self-duality for pregeometric C-field -- references

Formulation of the [[equations of motion]] of [[D=11 supergravity]] in [[superspace]] on fields including a flux density $G_7$ *a priori* independent of the flux density $G_4$ of the [[supergravity C-field]]:

* {#CastellaniDAuriaFré91} [[Leonardo Castellani]], [[Riccardo D'Auria]], [[Pietro Fré]], ch III.8.3-5 in vol 2 of: _[[Supergravity and Superstrings - A Geometric Perspective]]_, World Scientific (1991) \[doi:10.1142/0224, [epdf](https://epdf.pub/supergravity-and-superstrings-a-geometric-perspective-vol-2-supergravity.html), ch III.8: [[CastellaniDAuriaFre-ChIII8.pdf:file]]\]

  > (Using the [[D'Auria-Fre formulation of supergravity]].)

* Antonio Candiello, [[Kurt Lechner]], §6 in: *Duality in supergravity theories*, Nuclear Physics B **412** 3 (1994) 479-501 \[doi:10.1016/0550-3213(94)90389-1\]

  > (These authors seem not to be aware of [CDF91, III.8](#CastellaniDAuriaFré91) and, contrary to the result there, conclude that it is *not possible* without introducing non-local relations.)

Discussion of [[Lagrangian densities]] for [[D=11 supergravity]] with an *a priori* independent dual C-field field and introduction of the "[[duality-symmetric higher gauge theory|duality-symmetric]]" terminology:

* [[Igor Bandos]], [[Nathan Berkovits]], [[Dmitri Sorokin]], *Duality-Symmetric Eleven-Dimensional Supergravity and its Coupling to M-Branes*, Nucl. Phys. B **522** (1998) 214-233 \[doi:10.1016/S0550-3213(98)00102-3, [arXiv:hep-th/9711055](https://arxiv.org/abs/hep-th/9711055)\]

* {#CremmerJuliaLuPope} [[Eugene Cremmer]], [[Bernard Julia]], H. Lu, [[Christopher Pope]], Section 2 of *Dualisation of Dualities, II: Twisted self-duality of doubled fields and superdualities*, Nucl.Phys. B **535** (1998) 242-292 \[doi:10.1016/S0550-3213(98)00552-5, [arXiv:hep-th/9806106](https://arxiv.org/abs/hep-th/9806106)\]

* {#BandosNurmagambetovSorokin04} [[Igor Bandos]], [[Alexei Nurmagambetov]], [[Dmitri Sorokin]], Section 2 of: *Various Faces of Type IIA Supergravity*, Nucl. Phys. B **676** (2004) 189-228 \[doi:10.1016/j.nuclphysb.2003.10.036, [arXiv:hep-th/0307153](https://arxiv.org/abs/hep-th/0307153)\]

* [[Alexei J. Nurmagambetov]], *The Sigma-Model Representation for the Duality-Symmetric $D=11$ Supergravity*, eConf C0306234 (2003) 894-901 \[arXiv:hep-th/0312157, [inspire:635585](https://inspirehep.net/literature/635585)\]

Discussion in the context of [[shifted C-field flux quantization]]:

* [SS23, §3.4](duality-symmetric+higher+gauge+theory#SS23)


self-duality for pregeometric RR-fields -- references

Discussion of ([[Lagrangian densities]] for) [[D=10 type II supergravity]] with "[[duality-symmetric higher gauge theory|duality-symmetric]]"/"democratic"/"[[pregeometric RR-fields|pregeometric]]" for of the [[RR-fields]]:

* {#Dall'AgataLechnerTonin98} [[Gianguido Dall'Agata]], [[Kurt Lechner]], [[Mario Tonin]], *$D=10$, $N=IIB$ Supergravity: Lorentz-invariant actions and duality*, JHEP 9807:017 (1998) $[$[arXiv:hep-th/9806140](https://arxiv.org/abs/hep-th/9806140), [doi:10.1088/1126-6708/1998/07/017](https://doi.org/10.1088/1126-6708/1998/07/017)$]$

* {#CremmerJuliaLuPope98} [[Eugene Cremmer]], [[Bernard Julia]], H. Lu, [[Christopher Pope]], Section 3 of: *Dualisation of Dualities, II: Twisted self-duality of doubled fields and superdualities*, Nucl. Phys. B **535**(1998) 242-292 $[$[arXiv:hep-th/9806106](https://arxiv.org/abs/hep-th/9806106), doi:10.1016/S0550-3213(98)00552-5$]$

* {#BerhshoeffKalloshOrtinRoestVanProyen01} [[Eric Bergshoeff]], [[Renata Kallosh]], [[Tomas Ortin]], [[Diederik Roest]], [[Antoine Van Proeyen]], *New Formulations of D=10 Supersymmetry and D8-O8 Domain Walls*, Class. Quant. Grav. **18** (2001) 3359-3382 $[$[arXiv:hep-th/0103233](https://arxiv.org/abs/hep-th/0103233), [doi:10.1088/0264-9381/18/17/303](https://doi.org/10.1088/0264-9381/18/17/303)$]$

* {#BandosNurmagambetovSorokin04} [[Igor Bandos]], [[Alexei Nurmagambetov]], [[Dmitri Sorokin]], *Various Faces of Type IIA Supergravity*, Nucl. Phys. B **676** (2004) 189-228 $[$[arXiv:hep-th/0307153](https://arxiv.org/abs/hep-th/0307153), [doi:10.1016/j.nuclphysb.2003.10.036](https://doi.org/10.1016/j.nuclphysb.2003.10.036)$]$

* {#MkrtchyanValach23} [[Karapet Mkrtchyan]], [[Fridrich Valach]], *Democratic actions for type II supergravities*, Phys. Rev. D **107** 6 (2023) 066027 \[doi:10.1103/PhysRevD.107.066027, [arXiv:2207.00626](https://arxiv.org/abs/2207.00626)\]

Enhancement of the [[self-dual higher gauge field|self-duality]] constraint on [[pregeometric RR-field|pregeometric]] [[RR-fields]] from ([[twisted de Rham cohomology|twisted]]) [[de Rham cohomology]] to ([[twisted K-theory|twisted]]) [[topological K-theory]] (under the hypothesized [[K-theory classification of D-brane charge]])
in terms of a [[quadratic form]] on [[differential K-theory]]:

* {#MooreWitten99} [[Gregory Moore]], [[Edward Witten]], *Self-Duality, Ramond-Ramond Fields, and K-Theory*, JHEP 0005:032 (2000) $[$[arXiv:hep-th/9912279](http://arxiv.org/abs/hep-th/9912279)$]$

* {#Witten00} [[Edward Witten]], *Duality Relations Among Topological Effects In String Theory*, JHEP 0005:031 (2000) $[$[arXiv:hep-th/9912086](http://arxiv.org/abs/hep-th/9912086), [doi:10.1088/1126-6708/2000/05/032](https://doi.org/10.1088/1126-6708/2000/05/032)$]$

* {#FreedHopkins00} [[Daniel Freed]], [[Michael Hopkins]], *On Ramond-Ramond fields and K-theory*, JHEP 0005 (2000) 044 $[$[arXiv:hep-th/0002027](http://arxiv.org/abs/hep-th/0002027)$]$

* {#Freed02} [[Daniel Freed]], *[[Dirac charge quantization and generalized differential cohomology]]*, Surveys in Differential Geometry **7**, Int. Press (2000) 129-194 $[$[arXiv:hep-th/0011220](http://arxiv.org/abs/hep-th/0011220), [doi:10.4310/SDG.2002.v7.n1.a6](https://dx.doi.org/10.4310/SDG.2002.v7.n1.a6), [spire:537392](https://inspirehep.net/literature/537392)$]$

An indication of a more refined discussion in [[twisted differential K-theory|twisted differential]] [[KR-theory]]:

* {#DFM09} [[Jacques Distler]], [[Dan Freed]], [[Greg Moore]], _Orientifold Précis_ in: [[Hisham Sati]], [[Urs Schreiber]] (eds.) _[[schreiber:Mathematical Foundations of Quantum Field and Perturbative String Theory]]_, Proceedings of Symposia in Pure Mathematics, AMS (2011) $[$[arXiv:0906.0795](http://arxiv.org/abs/0906.0795), [slides](http://www.ma.utexas.edu/users/dafr/bilbao.pdf)$]$

See at _[[orientifold]]_ for more on this.

