of the primes dividing $\Delta$;

- Finally, (d) of IUTT-I, Def. 3.1 is satisfied upon excluding at most four of Masser's examples $E$. (See page 37 of IUTT-IV).

**Now**, take $\epsilon \coloneqq \big( \log{N} \big)^{-2}$ in Theorem 1.10 of IUTT-IV; this is certainly permissible for $N \gg 0$ large enough. *I claim that the conclusion of Theorem 1.10 contradicts \eqref{eqone} as soon as $N \gg 0$ is large enough.*

For note that Mochizuki's quantity $\log(\mathfrak{q})$ is precisely $\log{|\Delta|}$ (reference: see e.g. Szpiro's article in the Grothendieck Festschrift, vol. 3); his $\log{(\mathfrak{d}^{\mathrm{tpd}})}$ is zero; his $d_{\mathrm{mod}}$ is $1$; and his $\log{(\mathfrak{f}^{\mathrm{tpd}})}$ is our $\log{N}$. By construction, our choice $\epsilon \coloneqq \big( \log{N} \big)^{-2}$ then makes $1/\ell \lt \epsilon$ and $\ell \lt 3/\epsilon$, whence the finaly display of Theorem 1.10 would yield
$$
\frac{1}{6} \log{|\Delta|} \leq (1+29\epsilon) \cdot \log{N} + 2\log{(3\epsilon^{-8})}
\lt \log{N} + 16\log{\log{N}} + 32,
$$
where we have used $\epsilon \log{N} = (\log{N})^{-1} \lt 1$ for $N \gt 3$, and $2\log{3} \lt 3$.

*The last display contradicts \eqref{eqone} as soon as $N \gg 0$ is big enough.*

Thus Masser's examples yield infinitely many counterexamples to Theorem 1.10 of IUTT-IV (as presently written).

**Added on 10/15.** Mochizuki has commented on the apparent contradiction between Masser's examples and Theorem 1.10:

<http://www.kurims.kyoto-u.ac.jp/~motizuki/Inter-universal%20Teichmuller%20Theory%20IV%20(comments).pdf>

(Point (4.) in those Comments was subsequently modified twice. ) He writes that he will revise portions of IUTT-III and IUTT-IV, and will make them available in the near future. Taking into account the latest version of point (4.) from Mochizuki's Comments, here is the anticipated revised version of Theorem 1.10 (after taking $\epsilon \sim 1/\ell$ - which is essentially optimal - and not worrying about the best constants or the most general version):

Take the pair $(E,\ell)$, where $E/\mathbb{Q}$ is a semistable elliptic curve with (say, for the sake of simplifying) rational $2$-torsion (i.e., a Frey-Hellegouarch curve) of minimal discriminant $\Delta$ and conductor $N$ (square-free). Assume that:

- $\ell$ divides neither $N$ nor the any of the exponents in the prime factorization of $\Delta$;
- The Galois representation of $G_{\mathbb{Q}(\sqrt{-1})}$ on $E[\ell]$ has full image $\mathrm{GL}_2(\mathbb{Z}/\ell)$. (*Conjecturally, this condition should only exclude a finite list of values of $\ell$, independent of $E$. Therefore, it does not appear to be an essential condition here.*)
- The $2$-adic valuation of $\Delta$ is bounded. (*Assume this for simplicity in the statement below. It is not truly an essential condition either*).
- $\ell = O(\log{N})$ (Such a choice, compliant with the three previous conditions, can be ensured; see Section 2 in IUTT-IV, or the paper [GenEll]).

Let $M \coloneqq \prod_{p \mid N,  p \lt \ell} p$. *Then*, with the corrections outlined by Mochizuki, the revised Theorem 1.10 should essentially read (and certainly *imply*):
$$
\frac{1}{6} \log{\Delta} \lt \Big( 1 + \frac{200}{\ell} \Big) \log{N}  + \omega(M) \cdot \log{\log{N}} - \log{M} + O\big( \log{\log{N}} \big),
$$
where $\omega(\cdot)$ denotes "number of prime factors." If we take $\ell \sim \log{N}$ and $\omega(M)$ bounded (i.e., restrict to conductors $N$ which are only divisible by a bounded number of primes $\lt \log{N}$), then this consequence would yield $(1/6) \log{\Delta} \lt \log{N} + O(\log{\log{N}})$. (*Must this be true for $N$ a large enough square-free integer such that the number of primes $\lt \log{N}$ dividing $N$ is bounded*? A reminder: in terms of the $abc$-triple, $\Delta$ is *essentially* $(abc)^2$, and $N = \mathrm{rad}(abc)$).

A side remark: note that the inverse $1/\ell$ of the prime level from the de Rham-Etale correspondence $(E^{\dagger}, \lt \ell) \leftrightarrow E[\ell]$ in Mochizuki's "Hodge-Arakelov theory" ultimately figures as the $\epsilon$ in the ABC conjecture.

*(I have deleted the remainder of the 10/15 Addendum, since it is now obsolete after Mochizuki's revised comments.)*

**Added on 3-13-13.** Mochizuki has posted revisions of his second and third papers on Inter-Universal Teichmuller Theory. They can be found at the bottom of his webpage [here](http://www.kurims.kyoto-u.ac.jp/~motizuki/papers-english.html).