Expressing the [[self-dual higher gauge theory|self-duality]] of [[pregeometric RR-fields]] in terms of [[11d Chern-Simons theory]]:

* {#BelovMooreII} Dmitriy Belov, [[Greg Moore]], *Type II Actions from 11-Dimensional Chern-Simons Theories* $[$[arXiv:hep-th/0611020](http://arxiv.org/abs/hep-th/0611020)$]$

Some review:

* {#Szabo12} [[Richard Szabo]], section 3.6 and 4.6 of: *Quantization of Higher Abelian Gauge Theory in Generalized Differential Cohomology*, ESI 2385 (2012)  $[$[arXiv:1209.2530](http://arxiv.org/abs/1209.2530), [pdf](https://www.mat.univie.ac.at/~esiprpr/esi2385.pdf)$]$

Discussion in the context of [[flux quantization]] (here: [[D-brane charge quantization in K-theory]]):

* [SS23, §3.3](duality-symmetric+higher+gauge+theory#SS23)


Selmer group

\tableofcontents

## Idea

The Selmer group is a subgroup of the [[Galois cohomology]] satisfying certain local conditions.

It appears in the statement of the Bloch-Kato conjecture on [[special values of L-functions]], generalizing the [[Birch and Swinnerton-Dyer conjecture]].

## Definitions

The reference for this section is [#LoefflerZerbes18](#LoefflerZerbes18).

Let $K$ be a [[number field]]. Let $V$ be a [[Galois representation]], i.e. a representation of $\Gal(\overline{K}/K)$. We have the first [[Galois cohomology]] group, which we will denote with the following shortcut notation

$$H^{1}(K,V):=H^{1}(\Gal(\overline{K}/K),V)$$

For any prime $v$ of $K$, let $K_{v}$ be the completion. We have "localization" maps (we are also using similar notation as above for the local Galois cohomology groups)

$$\loc_{v}:H^{1}(K,V)\to H^{1}(K_{v},V)$$

\begin{definition}
A _local condition_ on $V$ at a prime $v$ is a subspace $\mathcal{F}_{v}\subseteq H^{1}(K_{v},V)$.
\end{definition}

An important example of a local condition is the _unramified local condition_. Letting $G_{K_{v}}$ be the [[absolute Galois group]] of $K_{v}$ and letting $I_{v}$ be its inertia subgroup, the unramified local condition is defined as follows:

$$\mathcal{F}_{v,\unr}:=\im(H^{1}(G_{K_{v}}/I_{v},V^{I_{v}})\to H^{1}(K_{v},V))$$

\begin{definition}
A _Selmer structure_ is a collection $\mathcal{F}=\lbrace\mathcal{F}_{v}\rbrace_{v}$ of local conditions $\mathcal{F}_{v}$, such that $\mathcal{F}_{v}=\mathcal{F}_{v,\unr}$ for all but finitely many primes $v$.

If $\mathcal{F}$ is a Selmer structure, we define the corresponding _Selmer group_ as

$$\mathrm{Sel}_{\mathcal{F}}(K,V)=\lbrace x\in H^{1}(K,V):\loc_{v}\in\mathcal{F}_{v}\forall v\rbrace.$$
\end{definition}

## References

{#LoefflerZerbes18} David Loeffler and Sarah Zerbes, _Euler Systems_, Arizona Winter School 2018 Notes ([pdf](https://swc-math.github.io/aws/2018/2018LoefflerZerbesNotes.pdf))


semantics

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Type theory
+-- {: .hide}
[[!include type theory - contents]]
=--
=--
=--

# Semantics
* table of contents
{: toc}

## Idea

Semantics is the interpretation of the [[syntax]] of a [[theory]] in a [[model]].

See a separate page for [[semantics of a programming language]].

## Examples

### Categorical semantics of dependent type theory

Notably the [[semantics]] of [[type theory|type theories]] is given by [[categories]]. For instance

* [[dependent type theory]] is precisely the formal language for which [[locally cartesian closed categories]] are the semantics,

* [[intensional type theory]] is the formal language for which [[locally cartesian closed (infinity,1)-categories]] are the semantics;

* [[homotopy type theory]] is the formal language for which [[elementary (infinity,1)-toposes]] are the semantics.

For more on this see at _[[categorical semantics]]_ and at _[[relation between type theory and category theory]]_.

## Related concepts

* [[categorical semantics]];

* [[denotational semantics]], [[operational semantics]]

* [[distributional semantics]]

* [[geometric models for modal logics]];

* [[syntax]], [[semiotics]]

* [[syntax - semantics duality]]

category:logic

[[!redirects semantics]]


semantics of a programming language

#Contents#
* table of contents
{:toc}

## Idea

Semantics in [[computer science]] is a field concerned with models of meaning ([[semantics]]) of [[programming language]] constructs, ie. with what the constructions actually do in execution. In fact, semantics in computer science (and cybernetics) nowdays is often considered also beyond language constructs and deals also with semantics of more general signals, codes and processes, especially within concurrency models.

## Related pages

* [[denotational semantics]],

* [[operational semantics]]

* [[categorical semantics]]

* [[domain theory]]

* [[computational trilogy]]

## Literature

* wikipedia [semantics (computer science)](https://en.wikipedia.org/wiki/Semantics_%28computer_science%29)
* Robert Harper, _Practical foundations for programming languages_, [pdf](http://www.cs.cmu.edu/~rwh/plbook/book.pdf)
* Shriram Krishnamurthi, _Programming languages, Application and interpretation_, [pdf](http://www.cs.brown.edu/courses/cs173/2012/book/book.pdf)
* Hanne Riis Nielson, Flemming Nielson, _Semantics with applications: a formal introduction_, Wiley 1992, 1999 [page with book and other files](http://www.daimi.au.dk/~bra8130/Wiley_book/wiley.html) (postscript recommended for the book)

* [[Frank J. Oles]], *A category-theoretic approach to the semantics of programming languages*, Syracuse University (1982) [[pdf](https://www.cs.cmu.edu/afs/cs.cmu.edu/project/fox-19/member/jcr/www/FrankOlesThesis.pdf)]

* cstheory.stackexchange discussion of books on semantics [html](http://cstheory.stackexchange.com/questions/3533/books-on-programming-language-semantics)
* Robert M. Amadio, Pierre-Louis Currien, _Domains and lambda-calculi_, Cambridge tracts in theoretical computer science 46, [gBooks](https://books.google.co.uk/books?id=2FycO5h5SocC)
* Tobias Nipkow, Gerwin Klein, _Concrete semantics with   Isabelle/HOL_, 2014 [book page](http://www21.in.tum.de/~nipkow/Concrete-Semantics)
* Glynn Winskel, __Denotational semantics__, University of Cambridge [pdf](http://www.cl.cam.ac.uk/~gw104/dens.pdf)

category: computer science

[[!redirects semantics of programming languages]]


semantics of W-types -- references

### Categorical semantics of $\mathcal{W}$-types
 {#CategoricalSemanticsOfWTypesReferences}

The observation that the [[categorical semantics]] of well-founded [[inductive types]] ([[W-types|$\mathcal{W}$-types]]) is given by [[initial algebra over an endofunctor|initial algebras over]] [[polynomial functor|polynomial]] [[endofunctors]] on the [[type system]]:

* {#Dybjer97} [[Peter Dybjer]], *Representing inductively defined sets by wellorderings in Martin-Löf's type theory*, Theoretical Computer Science **176** 1–2 (1997) 329-335 $[$doi:10.1016/S0304-3975(96)00145-4$]$

* {#MoerdijkPalmgren00} [[Ieke Moerdijk]], [[Erik Palmgren]], *Wellfounded trees in categories*, Annals of Pure and Applied Logic **104** 1-3 (2000) 189-218 $[$doi:10.1016/S0168-0072(00)00012-9$]$

Further discussion:

* [[Michael Abbott]], [[Thorsten Altenkirch]], [[Neil Ghani]], *Containers: Constructing strictly positive types*, Theoretical Computer Science **342** 1 (2005) 3-27 $[$[doi:10.1016/j.tcs.2005.06.002](https://doi.org/10.1016/j.tcs.2005.06.002)$]$

* [[Benno van den Berg]],  [[Ieke Moerdijk]], _$W$-types in sheaves_ [[arXiv:0810.2398](https://arxiv.org/abs/0810.2398)]

Generalization to [[inductive families]] (dependent $\mathcal{W}$-types):

* [[Nicola Gambino]], [[Martin Hyland]], *Wellfounded Trees and Dependent Polynomial Functors*, in: *Types for Proofs and Programs* TYPES 200,  Lecture Notes in Computer Science **3085**, Springer (2004) $[$[doi:10.1007/978-3-540-24849-1_14](https://doi.org/10.1007/978-3-540-24849-1_14)$]$

* {#AbbottAltenkirchGhani04} [[Michael Abbott]], [[Thorsten Altenkirch]], [[Neil Ghani]]: *Representing Nested Inductive Types using W-types*, in: *Automata, Languages and Programming*, ICALP 2004, Lecture Notes in Computer Science, **3142**, Springer (2004) $[$[doi:10.1007/978-3-540-27836-8_8](https://doi.org/10.1007/978-3-540-27836-8_8), [pdf](http://www.cs.nott.ac.uk/~psztxa/publ/icalp04.pdf)$]$

  exposition: _Inductive Types for Free -- Representing nested inductive types using W-types_, talk at ICALP (2004) $[$[pdf](http://www.cs.nott.ac.uk/~txa/talks/icalp-slides.pdf)$]$

Generalization to [[initial algebra of a presentable (infinity,1)-monad|homotopy-initial algebras]] as [[categorical semantics]] for [[W-types|$\mathcal{W}$-types]] in [[homotopy type theory]]:

* {#AwodeyGambinoSojakova12} [[Steve Awodey]], [[Nicola Gambino]], [[Kristina Sojakova]], *Inductive types in homotopy type theory*, LICS'12: (2012) 95–104 $[$[arXiv:1201.3898](http://arxiv.org/abs/1201.3898), [doi:10.1109/LICS.2012.21](https://doi.org/10.1109/LICS.2012.21), [Coq code](https://github.com/HoTT/HoTT/tree/master/Coq/IT)$]$

* {#vdBergMoerdijk13} [[Benno van den Berg]], [[Ieke Moerdijk]], _W-types in Homotopy Type Theory_, Mathematical Structures in Computer Science **25** Special Issue 5: *From type theory and homotopy theory to Univalent Foundations of Mathematics* (2015) 1100-1115  $[$[arXiv:1307.2765](http://arxiv.org/abs/1307.2765), [doi:10.1017/S0960129514000516](https://doi.org/10.1017/S0960129514000516)$]$

* [[Kristina Sojakova]], *Higher Inductive Types as Homotopy-Initial Algebras*, ACM SIGPLAN Notices **50** 1 (2015) 31–42 $[$[arXiv:1402.0761](http://arxiv.org/abs/1402.0761), [doi:10.1145/2775051.2676983](https://doi.org/10.1145/2775051.2676983)$]$

* [[Steve Awodey]], [[Nicola Gambino]], [[Kristina Sojakova]], *Homotopy-initial algebras in type theory*, Journal of the ACM **63** 6 (2017) 1–45 $[$[arXiv:1504.05531](http://arxiv.org/abs/1504.05531), [doi:10.1145/3006383](https://doi.org/10.1145/3006383)$]$

* {#vdBergMoerdijk13} [[Benno van den Berg]], [[Ieke Moerdijk]], _W-types in Homotopy Type Theory_, Mathematical Structures in Computer Science **25** Special Issue 5: *From type theory and homotopy theory to Univalent Foundations of Mathematics* (2015) 1100-1115  [[arXiv:1307.2765](http://arxiv.org/abs/1307.2765), [doi:10.1017/S0960129514000516](https://doi.org/10.1017/S0960129514000516)]

Towards combining generalization to dependent and homotopical W-types:

* [[Christian Sattler]], _On relating indexed W-types with ordinary ones_, in *[Types for Proofs and Programs -- TYPES 2015](https://cs.ioc.ee/types15/abstracts-book/)* (2015) 71-72 $[$[pdf](http://cs.ioc.ee/types15/abstracts-book/contrib31.pdf), [[Sattler-WTypes.pdf:file]]$]$


semi-abelian category

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

# Semiabelian categories
* table of contents
{: toc}

## Idea

The notion of _semi-abelian category_ is supposed to capture the properties of [[categories]] such as that of [[groups]], [[rng|rings without unit]], [[associative algebras]] without [[unit]], [[Lie algebras]], etc.; in generalization of how the notion of _[[abelian categories]]_ captures the properties of the categories of [[abelian groups]] and of [[modules]], etc.

\begin{remark}
Here it is important to consider rings and algebras without [[unit]] (really: not necessarily having a [[unit]]), since otherwise there is no [[zero object]], and also to allow [[ideals]] to appear as subrings-without-unit.

Note that the category of rings with unit  is still [[protomodular category|protomodular]].
\end{remark}

## Definition

A [[category]] $C$ is **semi-abelian** if it

* is [[exact category|Barr-exact]] (hence [[regular category|regular]] and in particular has [[finite limits]]);

* has a [[zero object]];

* has [[finite product|finite]] [[coproducts]]; and

* is [[protomodular category|protomodular]].

In other words, it is a [[homological category]] which is [[Barr-exact]] and has finite [[coproducts]].