***

> end of verbatim quote from [Dimitrov 2012](#Dimitrov12).

### Scholze & Stix's comment

See [Scholze & Stix 2018](#ScholzeStix18).

## Related entries

* [[abc conjecture]]

* [[Mochizuki's corollary 3.12]]

## References

For [[Shinichi Mochizuki]]'s proposed proof see the references at *[[inter-universal Teichmüller theory]]*.

Commentary on this proof by other authors:

* {#Dimitrov12} Vesselin Dimitrov, MO comment 2012 ([MO:a/107279](https://mathoverflow.net/a/107279/381))

* {#ScholzeStix18} [[Peter Scholze]], [[Jakob Stix]], _Why abc is still a conjecture_, 2018 ([pdf](https://ncatlab.org/nlab/files/why_abc_is_still_a_conjecture.pdf))


Mod

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
=--
=--

# The category $Mod$ of modules #
* table of contents
{:toc}

## Idea

Given a [[monoid]] $R$ in a [[monoidal category]] $(\mathcal{C}, \otimes)$, $R$[[Mod]] is the [[category]] whose [[objects]] are $R$-[[modules]] in $\mathcal{C}$ and whose morphisms are module homomorphisms.

Specifically if $(\mathcal{C}, \otimes)$ is the category [[Ab]] of [[abelian groups]] and $\otimes$ the [[tensor product of abelian groups]], then $R$ is a [[ring]].

We write just $Mod$ for the category whose objects are pairs $(R,N)$ consisting of a monoid $R$ and an $R$-module, and whose morphisms may also map between different monoids.

## Definition of $Mod$

We assume that the ambient [[monoidal category]] is [[Ab]] with the [[tensor product of abelian groups]]. But the definition works more generally

+-- {: .num_defn #ModSpelledOut}
###### Definition

An [[object]] in $Mod$ is a pair $(R,N)$ consisting of a commutative [[ring]] $R$ and an $R$-[[module]] $N$.

A [[morphism]]

$$
 (\phi,\kappa) :  (R,N) \to (R',N')
$$

is a pair consisting of a ring [[homomorphism]] $\phi : R \to R'$ and a morphism $\kappa : N \to \phi^* N'$ of $R$-modules, where $\phi^* N'$ is the [[restriction of scalars]].

=--

## Properties

### $Mod$ as a bifibration
 {#ModAsBifibration}

Projecting out the first items in the pairs appearing in def. \ref{ModSpelledOut} yields a canonical functor

$$
  p :Mod \to CRing
$$

$$
  (R,N) \mapsto R
  \,.
$$

that exhibits $Mod$ as a [[bifibration]] over $R$.

The fiber of this projection over a ring $R$ is $Mod_R$, the category of $R$-modules.

In particular the fiber over the initial commutative ring $R = \mathbb{Z}$ is

$$
  Mod_{\mathbb{Z}} = Ab
$$

the category [[Ab]] of abelian groups.

### Tangents and deformation theory
 {#TangentsAndDeformationTheory}

By an old observation of Quillen -- reviewed at [[module]] -- the bifibration $Mod \to CRing$ is [[equivalence of categories|equivalent]] to the category of fiberwise abelian [[group object]]s in the [[codomain fibration]] $[I,CRing] \to CRing$:

$$
  (Mod \to CRing) \simeq Ab([I,CRing] \to CRing)
  \,.
$$

For a fixed ring $R$, the category $Mod_R$ of $R$-modules is canonically equivalent to $Ab(CRing/R)$, the category of abelian [[group object]]s in the [[overcategory]] $CRing/R$:

$$
  Mod_R \simeq Ab(CRing/R)
  \,.
$$

This says that $Mod \to Ring$ is the [[tangent category]] of $CRing$: the above equivalence regards an $R$-module $N$ equivalently as the [[square-0 extension]] ring $R \oplus N$ (with multiplication $(r_1,n_1) \cdots (r_2,n_2) = (r_1 r_2, r_1 n_2 + r_2 n_1)$), which may be thought of as the ring of functions on the infinitesimal neighbourhood of the 0-section of the [[vector bundle]] (or rather [[quasicoherent sheaf]]) over $Spec R$ that is given by $N$.

There is thus another natural projection from $Mod$ to rings, namely the functor that remembers these [[square-0 extensions]]

$$
  f : Mod \to CRing
$$

$$
  (R,N) \mapsto R \oplus N
  \,.
$$

This functor has a [[left adjoint]] $\Omega : CRing \to Mod$ which is also a [[section]]: this is the functor that sends a ring to its module of [[Kähler differential]]s.

$$
  (\Omega \dashv f) :  Mod \stackrel{\overset{\Omega}{\leftarrow}}{\underset{f}{\to}}
  CRing
  \,.
$$

### $R Mod$ is an abelian category
 {#RModIsAbelian}

Let the ambient [[monoidal category]] be [[Ab]] equipped with the [[tensor product of abelian groups]].

+-- {: .num_theorem #RModIsAbelian}
###### Theorem

Let $R$ be a [[commutative ring]]. Then $R Mod$ is an [[abelian category]].