Equivalently, $C$ is semi-abelian if:

* it has finite [[product]]s and [[coproduct]]s and a [[zero object]];

* it has [[pullback]]s of [[monomorphism]]s (or even only of [[split monomorphism]]s);

* it has [[coequalizer]]s of [[kernel pair]]s;

* [[regular epimorphism]]s are stable under [[pullback]];

* [[congruence|equivalence relations]] are effective; and

* the _Split Short [[five lemma|Five Lemma]]_ holds:

+-- {: .un_defn #SplitShortFiveLemma}
###### Definition
**(split short five lemma)**

Given a [[commutative diagram]]

$$
 \array{
    L & \overset{l}{\to} & F & \overset{q}{\to} & C
   \\
   {}^{\mathllap{u}}\downarrow
     && \downarrow^{\mathrlap{w}} && \downarrow^{\mathrlap{v}}
   \\
    K & \underset{k}{\to} & E& \underset{p}{\to} & B
   }
$$

where

* $p$ and $q$ are [[split epimorphism]]s

* and $l$ and $k$ are their [[kernel]]s,

then if $u$ and $v$ are [[isomorphism]]s so is $w$.

=--

To see that the second list of axioms implies the existence of finite limits, observe that the pullback
$$\array{P & \to & A\\
  \downarrow && \downarrow^f\\
  B& \underset{g}{\to} & C}$$
can be computed as the pullback
$$\array{P & \to & A\times B\\
  \downarrow && \downarrow^{(1,1,f)}\\
  A\times B& \underset{(1,1,g)}{\to} & A\times B\times C}$$
in which both legs are split monics.  Filling in one of the equivalent definitions of Barr-exactness, the equivalence of the two lists of axioms reduces to showing that in a Barr-exact category with coproducts and a zero object, protomodularity is equivalent to the Split Short Five Lemma; see the paper referenced below for a proof.

## Examples

* Every [[abelian category]] is semi-abelian.  Conversely, a semi-abelian category is abelian if and only if it is [[additive category|additive]] (since any exact additive category is abelian), and if and only if its opposite is semi-abelian.

* The category [[Grp]] of not-necessarily-abelian [[group]]s is semi-abelian but not [[abelian category|abelian]].  So are the categories of rings without units, algebras without units, Lie algebras, and many other sorts of algebras.  (The category of rings with unit is not semi-abelian since it lacks a zero object.)

* More generally, the category of internal [[group object]]s in any exact category is semi-abelian as soon as it has finite coproducts.  For instance, this applies to internal groups in any [[topos]] with a [[natural numbers object|NNO]].

* The opposite of any [[topos]], such as $Set^{op}$, is Barr-exact and protomodular, but obviously lacks a zero object.

* The category of Heyting semilattices

* The category of (ordinary) Lie algebras

* The category $Set_*$ of [[pointed set]]s is Barr-exact with finite coproducts and a zero object, but is not semi-abelian: protomodularity and the Split Short Five Lemma fail to hold.

* If $C$ is exact and protomodular with finite colimits, then for any $x\in C$ the [[over category|over]]-[[under category|under]] category $(x/C/x)$ is semi-abelian.  For example, the opposite of the category of [[pointed object|pointed objects]] in a [[topos]] is semi-abelian, and in particular, $Set_*^{op}$ is semi-abelian.

* The categories of [[crossed module|crossed modules]], [[crossed complex]]es, and their friends are semi-abelian; see example 4.2.6 of the Van der Linden paper referenced below.

* the category of cocommutative Hopf algebras over a field. Was proven in papers by Gran/Kadjo/Vercruysse and Gran/Sterck/Vercruyse

### Dold--Kan correspondence

* The analogue of a [[Dold–Kan correspondence]] holds for [[simplicial object]]s in semi-abelian categories.

## Related concepts

* [[pseudo-abelian category]]

* [[quasi-abelian category]]

* [[abelian category]]

## References

* [[George Janelidze]], László Márki, [[Walter Tholen]], _Semi-abelian categories_, J. Pure Appl. Alg. __168__, 2-3 (2002) 367-386, doi

* [[Dominique Bourn]], [[Francis Borceux]], *[[Mal'cev, protomodular, homological and semi-abelian categories]]*, Kluwer 2004 ([doi:10.1007/978-1-4020-1962-3](https://link.springer.com/book/10.1007/978-1-4020-1962-3))

* Dominique Bourn, [[Maria Manuel Clementino]], _Categorical and topological aspects of semi-abelian theories_ , lecture notes Haute Bodeux 2007. ([pdf](http://www.math.yorku.ca/~tholen/HB07BournClementino.pdf))

* [[Tim Van der Linden]], _Homology and homotopy in semi-abelian categories_, [math/0607100](http://arxiv.org/abs/math/0607100).

[[!redirects semi-abelian category]]
[[!redirects semi-abelian categories]]
[[!redirects semiabelian category]]
[[!redirects semiabelian categories]]


semi-atom

A __semi-atom__ is a generalization of the notion of [[atom]] in a [[bottom]] bounded [[partial order]].

An object $a$ is a semi-atom when the [[interval]] $[\bot, a ]$ is a [[partial order chain|chain]], i.e. any two objects $x$ and $y$ in that interval are [[comparable]]  ($x \le y$ or $y \leq x$).

All atoms are semi-atoms and usually the bottom is not considered one.

If $a$ and $b$ are semi-atoms then their [[meet]] exists and we have $a \wedge b \in \{\bot, a, b\}$

## Examples

The [[divisor lattice]] for some number $n$ contains prime numbers as atoms and may contain powers of primes as semi-atoms that are not atoms.


semi-decidable proposition

\tableofcontents

## Definition

A [[proposition]] or [[truth value]] $P$ is **semi-decidable** or **semidecidable** if and only if [[existential quantifier|there exists]] a [[sequence]] of [[booleans]] $f \in 2^\mathbb{N}$ such that $P$ if and only if there exists a natural number $x \in \mathbb{N}$ such that $f(x) = 1$.

$$\mathrm{isSemiDecidable}(P) \coloneqq \exists f \in 2^\mathbb{N}.P \iff \exists x \in \mathbb{N}.f(x) = 1$$

The set of all semi-decidable propositions is typically called the **[[Rosolini dominance]]**, though it may not be a [[dominance]] without certain assumptions such as [[countable choice]] or [[excluded middle]].

If the [[foundations of mathematics]] has the [[Cauchy real numbers]] $\mathbb{R}_C$ as well, then $P$ is semideciable if and only if there exists a Cauchy real number $x \in \mathbb{R}_C$ such that $P$ if and only if $x \gt 0$.

$$\mathrm{isSemiDecidable}(P) \coloneqq \exists x \in \mathbb{R}_C.P \iff x \gt 0$$

The [[limited principle of omniscience]] for the [[natural numbers]] $\mathrm{LPO}_\mathbb{N}$ implies that every semi-decidable proposition is a [[decidable proposition]].

## Related concepts

* [[decidable proposition]]

* [[proposition]], [[truth value]]

* [[synthetic topology]], [[dominance]]

## References

* [[Andrej Bauer]], [[Davorin Lešnik]], _Metric Spaces in Synthetic Topology_, 2010 ([pdf](http://math.andrej.com/wp-content/uploads/2010/01/csms_in_synthtop.pdf))

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semi-direct product > history

< [[semi-direct product]]


semi-free action

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Group Theory
+-- {: .hide}
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=--
#### Representation theory
+-- {: .hide}
[[!include representation theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition

An [[action]] of a [[group]] is called _semi-free_ if it is a [[free action]] after restriction to the [[complement]] of the [[fixed points]]. In other words, it is semi-free if all its [[stabilizer groups]] are either the full group itself or the [[trivial group]].

## Related concepts

* [[transitive action]], [[free action]], [[semi-free action]], [[regular action]],

* [[proper action]], [[properly discontinuous action]]

## References

* {#Stong70} [[Robert Stong]], _Semi-free group actions_, Illinois Journal of Mathematics, Volume 23, Number 4, 1979 ([pdf](https://projecteuclid.org/download/pdf_1/euclid.ijm/1256047939))

* Mikiya Masuda, Taras Panov, _Semifree circle actions, Bott towers, and quasitoric manifolds_, Sbornik Math. 199 (2008), no.7-8, 1201-1223 ([arXiv:math/0607094](https://arxiv.org/abs/math/0607094))

* Ronald Dotzel, _Semifree finite group actions on homotopy spheres_, Pacific Journal of Mathematics, Vol 103, No. 1, 1982

[[!redirects semi-free actions]]

[[!redirects semifree action]]
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semi-graph

\tableofcontents

\section{Introduction}

A semi-graph is like a _graph_ (more precisely, an undirected, simple graph, possibly with loops) but where certain edges may be 'open' or 'half-open': they may have a vertex at only one end, or at neither end. As usual, edges can only be joined at vertices. A vertex is typically indicated by a solid dot (possibly with a label), an end of edge without a vertex is typically indicated by a hollow dot (never with a label).

Here is a quick example, which has one edge without any vertices, and two edges with only one vertex.

\begin{centre}
  \begin{tikzpicture}
    \draw (0,0) -- (0,2) -- (2,2) -- (2,0) -- (0,0);
    \draw (2,2) -- (2,4);
    \draw (2,0) -- (4,0);
    \draw (0,-2) -- (2,-2);
    \fill (0,0) circle[radius=0.1];
    \fill (0,2) circle[radius=0.1];
    \fill (2,2) circle[radius=0.1];
    \fill (2,0) circle[radius=0.1];
    \draw (2,4) circle[radius=0.1];
    \draw (4,0) circle[radius=0.1];
    \draw (0,-2) circle[radius=0.1];
    \draw (2,-2) circle[radius=0.1];
  \end{tikzpicture}
\end{centre}

\section{Preliminaries}

Throughout this page, we pick a set with exactly one element, and denote it $1$.

\section{Details}

The following definition follows [Mochizuki2006](#Mochizuki2006).

\begin{defn} A _semi-graph_ consists of a set $V$, a set $E$ whose elements are sets with exactly two elements, and, for every element $e$ of $E$, a map $\zeta_{e}: e \rightarrow V \sqcup 1$.
 \end{defn}

\begin{terminology} Let $G$ be a semi-graph. A _vertex_ of $G$ is an element of $V$. An _edge_ of $G$ is an element of $E$. An edge $e$ of $G$ _abuts_ to a vertex $v$ of $G$ if $v$ belongs to the [[image]] of $\zeta_{e}$. We refer to an element of an edge $e$ of $G$ as a _branch_ of $e$.   \end{terminology}

\begin{rmk} In particular, every edge of a semi-graph has exactly two branches, but it is possible that neither, both, or only one of the branches abuts a vertex. Thus we think of an edge $e$ as subdivided into two branches $b_{1}$ and $b_{2}$, and do not think of the meeting point as a vertex of the graph; but always consider the other end of each branch as a vertex, either 'closed' (the branch maps to an actual vertex of $G$ under $\zeta_{e}$) or 'open' (the branch maps to $1$ under $\zeta_{e}$).

\begin{centre}
  \begin{tikzpicture}
    \draw (0,0) to node[auto] {\large $b_{1}$} (2,0) to node[auto] {\large $b_{2}$} (4,0);
    \fill (0,0) circle[radius=0.1];
    \draw (4,0) circle[radius=0.1];
    \node[anchor=north] at (0,-0.2) {\large $v$};
  \end{tikzpicture}
\end{centre}

\end{rmk}

\section{References}

* {#Mochizuki2006} _Semi-graphs of anabelioids_, [[Shinichi Mochizuki]], Publ. Res. Inst. Math. Sci., 42, No. 1, 221-322 (2006). [paper](http://www.kurims.kyoto-u.ac.jp/~motizuki/Semi-graphs%20of%20Anabelioids.pdf) [Zentralblatt review](https://zbmath.org/?q=an%3A1113.14025)


# Semi-left-exact left Bousfield localizations

* table of contents
{: toc}

## Idea

When constructing a general [[left Bousfield localization]] of a [[cofibrantly generated model category]] $C$ at a set of maps $S$, it is fairly straightforward to construct a [[localization]] functor, i.e. a [[fibrant replacement]] functor for the putative localized [[model category]] $L_S C$, by a [[small object argument]].  We represent the maps in $S$ by cofibrations between [[cofibrant]] objects, take their [[pushout]] products with all boundary inclusions $\partial \Delta^n \hookrightarrow \Delta^n$ (assuming for simplicity that $C$ is a [[simplicial model category]]), and add all the generating [[acyclic cofibrations]] of $C$.  Then an object $Z$ has the right lifting property with respect to this set if and only if it is [[fibrant]] in $C$ (because we added the generating acyclic cofibrations of $C$) and for all $f:A\to B$ in $S$ the induced map $Map(B,Z) \to Map(A,Z)$ of simplicial mapping spaces (which is a fibration since $f$ is a cofibration and $Z$ is fibrant) is an acyclic fibration, i.e. $Z$ is $S$-local.  However, it is much harder to construct a factorization of an *arbitrary* map as an $S$-acyclic cofibration followed by an $S$-fibration --- one has to use a cardinality argument to obtain a set of generating $S$-acyclic cofibrations --- and accordingly the $S$-fibrations have no explicit description.

This can be seen as a homotopy-theoretic analogue of the construction of a [[reflective factorization system]].  The localization functor exhibits the $S$-local objects as a [[reflective subcategory]], while the $S$-acyclic cofibrations and $S$-fibrations are a model-categorical representation of the corresponding reflective [[factorization system]], whose left class consists of the morphisms inverted by the [[reflector]] (here, the $S$-local equivalences) and whose right class is defined by orthogonality to these.  Even in [[1-category theory]], constructing reflective factorizations requires finicky cardinality or size-based arguments as well.  However, there are some reflections, such as the [[semi-left-exact reflections]], for which the [[reflective factorization system]] can be constructed by a direct argument in one step.  The homotopy-theoretic analogue of these is a **semi-left-exact left Bousfield localization**.

## Construction

The following is Theorem 1.1 of [Stanculescu](#Stanculescu), which is an improved version of Theorems 9.3 and 9.7 from [Bousfield 2001](#Bousfield01), which are in turn an improved version of Appendix A of [Bousfield-Friedlander 78](#BousfieldFriedlander78).

\begin{theorem}
  Let $C$ be a [[model category]] and $Q:C\to C$ a functor equipped with a natural transformation $\alpha:Id\to Q$ such that

  1. $Q$ is a [[homotopical functor]], i.e. preserves weak equivalences.
  2. For each $X\in C$, the map $Q \alpha_X:Q X \to Q Q X$ is a weak equivalence, and the map $\alpha_{Q X}:Q X \to Q Q X$ becomes a [[monomorphism]] in the [[homotopy category]].
  3. Define an object $X$ to be $Q$-local if it is fibrant and $X\to Q X$ is a weak equivalence, and define a morphism $f$ to be a $Q$-equivalence if $Q f$ is a weak equivalence.  Then pullback along fibrations between $Q$-local objects preserves $Q$-equivalences.