In fact $R Mod$ is a [[Grothendieck category]].

=--

We discuss now all the ingredients of this statement in detail.

Let $U : R Mod \to Set$ be the [[forgetful functor]] to the underlying sets.

+-- {: .num_prop #RModHasZeroObject}
###### Proposition

$R Mod$ has a [[zero object]], given by the 0-module, the [[trivial group]] equipped with trivial $R$-action.

=--

+-- {: .proof}
###### Proof

Clearly the 0-module $0$ is a [[terminal object]], since every morphism $N \to 0$ has to send all elements of $N$ to the unique element of $0$, and every such morphism is a [[homomorphism]].
Also, 0 is an [[initial object]] because a morphism $0 \to N$ always exists and is unique, as it has to send the unique element of 0, which is the neutral element, to the neutral element of $N$.

=--

+-- {: .num_prop #RModHasKernelsAndCokernels}
###### Proposition

1. $R Mod$ has all [[kernels]]. The kernel of a homomorphism $f : N_1 \to N_2$ is the set-theoretic [[preimage]] $U(f)^{-1}(0)$ equipped with the induced $R$-module structure.

1. $R Mod$ has all [[cokernels]]. The cokernel of a homomorphism $f : N_1 \to N_2$ is the [[quotient]] abelian group

   $$
     coker f = \frac{N_2}{im(f)}
   $$

   of $N_2$ by the [[image]] of $f$.

=--

+-- {: .proof}
###### Proof

The defining [[universal property]] of kernel and cokernels is immediately checked.

=--

+-- {: .num_prop }
###### Proposition

$U : R Mod \to Set$ preserves and reflects [[monomorphisms]] and [[epimorphisms]]:

A homomorphism $f : N_1 \to N_2$ in $R Mod$ is a [[monomorphism]] / [[epimorphism]] precisely if $U(f)$ is an [[injection]] / [[surjection]].

=--

+-- {: .proof}
###### Proof

Suppose that $f$ is a [[monomorphism]], hence that $f : N_1 \to N_2$ is such that for all morphisms $g_1, g_2 : K \to N_1$ such that $f \circ g_1 = f \circ g_2$ already $g_1 = g_2$. Let then $g_1$ and $g_2$ be the inclusion of [[submodules]] generated by a single element $k_1 \in K$ and $k_2 \in K$, respectively. It follows that if $f(k_1) = f(k_2)$ then already $k_1 = k_2$ and so $f$ is an [[injection]].  Conversely, if $f$ is an injection then its image is a [[submodule]] and it follows directly that $f$ is a monomorphism.

Suppose now that $f$ is an [[epimorphism]] and hence that $f : N_1 \to N_2$ is such that for all morphisms $g_1, g_2 : N_2 \to K$ such that $f \circ g_1 = f \circ g_2$ already $g_1 = g_2$. Let then $g_1 : N_2 \to \frac{N_2}{im(f)}$ be the natural projection. and let $g_2 : N_2 \to 0$ be the [[zero morphism]]. Since by construction $f \circ g_1 = 0$ and $f \circ g_2 = 0$ we have that $g_1 = 0$, which means that $\frac{N}{im(f)} = 0$ and hence that $N = im(f)$ and so that $f$ is surjective. The other direction is evident on elements.

=--

+-- {: .num_defn #AbelianGroupStructureOnHoms}
###### Definition

For $N_1, N_2 \in R Mod$ two modules, define on the [[hom set]] $Hom_{R Mod}(N_1,N_2)$ the structure of an [[abelian group]] whose addition is given by argumentwise addition in $N_2$: $(f_1 + f_2) : n \mapsto f_1(n) + f_2(n)$.

=--

+-- {: .num_prop #RModIsAbEnriched}
###### Proposition

With def. \ref{AbelianGroupStructureOnHoms} $R Mod$ composition of morphisms

$$
  \circ : Hom(N_1, N_2) \times Hom(N_2, N_3) \to Hom(N_1,N_3)
$$

is a [[bilinear map]], hence is equivalently a morphism

$$
  Hom(N_1, N_2) \otimes Hom(N_2,N_3) \to Hom(N_1, N_3)
$$

out of the [[tensor product of abelian groups]].

This makes $R Mod$ into an [[Ab-enriched category]].

=--

+-- {: .proof}
###### Proof

Linearity of composition in the second argument is immediate from the pointwise definition of the abelian group structure on morphisms.
Linearity of the composition in the first argument comes down to linearity of the second module homomorphism.

=--

+-- {: .num_remark }
###### Remark

In fact $R Mod$ is even a [[closed category]], see prop. \ref{RModIsClosedMonoidal} below, but this we do not need for showing that it is abelian.

=--

Prop. \ref{RModHasZeroObject} and prop. \ref{RModIsAbEnriched} together say that:

+-- {: .num_cor #RModIsAdditive}
###### Corollary

$R Mod$ is an [[pre-additive category]].

=--

+-- {: .num_prop #RModHasProductsAndCoproducts}
###### Proposition

$R Mod$ has all [[products]] and [[coproducts]], being [[direct products]] $\prod_{i \in I} N_i$ and  [[direct sums]] $\oplus_{i \in I} N_i$.