  Then there is a new model structure $C^Q$ on $C$ whose cofibrations are those of $C$, whose weak equivalences are the $Q$-equivalences, and whose fibrations are the maps whose $\alpha$-naturality-square is a [[homotopy pullback]].  Moreover, $C^Q$ is [[proper model category|right proper]], and simplicial if $C$ is.
\end{theorem}

The first two conditions say essentially that $Q$ is a homotopical reflection into some subcategory (namely the $Q$-local objects).  The third condition says that it is semi-left-exact (a 1-categorical reflection of $C$ into $B\subseteq C$ is semi-left-exact if and only if pullback along morphisms in $B$ preserves morphisms that are inverted by the reflector).

For details of the proof, see the references.  The central point is the construction of the factorization into an $S$-acyclic cofibration and an $S$-fibration, which proceeds by first applying $Q$ along with fibrant replacement, then taking a homotopy pullback: the same way that a [[reflective factorization system]] is constructed from a [[semi-left-exact reflection]] in 1-category theory.

## Remarks

* Right properness of semi-left-exact left Bousfield localizations was also shown in [Gepner-Kock, Prop. 7.8](#GK), with special attention paid to [[type-theoretic model categories]].

* Nullification, i.e. localization at a family of maps $A\to \ast$, is always semi-left-exact.  (Indeed, nullification is what in [[homotopy type theory]] is called a [[higher modality]], and has [[stable units|reflection with stable units]], a stronger condition.)

* [[left-exact reflection|Left exact]] reflections are always semi-left exact.  In particular, the left-exact localizations that present [[Grothendieck (∞,1)-toposes]] can be constructed in this way.

## References

* {#BousfieldFriedlander78} [[Aldridge Bousfield]], [[Eric Friedlander]], def. 1.1.6 in _Homotopy theory of $\Gamma$-spaces, spectra, and bisimplicial sets_, Springer Lecture Notes in Math., Vol. 658, Springer, Berlin, 1978, pp. 80-130. ([pdf](https://www.math.rochester.edu/people/faculty/doug/otherpapers/bousfield-friedlander.pdf))

* {#Bousfield01} [[Aldridge Bousfield]], *On the telescopic homotopy theory of spaces*, Trans. Amer. Math. Soc. 353 (2001), 2391-2426, [web with fulltext](https://www.ams.org/journals/tran/2001-353-06/S0002-9947-00-02649-0/)

* {#Stanculescu} [[Alexandru E. Stanculescu]], *Note on a theorem of Bousfield and Friedlander*, Topology and its Applications **155** 13 (2008) 1434-1438 [[arxiv:0806.4547](https://arxiv.org/abs/0806.4547), [doi:10.1016/j.topol.2008.05.003](https://doi.org/10.1016/j.topol.2008.05.003)]

* {#Hirschhorn} [[Philip Hirschhorn]], chapter 13 of _Model Categories and Their Localizations_, 2003 ([AMS](http://www.ams.org/bookstore?fn=20&arg1=whatsnew&item=SURV-99), [pdf toc](http://www.gbv.de/dms/goettingen/360115845.pdf), [pdf](http://www.maths.ed.ac.uk/~aar/papers/hirschhornloc.pdf))

* {#CHK} Cassidy and Hébert and [[Max Kelly|Kelly]], "Reflective subcategories, localizations, and factorization systems".  *J. Austral. Math Soc. (Series A)* 38 (1985), 287--329 ([pdf](http://journals.cambridge.org/download.php?file=%2FJAZ%2FJAZ1_38_03%2FS1446788700023624a.pdf&code=5796045be8904c5183c2e95bce65491e))

* {#GK} [[David Gepner]] and [[Joachim Kock]], *Univalence in locally cartesian closed categories*, [arxiv:1208.1749](https://arxiv.org/abs/1208.1749)

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semi-left-exact reflection

# Semi-left-exact reflections

* table of contents
{: toc}

## Idea

A **semi-left-exact reflection** (also called a *locally cartesian reflection*) is a [[reflector]] into a [[reflective subcategory]] that preserves some [[pullbacks]].  It fits into a hierarchy of left-exactness properties for reflectors:

$\;\;\;\;\;\;\;\;$
[left exact](reflective+subcategory#ExactReflectiveSubcategories) $\;\;\Rightarrow\;\;$
[[reflection with stable units|stable units]]
$\;\;\Rightarrow\;\;$
**semi-left-exact**
$\;\;\Rightarrow\;\;$
[[simple reflection|simple]]

In particular, semi-left-exactness is sufficient to imply that the corresponding [[reflective factorization system]] exists and can be constructed in one step, and that the reflective subcategory inherits [[local cartesian closure]] from the ambient category.

## Definitions

Let $C$ be a [[category]] with [[finite limits]], and $B\subseteq C$ a [[full subcategory|full]] [[reflective subcategory]] with reflector $L:C\to B$.  Let $E$ be the class of morphisms inverted by $L$, and let $M$ be the class of morphisms right orthogonal to $E$, i.e. $(E,M)$ is the associated [[reflective prefactorization system]].

The following is a combination of Theorems 4.1 and 4.3 from [CHK](#CHK) with Proposition 1.3 from [GK](#GK).

\begin{theorem}
  The following are equivalent:

  1. Every pullback of an $E$-morphism along an $M$-morphism is an $E$-morphism.
  2. Every pullback of an $E$-morphism along a morphism in $B$ is an $E$-morphism.
  3. Every pullback of a reflection unit $\eta_x : x \to L x$ along a morphism in $B$ is an $E$-morphism.
  4. $L$ preserves pullbacks along $M$-morphisms.
  5. $L$ preserves pullbacks along any morphism in $B$.
  6. If $f:x\to y$ is a morphism in $B$, then $f^* : C/y \to C/x$ preserves $E$-morphisms.
  7. (If $C$ is locally cartesian closed) If $f:x\to y$ is a morphism in $B$, then $f_* : C/x \to C/y$ maps $B/x$ into $B/y$.

  They all imply that $(E,M)$ is a factorization system and that the factorization of an arbitrary morphism $f:x\to y$ can be constructed as the following pullback:

  \begin{center}
    \begin{tikzcd}
      x \ar[dr] \ar[drr,"{\eta_x}"] \ar[ddr,"f"'] \ar[dr,"{\lambda_f}" description] \\
      & \bullet \ar[r,"g"'] \ar[d,"{\rho_f}"] & L x \ar[d,"L f"] \\
      & y \ar[r,"{\eta_y}"'] & L y
    \end{tikzcd}
  \end{center}
  If these conditions hold, we say that the reflector is **semi-left-exact**.
\end{theorem}
\begin{proof}
  Since all morphisms in $B$ are $M$-morphisms, (1)$\Rightarrow$(2) and (4)$\Rightarrow$(5), while (2)$\Rightarrow$(3) since the reflection units are in $E$.  And since $E$ consists of the $L$-inverted morphisms, (5)$\Rightarrow$(2) and (4)$\Rightarrow$(1).  And clearly (6)$\Rightarrow$(2); while for $f\in B$ the action of $f^*$ on a morphism in $C/y$ is the latter's pullback along a pullback of $f$, and any pullback of $f$ lies in $M$; thus (1)$\Rightarrow$(6).  So to complete the equivalence of (1)-(6) it will suffice to show that (3) implies (4).