The products are given by [[cartesian product]] of the underlying sets with componentwise addition and $R$-action.

The direct sum is the [[submodule]] of the direct product consisting of tuples of elements such that only finitely many are non-zero.

=--

+-- {: .proof}
###### Proof

The defining [[universal properties]] are directly checked. Notice that the direct product $\prod_{i \in I} N_i$ consists of arbitrary tuples because it needs to have a projection map

$$
  p_j : \prod_{i \in I} N_i \to N_j
$$

to each of the modules in the product, reproducing all of a possibly infinite number of non-trivial maps $\{K \to N_j\}$. On the other hand, the direct sum just needs to contain all the modules in the sum

$$
  \iota_j : N_j \to \oplus_{i \in I} N_i
$$

and since, being a module, it needs to be closed only under addition of _finitely_ many elements, so it consists only of [[linear combinations]] of the elements in the $N_j$, hence of finite formal sums of these.

=--

Together cor. \ref{RModIsAdditive} and prop. \ref{RModHasProductsAndCoproducts} say that:

+-- {: .num_cor #RModIsAdditive}
###### Corollary

$R Mod$ is an [[additive category]].

=--

+-- {: .num_prop #InRModMonosAreKernelOfTheirCokernel}
###### Proposition

In $R Mod$

* every [[monomorphism]] is the [[kernel]] of its [[cokernel]];

* every [[epimorphism]] is the [[cokernel]] of its [[kernel]].

=--

+-- {: .proof}
###### Proof

Using prop. \ref{RModHasKernelsAndCokernels} this is directly checked on the underlying sets: given a monomorphism $K \hookrightarrow N$, its cokernel is $N \to \frac{N}{K}$, The kernel of that morphism is evidently $K \hookrightarrow N$.

=--

Now cor. \ref{RModIsAdditive} and prop. \ref{InRModMonosAreKernelOfTheirCokernel} imply theorem \ref{RModIsAbelian}, by definition.

+-- {: .num_prop #FilteredColimitsInRModAreExact}
###### Proposition

The operation of forming [[filtered colimits]] in $R Mod$
is an [[exact functor]].

=--

(e.g. [Weibel 1994, Lem. 2.6.14](#Weibel94) [Kiersz, prop. 4](#Kiersz)).

### $R Mod$ is a closed monoidal category
 {#RModIsClosedMonoidalCategory}

Let $R$ be a [[commutative ring]].

+-- {: .num_defn #HomModule}
###### Definition

For $N_1, N_2 \in R Mod$, equip the [[hom-set]] $Hom_{R Mod}(N_1, N_2)$ with the structure of an $R$-module as follows: for all $f,g \in Hom_{R Mod}(N_1, N_2)$, all $n_1 \in N_1$ and all $r \in R$ set

* $(f + g) \colon n_1 \mapsto f(n_1) + g(n_2)$

* $r \cdot f \colon n_1 \mapsto r\cdot (f(n_1))$.

Write $[N_1,N_2] \in R Mod$ for the resulting $R$-module structure.

=--

+-- {: .num_prop #RModIsClosedMonoidal}
###### Proposition

Equipped with the [[tensor product of modules]], $R Mod$ becomes a [[monoidal category]] (in fact a [[distributive monoidal category]]). The tensor unit is $R$ regarded canonically as an $R$-module over itself.

This is a [[closed monoidal category]], the [[internal hom]] is given by the hom-modules of def. \ref{HomModule}.

=--

+-- {: .proof}
###### Proof

Either by definition or by a basic property of the [[tensor product of modules]], a module [[homomorphism]] $\phi \colon N_1 \otimes_R N_2 \to N_3$ is precisely an $R$-[[bilinear function]]  of the underlying sets. For fixed elements $n_1 \in N_1$ and  $n_2 \in N_2$ write

$$
  \overline{\phi}(n_1) \coloneqq \phi(n_1, -) \colon N_2 \to N_3
$$

and

$$
  \phi(-,n_2) \colon N_1 \to N_3
$$

for the hom-[[adjuncts]] on the underlying sets. By the bilinearity of $\phi$ both of these are $R$-linear maps. The first being linear means that $\overline{\phi}$ is a function $\overline{\phi} \colon N_1 \to [N_2, N_3]$ to the set of module homomorphisms, and the second being linear says that it is itself a mododule homomorphisms by def. \ref{HomModule}, since

$$
  \overline{\phi}(r\cdot n_1) = (n_2 \mapsto \phi(r\cdot n_1, n_2) = r \phi(n_1, n_2)) = r \cdot \left(\overline{\phi}(n_1)\right)
  \,.
$$

The map $\phi \mapsto \overline{\phi}$ establishes a [[natural transformation]]

$$
  Hom_{R Mod}(N_1 \otimes_R N_2, N_3) \stackrel{}{\to} Hom_{R Mod}(N_1, [N_2, N_3])
  \,.
$$

Conversely, every element of $Hom_{R Mod}(N_1, [N_2, N_3])$ defines [[bilinear map]], hence a homomorphism $N_1 \otimes_R N_2 \to N_3$ and this construction is inverse to the above, showing that it is a [[natural isomorphism]].
This exhibits the [[internal hom]]-[[adjunction]] $(-) \otimes_R N_2 \vdash [N_2,-]$.

=--

### Exact functors between categories of modules

The _[[Eilenberg-Watts theorem]]_ says that sufficiently [[exact functors]] between categories of modules are necessarily given by forming [[tensor products of modules]].

### Limits and colimits
 {#LimitsAndColimits}

Let $R$ be a [[ring]].

+-- {: .num_prop #ModuleIsFilteredColimitOverFinitelyGeneratedSubmodules}
###### Proposition

Every $R$-module is the [[filtered colimit]] over its [[finitely generated module|finite generated]] [[submodules]].

=--

See for instance ([Kiersz, prop. 3](#Kiersz)).

### Tiny objects

For discussion of [[tiny objects]] in $Mod$, see at _[Tiny object -- In categories of modules over rings](tiny+object#InCategoriesOfModulesOverRings)_.

### Tannaka duality

[[!include structure on algebras and their module categories - table]]

## Related concepts

* [[Ring]], [[CRing]]

* [[Vect]], [[VectBund(B)]], [[VectBund]]

* [[(∞,1)Mod]]

* [[2Mod]]

* [[nMod]]

* [[2-category of module categories]]

## References

Discussion of $R Mod$ in $(Ab, \otimes)$ being an [[abelian category]] is for instance in

* Rankeya Datta, _The category of modules over a commutative ring and abelian categories_ ([pdf](http://www.math.columbia.edu/~ums/pdf/Rankeya_R-mod_and_Abelian_Categories.pdf))

Discussion of [[limits]] and [[colimits]] in $R Mod$:

* {#Weibel94} [[Charles Weibel]], *[[An introduction to homological algebra]]*,  Cambridge Studies in Adv. Math. **38**, Cambridge University Press (1994) [[doi:10.1017/CBO9781139644136](https://doi.org/10.1017/CBO9781139644136), [pdf](https://web.math.rochester.edu/people/faculty/doug//otherpapers/weibel-hom.pdf)]

* {#Kiersz} Andy Kiersz, _Colimits and homological algebra_, 2006 ([pdf](http://www.math.uchicago.edu/~may/VIGRE/VIGRE2006/PAPERS/Kiersz.pdf))

Discussion of $Mod_R$ in the generality of [[module objects]] over a [[commutative monoid in a symmetric monoidal category|commutative monoid object]] $R$ [[internalization|internal]] to a [[bicomplete category|bicomplete]] [[closed symmetric monoidal category]] and proof that it is itself [[bicomplete category|bicomplete]] [[closed symmetric monoidal category|closed symmetric monoidal]]:

* [[Mark Hovey]], [[Brooke Shipley]], [[Jeff Smith]], Lemma 2.2.2 & 2.2.8 in: *Symmetric spectra*, J. Amer. Math. Soc. **13** (2000) 149-208 [[arXiv:math/9801077](http://arxiv.org/abs/math/9801077), [doi:10.1090/S0894-0347-99-00320-3](https://doi.org/10.1090/S0894-0347-99-00320-3)]

* [[Florian Marty]], Prop. 1.2.14, 1.2.16, 1.2.17 in: _Des Ouverts Zariski et des Morphismes Lisses en Géométrie Relative_, Ph.D. Toulouse (2009) [[theses:2009TOU30071](https://www.theses.fr/2009TOU30071), [[Marty-DesOuverts.pdf:file]]]

* [[Martin Brandenburg]], Prop. 4.1.10: *Tensor categorical foundations of algebraic geometry* [[arXiv:1410.1716](http://arxiv.org/abs/1410.1716)]

See also:

* [MO:180673](http://mathoverflow.net/questions/180673/category-of-modules-over-commutative-monoid-in-symmetric-monoidal-category).

Lecture notes:

* [[Urs Schreiber]]: *[Algebras and Modules](Introduction+to+Stable+homotopy+theory+--+1-2#AlgebrasAndModules)*, in *[[Introduction to Stable Homotopy Theory]]* 1.2: *[[Introduction to Stable homotopy theory -- 1-2|Structured Spectra]]*

Discussion of homotopy theoretic modules via [[stabilization]] of [[slice model structures]] is in

* {#Schwede97} [[Stefan Schwede]], _Spectra in model categories and applications to the algebraic cotangent complex_, Journal of Pure and Applied Algebra 120 (1997) 104

A summary of the discussion in [Mod as a bifibration](#ModAsBifibration) and [Tangents and deformation theory](#TangentsAndDeformationTheory) together with their embedding into the bigger picture of [[tangent (∞,1)-category|tangent (∞,1)-categories]] is in

* [[Jacob Lurie]], _[[Deformation Theory]]_

Formalization of [[abelian category|abelian]] [[univalent categories]] of [[ring]]-[[modules]], in [[homotopy type theory]] ([[univalent foundations of mathematics]]):

* [[Jarl G. Taxerås Flaten]], *Univalent categories of modules* [[arXiv:2207.03261](https://arxiv.org/abs/2207.03261)]

category: category

[[!redirects Mod]]