  First we prove that (3) implies the factorization exists.  For in the diagram above $g$ is a pullback of the unit $y\to L y$ along the morphism $L x \to L y$ in $B$, hence $g\in E$; so by 2-out-of-3 $\lambda_f\in E$, while $\rho_f\in M$ since $M$ is closed under pullback and $L f \in M$.

  Now this construction implies that if $f\in M$ then $\lambda_f$ is invertible, hence the naturality square for $\eta$ at $f$ is a pullback.  Now if we want to pull back some $h:c\to k$ along an $M$-morphism $f:a\to k$, we can paste two pullback squares to obtain a large pullback rectangle:

\begin{center}
  \begin{tikzcd}
    d \ar[r] \ar[d] & a \ar[d,"f"] \ar[r,"{\eta_a}"] & L a \ar[d,"{L f}"]\\
    c \ar[r,"h"'] & k \ar[r,"{\eta_k}"'] & L k.
  \end{tikzcd}
\end{center}

  Since $\eta_k \circ h = L h \circ \eta_c$ by naturality, we can now factor this pullback rectangle through the pullback of $f$ along $L h$:

\begin{center}
  \begin{tikzcd}
    d \ar[r,dashed] \ar[d] & e \ar[r] \ar[d] & L a \ar[d,"{L f}"]\\
    c \ar[r,"{\eta_c}"'] & L c \ar[r,"{L h}"'] & L k
  \end{tikzcd}
\end{center}

  so that the left-hand square is also a pullback.  But this is a pullback of the reflection unit $\eta_c$ along the map $e\to L c$ that lies in $B$ (since $B$ is closed under pullbacks).  Thus by (3) the map $d \to e$ lies in $E$, and hence exhibits $e$ as $L d$.  The right-hand pullback square above is then $L$ applied to the given pullback square, so that $L$ indeed preserves this pullback, i.e. (4) holds.

  Finally, if $C$ is locally cartesian closed, then (6) implies (7) by adjointness, while (7) implies (5) in the form $L/x \circ f^* \cong L/y \circ f^*$ by passage to right-adjoint mates.
\end{proof}

Of course, if $L$ preserves all pullbacks (i.e. it is [[left exact reflection|left exact]]), then it is semi-left-exact.  The statement about the existence of the reflective factorization system implies that any semi-left-exact reflection is [[simple reflection|simple]].

\begin{remark}
  Note that for any $x\in C$, the functor $L/x\colon C/x \to B/L x$ has a right adjoint given by pullback along $\eta_x : x \to L x$.  Condition (3) above then says that the counit of this adjunction is an isomorphism, which is to say that this right adjoint is fully faithful.  In this form, semi-left-exactness is equivalent to (a particular case of) the notion of *admissible* reflection in [[categorical Galois theory]].
\end{remark}

## Related pages

* [[semi-left-exact left Bousfield localization]]

## References

* {#CHK} Cassidy and Hébert and [[Max Kelly|Kelly]], "Reflective subcategories, localizations, and factorization systems".  *J. Austral. Math Soc. (Series A)* 38 (1985), 287--329 ([pdf](http://journals.cambridge.org/download.php?file=%2FJAZ%2FJAZ1_38_03%2FS1446788700023624a.pdf&code=5796045be8904c5183c2e95bce65491e))

* {#GK} [[David Gepner]] and [[Joachim Kock]], *Univalence in locally cartesian closed categories*, [arxiv:1208.1749](https://arxiv.org/abs/1208.1749)

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

#Contents#
* table of contents
{:toc}

## Idea

A [[topological space]] $X$ is _(semi-)locally simply connected_ if every neighborhood of a point has a subneighbourhood in which loops based at the point in the subneighborhood can be contracted in $X$. It is similar to but weaker than the condition that every neighborhood of a point has a subneighborhood that is [[simply connected]]. This latter condition is called _local simple-connectedness_.

## Definition

A [[topological space]] $X$ is **semi-locally simply-connected** if it has a [[basis of the topology|basis]] of [[neighbourhood | neighbourhoods]] $U$ such that the inclusion $\Pi_1(U) \to \Pi_1(X)$ of [[fundamental groupoid | fundamental groupoids]] factors through the canonical functor $\Pi_1(U) \to codisc(U)$ to the [[codiscrete groupoid]] whose objects are the elements of $U$. The condition on $U$ is equivalent to the condition that the homomorphism $\pi_1(U, x) \to \pi_1(X, x)$ of [[fundamental group | fundamental groups]] induced by inclusion $U \subseteq X$ is trivial.

## Examples

+-- {: .num_example #LocallySimplyConnectedCircle}
###### Example
**([[circle]] is locally simply connected)**

The [[Euclidean space|Euclidean]] [[circle]]

$$
  S^1 = \left\{ x \in \mathbb{R}^2 \;\vert\;  {\Vert x\Vert} = 1\right\}
  \subset \mathbb{R}^2
$$

is locally simply connected

=--

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

By definition of the [[subspace topology]] and the defining [[topological base]] of the [[Euclidean plane]], a [[base for a topology|base for the topology]] of $S^1$ is given by the [[images]] of [[open intervals]] under the [[local homeomorphism]]

$$
  (cos(-), sin(-)) \;\colon\; \mathbb{R}^1 \to S^1
  \,.
$$

But these open intervals are simply connected ([this example](fundamental+group#EuclideanSpaceFundamentalGroup)).

=--

* A semi-locally simply connected space need not be locally simply connected. For a simple counterexample, take the cone on the [[Hawaiian earring space]].

## Application

Semi-local simple connectedness is the crucial condition needed to have a good theory of [[covering space | covering spaces]], to the effect that the topos of permutation representations of the fundamental groupoid of $X$ is equivalent to the category of covering spaces of $X$.

This is the _[[fundamental theorem of covering spaces]]_, see there for more.

## Generalizations

For a [[topos]]-theoretic notion of locally $n$-connected see _[[locally n-connected (infinity,1)-topos]]_.

## Related concepts

* [[locally path-connected topological space]]

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semi-locally simply-connected topological space

[[!redirects semi-locally simply topological connected space]]

#Contents#
* automatic table of contents goes here
{:toc}

## Idea

A [[topological space]] $X$ is _(semi-)locally simply connected_ if every [[neighborhood]] of a point has a subneighbourhood in which [[loops]] based at the point in the subneighborhood can be contracted in $X$. It is similar to but weaker than the condition that every neighborhood of a point has a subneighborhood that is [[simply connected]]. This latter condition is called _local simple-connectedness_.

## Definition

A [[topological space]] $X$ is **semi-locally simply-connected** if it has a basis of [[neighbourhood]]s $U$ such that the inclusion $\Pi_1(U) \to \Pi_1(X)$ of [[fundamental groupoid]]s factors through the canonical functor $\Pi_1(U) \to codisc(U)$ to the [[codiscrete groupoid]] whose objects are the elements of $U$. The condition on $U$ is equivalent to the condition that the homomorphism $\pi_1(U, x) \to \pi_1(X, x)$ of [[fundamental group]]s induced by inclusion $U \subseteq X$ is trivial.

* A semi-locally simply connected space need not be locally simply connected. For a simple counterexample, take the cone on the [[Hawaiian earring space]].