[[!redirects RMod]]
[[!redirects R Mod]]
[[!redirects R-Mod]]
[[!redirects KMod]]
[[!redirects K Mod]]
[[!redirects K-Mod]]
[[!redirects kMod]]
[[!redirects k Mod]]
[[!redirects k-Mod]]

[[!redirects category of modules]]
[[!redirects categories of modules]]


mod p local Langlands correspondence

#Contents#
* table of contents
{:toc}

## Idea

The mod $p$ local Langlands correspondence is a version of the [[local Langlands correspondence]] where the representations are over a finite field of characteristic $p$, i.e. on one side we have certain $k$-representations of $G(F)$ (for $G$ some reductive group), and on the other we have certain $k$-representations of $\mathrm{Gal}(\overline{F}/F)$, where $F$ is a finite extension of $\mathbb{Q}_{p}$, and $k$ is a finite field of characteristic $p$. It should be compatible with the [[p-adic local Langlands correspondence]].The only case that is known is when $G=\mathrm{GL}_{2}$ and $F=\mathbb{Q}_{p}$.

## References

* {#Breuil2010} [[Christophe Breuil]], *The emerging p-adic Langlands program*, Proceedings of the International Congress of Mathematicians, Hyderabad, India, 2010 ([pdf](https://www.imo.universite-paris-saclay.fr/~breuil/PUBLICATIONS/ICM2010.pdf))


mod p Whitehead theorem

#Contents#
* table of contents
{:toc}

## Idea

A basic fact of [[p-adic homotopy theory]], roughly analogous to the plain [[Whitehead theorem]] in plain [[homotopy theory]].

Let $p$ be a [[prime number]] and write $\mathbb{F}_p \coloneqq \mathbb{Z}/(p)$ for its [[cyclic group]]/[[finite field]] and $\mathbb{Z}_p$ for the [[p-adic integers]].

**Statement** For $X$ and $Y$ [[simply connected]] [[finite homotopy types]] and a morphism $f \colon X \longrightarrow Y$ between them, the following are equivalent:

1. $f$ is an $\mathbb{F}_p$-[[homology equivalence]], i.e. $H_\ast(f,\mathbb{F}_p)$ is an equivalence;

1. $f$ is a $\mathbb{Z}_p$-[[homology equivalence]], i.e. $H_\ast(f,\mathbb{Z}_p)$ is an equivalence;

1. $f$ is a $\mathbb{Z}_p$-[[weak homotopy equivalence]], i.e. $\pi_\ast(f) \otimes \mathbb{Z}_p$ is an equivalence.

([Schiffmann 81](#Schiffmann81))

## Applications

This theorem controls the relation between [[localization of a space]] at $\mathbb{F}_p$ and its [[p-completion]].

## References

* {#Schiffmann81} Stephen J. Schiffman, _A mod p Whitehead Theorem_, Proceedings of the American Mathematical Society Vol. 82, No. 1 (May, 1981), pp. 139-144

[[!redirects mod-p Whitehead theorem]]


modal homotopy type theory

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### Context
#### Type theory
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[[!include type theory - contents]]
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#### Modalities, Closure and Reflection
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[[!include modalities - contents]]
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#### Homotopy theory
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[[!include homotopy - contents]]
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=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The refinement of [[modal type theory]] to [[homotopy type theory]]: hence homotopy type theory equipped with [[modalities]] as in [[modal logic]] (and in [[modal type theory]]).

## Definitions

There are a number of different but equivalent ways to define a [[modality]] in [[homotopy type theory]]. From [Rijke, Shulman, Spitters 17](#RSS):

* Higher modality: A modality consists of a modal operator $L : {Type} \to {Type}$ and for every type $X$ a modal unit $(-)^L : X \to LX$ together with

1. an induction principle of type
$$\text{ind} : ((a : A) \to LB(a^L)) \to ((u : LA) \to LB(u))$$
for any type $A$ and type $B$ depending on $LA$

1. a computation rule
$$\text{com} : \text{ind}(f)(a^L) = f(a).$$

1. a witness that for any $x, y : LA$, the modal unit $(-)^L : (x = y) \to L(x = y)$ is an equivalence.

* Uniquely eliminating modality: A modality consists of a modal operator and modal unit such that operation
$$f \mapsto f \circ (-)^L : ((u : LA) \to LB(u)) \to ((a : A) \to LB(a^L))$$
is an equivalence for all types $A$ and types $B$ depending on $LA$.

* $\Sigma$-closed reflective subuniverse (aka localization): A reflective subuniverse (a.k.a. localization) consists of a proposition $isLocal : {Type} \to {Prop}$ together with an operator $L : {Type} \to {Type}$ and a unit $(-)^L : X \to LX$ such that $LX$ is local for any type $X$ and for any local type $Z$, then function
$$f \mapsto f \circ (-)^L : (LA \to Z) \to (A \to Z)$$
is an equivalence. A localization is $\Sigma$-closed if whenever $A$ is local and for all $a : A$, $B(a)$ is local, then the dependent sum $\Sigma_{a : A} B(a)$ is local.

* Stable factorization systems: A modality consists of an orthogonal factorization system $(\mathcal{L}, \mathcal{R})$ in which the left class is stable under pullback.

* A localization (reflective subuniverse) whose units $(-)^L : X \to LX$ are $L$-connected.

We say that a type $X$ is $L$-modal if the unit $(-)^L : X \to LX$ is an equivalence. We say a type $X$ $L$-connected if $LX$ is contractible.

In terms of the other structure, the stable factorization system associated to a modality $L$ has

* left class the $L$-connected maps, whose fibers are $L$-connected.

* right class the $L$-modal maps, whose fibers are $L$-modal.

Conversely, given a stable factorization system, the modal operator and unit are given by factorizing the terminal map.

## Related concepts

* [[reflective subuniverse]]

* [[stable homotopy type]]

* [[equivariant homotopy type]]

* [[geometric homotopy type theory]]

* [[cohesive homotopy type theory]]

* [[directed homotopy type theory]]

## References

See also the references at _[[modal type theory]]_.

### Review

* {#LicataWellen18} [[Dan Licata]], [[Felix Wellen]], _Synthetic Mathematics in Modal Dependent Type Theories_, tutorial at *[[HoTT in Bonn2018|Types, Homotopy Theory and Verification]]* (2018)

  Tutorial 1, Dan Licata:  _A Fibrational Framework for Modal Simple Type Theories_
([recording](https://www.youtube.com/watch?v=uLLbSAYd3yI))

  Tutorial 2, Felix Wellen: _The Shape Modality in Real cohesive HoTT and Covering Spaces_ ([recording](https://www.youtube.com/watch?v=ACGjJDarEc4))

  Tutorial 3, Dan Licata: _Discrete and Codiscrete Modalities in Cohesive HoTT_ ([recording](https://www.youtube.com/watch?v=OHN9vEVPBN8))

  Tutorial 4, Felix Wellen, _Discrete and Codiscrete Modalities in Cohesive HoTT, II_ ([recording](https://www.youtube.com/watch?v=KXx9UDQdWPw))

  Tutorial 5, Dan Licata: _A Fibrational Framework for Modal Dependent Type Theories_ ([recording](https://www.youtube.com/watch?v=R4SsxUnIcng))

  Tutorial 6, Felix Wellen: _Differential Cohesive HoTT_, ([recording](https://www.youtube.com/watch?v=uEZXHPdwvJU&t=226s))

* {#Licata18} [[Dan Licata]], _Synthetic Mathematics in Modal Dependent Type Theories_, notes for tutorial at _[Types, Homotopy Theory and Verification](https://www.him.uni-bonn.de/programs/past-programs/past-trimester-programs/types-sets-constructions/workshop-types-homotopy-type-theory-and-verification/)_, 2018 ([pdf](http://dlicata.web.wesleyan.edu/pubs/lsr17multi/him-tutorial.pdf))

### General

* [[UF-IAS-2012]], _[Modal type theory](http://uf-ias-2012.wikispaces.com/Modal+type+theory)_

* {#HoTTBook} [Univalent Foundations Project](http://ncatlab.org/nlab/show/UF-IAS-2012), section 7.7 of _[[Homotopy Type Theory -- Univalent Foundations of Mathematics]]_

* {#Shulman} [[Mike Shulman]], _Higher modalities_, talk at [[UF-IAS-2012]], October 2012  ([pdf](http://uf-ias-2012.wikispaces.com/file/view/modalitt.pdf))

* {#RSS} [[Egbert Rijke]], [[Mike Shulman]], [[Bas Spitters]], *Modalities in homotopy type theory*,  Logical Methods in Computer Science, **16** 1 (2020)  [[arXiv:1706.07526](https://arxiv.org/abs/1706.07526), doi:10.23638/LMCS-16(1:2)2020]

* [[Mike Shulman]], _[http://homotopytypetheory.org/2015/07/05/modules-for-modalities/](http://homotopytypetheory.org/2015/07/05/modules-for-modalities/)_  (2015)

* [[Egbert Rijke]], [[Bas Spitters]], around def. 1.11 of _Sets in homotopy type theory_ ([arXiv:1305.3835](http://arxiv.org/abs/1305.3835))

* Kevin Quirin and Nicolas Tabareau, _Lawvere-Tierney Sheafification in Homotopy Type Theory_, [Workshop](http://hott-uf.gforge.inria.fr/2015/) talk 2015 ([pdf](http://hott-uf.gforge.inria.fr/2015/HOTTUF_Kevin.pdf)), Kevin Quirin [thesis](https://kevinquirin.github.io/thesis/main.pdf)

For a discussion of the [[reflective factorization system]] generated by a modality, see

* [[Felix Cherubini]], [[Egbert Rijke]], *Modal Descent*, Mathematical Structures in Computer Science **31** 4 (2021) 363-391  [[arXiv:2003.09713](https://arxiv.org/abs/2003.09713), [doi:10.1017/S0960129520000201](https://doi.org/10.1017/S0960129520000201)]

See also

* {#ChristensenRijke20} [[J. Daniel Christensen]], [[Egbert Rijke]], _Characterizations of modalities and lex modalities_ ([arXiv:2008.03538](https://arxiv.org/abs/2008.03538))

Outlook in view of [[cohesive homotopy type theory]]:

* {#Schreiber15} [[Urs Schreiber]], _[[schreiber:Some thoughts on the future of modal homotopy type theory]]_ , talk at: _[Homotopy Type Theory and Univalent Foundations - Mini-Symposium](https://sites.google.com/site/dmv2015hott/)_, within the _[German Mathematical Society meeting 2015](http://www.math.uni-hamburg.de/DMV2015/)_ 21st - 25th Sept 2015, Hamburg ([pdf notes](https://ncatlab.org/schreiber/files/SchreiberDMV2015b.pdf))

Introduction from the perspective of [[philosophy]]:

* {#Corfield17} [[David Corfield]], _Modal homotopy type theory_, 2017 ([PhilSci archive:15260](http://philsci-archive.pitt.edu/15260/))

* {#Corfield20} [[David Corfield]], _Modal homotopy type theory_, Oxford University Press 2020 ([ISBN: 9780198853404](https://global.oup.com/academic/product/modal-homotopy-type-theory-9780198853404))

Specifically for [[localization of a space|localization]] in the classical sense of [[algebraic topology]]:

* [[J. Daniel Christensen]], [[Morgan Opie]], [[Egbert Rijke]], [[Luis Scoccola]], _Localization in Homotopy Type Theory_, Higher Structures, 4(1) (2020), 1-32 ([arXiv:1807.04155](https://arxiv.org/abs/1807.04155))

Specifically for [[cohesive homotopy type theory]] see:

* [[Urs Schreiber]], [[Michael Shulman]], _[[schreiber:Quantum gauge field theory in Cohesive homotopy type theory]]_, in [[Ross Duncan]], [[Prakash Panangaden]] (eds.) _Proceedings 9th Workshop on Quantum Physics and Logic_, Brussels, Belgium, 10-12 October 2012 ([arXiv:1408.0054](http://arxiv.org/abs/1408.0054))

* {#Shulman15} [[Mike Shulman]], _Brouwer's fixed-point theorem in real-cohesive homotopy type theory_, Mathematical Structures in Computer Science Vol 28 (6) (2018): 856-941 ([arXiv:1509.07584](https://arxiv.org/abs/1509.07584), [doi:10.1017/S0960129517000147](https://doi.org/10.1017/S0960129517000147))

* {#Wellen18} [[Felix Wellen]], _[[schreiber:thesis Wellen|Cartan Geometry in Modal Homotopy Type Theory]]_ ([arXiv:1806.05966](https://arxiv.org/abs/1806.05966), [thesis pdf](http://www.math.kit.edu/iag3/~wellen/media/diss.pdf) following [Schreiber 15](#Schreiber15))

* {#Myers08} [[David Jaz Myers]], _Good Fibrations through the Modal Prism_ ([arXiv:1908.08034](https://arxiv.org/abs/1908.08034))

Formalization of the [[shape modality|shape]]/[[flat modality|flat]]-[[fracture square]] ([[differential cohomology hexagon]]) in [[cohesive homotopy type theory]]:

* [[David Jaz Myers]], _Modal Fracture of Higher Groups_,  talk at _[CMU-HoTT Seminar](https://www.cmu.edu/dietrich/philosophy/hott/seminars/index.html), 2021 ([pdf](http://davidjaz.com/Talks/CMU_March_2021.pdf), [[MyersModalFracture2021.pdf:file]])

* [[David Jaz Myers]], _Modal Fracture of Higher Groups_ ([arXiv:2106.15390](https://arxiv.org/abs/2106.15390))

See also the proof of the [[Blakers-Massey theorem]] for modalities in this generality:

* {#AnelBiedermanFinsterJoyal17a} [[Mathieu Anel]], [[Georg Biedermann]], [[Eric Finster]], [[André Joyal]], _A Generalized Blakers-Massey Theorem_, Journal of Topology **13** 4 (2020) 1521-1553 $[$[arXiv:1703.09050](https://arxiv.org/abs/1703.09050), [doi:10.1112/topo.12163](https://doi.org/10.1112/topo.12163)$]$

[[!redirects modal homotopy type theories]]

[[!redirects modal homotopy type]]
[[!redirects modal homotopy types]]


modal hyperdoctrine

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Modalities, Closure and Reflection
+-- {: .hide}
[[!include modalities - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A **modal hyperdoctrine** is a framework for [[modal logic]] in the presence of different [[contexts]].

It is an [[hyperdoctrine]] that is equipped with a compatible [[modal operator]], often a compatible [[closure operator]], hence a [[universal closure operator]].

## Related concepts

* [[modal type theory]]

## References

Around p. 597 of

* P. Blacksburn, J. van Benthem, F. Wolter, _Handbook of modal logic_, Studies in Logic and Practical reasoning volume 3 (2007)

p. 43 of

* M. Kracht, O. Kutz, _Logically Possible Worlds and Counterpart Semantics for Modal Logic_, ([pdf](https://www.semanticscholar.org/paper/Logically-Possible-Worlds-and-Counterpart-Semantics-Kracht-Kutz/12295c55f5da4d69229d2b7a02f0376d8520ce6e))

[[!redirects modal hyperdoctrines]]