## Application

Semi-local simple connectedness is the crucial condition needed to have a good theory of [[covering space]]s, to the effect that the topos of permutation representations of the fundamental groupoid of $X$ is equivalent to the category of covering spaces of $X$.

## In topos theory

For a [[topos]]-theoretic notion of locally $n$-connected see [[locally n-connected (infinity,1)-topos]].

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semi-metal

+-- {: .rightHandSide}
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#### Solid state physics
+-- {: .hide}
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#### Topological physics
+--{: .hide}
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=--
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#Contents#
* table of contents
{:toc}

## Idea

In [[condensed matter physics]], a *semi-metal* is a [[crystal|crystalline]] material whose [[electronic band structure]] is such that there is a sizeable gap between the [[valence band]] and the [[conduction band]] (as for an [[insulator]]) *except* over a [[submanifold]] of "nodal loci" inside the [[Brillouin torus]] which is of [[positive number|positive]] [[codimension]] $\geq 2$.

This terminology, which was established in the early 2000s, is meant to follow the  older terminology *[[semiconductor]]*, which refers to the case where there is globally a gap between valence and conduction band, but just a small one. Notice that a global and large gap corresponds to an [[insulator]], while the absence of a gap, hence the broad overlap of valence and conductions band, corresponds to a [[metal]]. Therefore a semi-metal behaves like an [[insulator]] for the bulk of its excitation modes, like a [[semiconductor]] for excitations close to the nodal loci, and almost like a [[metal]] for excitations right at the nodal loci; whence the terminology.

(graphics from [[schreiber:Topological Quantum Computation in TED-K|SS 22]])

If the gap closure happens over an isolated point, one speaks of a *nodal point* proper, or more specifically of a *Dirac point* or *Weyl point*, indicating the "degeneracy" (multiplicity) of the [[energy]] [[eigenstates]] at the point (2 for Weyl, 4 for Dirac, see [below](#MassTerms)). If the gap closes over a 1-dimensional manifold (a [[curve]] or a [[circle]]) one speaks of a *nodal line*.

Due to the gap on (a [[neighbourhood retract]] of) the [[complement]] of the nodal points/lines inside the [[Brillouin torus]], the [[Berry connection]] is well-defined away from the nodal loci, and it tends to be [[flat connection|flat]] in suitable hyperplanes there. It's [[holonomy]] around [[codimension]]=2 nodal loci (ie. around nodal lines in 3d materials and nodal points in effectively 2d materials such as [[graphene]], see also at *[[defect brane]]*) -- known as a *[[Berry phase]]* -- is then a property of the nodal locus alone, and to be interpreted as a measure for the "topological protection" of the gap over the nodal loci (e.g. [FWDF 16, II.B](#FWDF16), [Vanderbilt 18, 5.5.2](#Vanderbilt18)): Under [[quantum adiabatic theorem|adiabatic]] deformations of the material that do not excite modes above the gap, these nodal points may move and merge or split, but their total topological charge cannot change.

Therefore semi-metals with such non-trivial Berry phases are examples of [[topological phases of matter]], and are sometimes called *topological semi-metals*, for emphasis. In fact, if the topological charges of the nodal loci in a semi-metal do happen to be trivial, then it is (in the deformation class of) a fully gapped [[insulator]] and may then be a *[[topological insulator]]* (if the [[twisted equivariant K-theory|twisted equivariant]] [[topological K-theory]] class of its [[valence bundle]] is non-trivial).

## Properties

### Phase decay via mass terms
 {#MassTerms}

Near nodal points in the Brillouin torus of a semi-metal, the [[dispersion
relation]] $k \mapsto E(
k)$ exhibited by the [[energy bands]] is thought to be approximated by that of a massless [[Dirac equation|Dirac equation]]/[[Weyl spinor|Weyl equation]], whence the terminology *Dirac point* or *Weyl point*.

When the material's parameters can be and are [[quantum adiabatic theorem|adiabatically]] tuned such that this [[dispersion relation]] turns into a *massive* [[Dirac equation]], then the band gap at the former nodal crossing will "open up" in proportion to the [[coefficient]] $m$ of the effective *[[mass term]]* $m \gamma_0$ in the Dirac equation. If this happens to all nodal points (while keeping the band gap open everywhere else), one expects that the topological semi-metal phase decays into a [[topological insulator]]-phase.

A necessary condition for such a *[[mass term]]* to exist at all is that a further [[Clifford algebra|Clifford generator]] $\gamma_0$ is represented on the Bloch-Hilbert space of electrons such that it skew-commutes with all [[Feynman slash notation|Clifford momenta]] $k\!\!\!/$ (e.g. [Schnyder 18, Sec. II.A](#Schnyder18), [Schnyder 20, Sec. 2.1](#Schnyder20), see also [Freed & Hopkins 21, around Lem. 9.55](#FreedHopkins21)).

\begin{imagefromfile}
    "file_name": "MassTermsForSemiMetalNodalPoints-220517.jpg",
    "width": 600,
    "unit": "px",
    "margin": {
        "top": -20,
        "bottom": 20,
        "right": 0,
        "left": 10
    },
    "caption": "(graphics from [SS22](#SS22))"
\end{imagefromfile}

By [[Karoubi K-theory|Karoubi/Atiyah-Singer-type theorems]], such gap-opening mass terms are expected to again be classified by topological K-theory groups ([Chiu & Schnyder 14, Sec. A.2](#ChiuSchnyder14), reviewed in [Schnyder 18, Sec. II.A](#Schnyder18), [Schnyder 20, Sec. 2.1](#Schnyder20), following [Morimoto & Furusaki 13, Sec. V](#MorimotoFurusaki13), [CTSR 15, Sec. III.C](#CTSR15)).

Beware:

1. This means that the topological *semi-metal*-phase must be classified in a group *modulo* the group of [[mass terms]] (cf. [Freed & Hopkins 21, Thm. 9.63](#FreedHopkins21)). The K-group of mass terms itself at best classifies aspects of the [[topological insulator]]-phase that the semi-metal phase may decay to.

1. The argument about single [[mass terms]] in a K-theory group of a point ignores the global topology of the valence bundle.

   This is manifest already in the canonical example of the [[Haldane model]], where for *constant* mass terms (as usually understood) the total topological charge of all nodal points combined is constrained to vanish.

## Examples

The archetypical and original example of a semi-metal is *[[graphene]]*, where the gap closes over two [[Dirac points]]. (Or almost: a tiny [[spin-orbit coupling]] in graphene actually produces a tiny gap even at the would-be Dirac points, which however tends to be too small to be visible in practice. But if one will or would analyze graphene with an accuracy that does resolve this tiny gap-opening, then graphene will or would appear as a [[topological insulator]] instead of a semi-metal.)

After the synthesis of graphene, which is an effectively 2-dimensional material (an atomic mono-layer) much attention has shifted to the synthesis and understanding of 3-dimensional semi-metals.

## Related concepts

* [[topological insulator]]

* [[photonic semimetal]], [[phononic semimetal]]

## References

### General

General monographs:

* [[Tomáš Bzdušek]], *Symmetry and topology of nodal semimetals*, ETH (2017) [[doi:10.3929/ethz-b-000216959](https://doi.org/10.3929/ethz-b-000216959)]

* {#Vanderbilt18} [[David Vanderbilt]],  Section 5 of: *Berry Phases in Electronic Structure Theory -- Electric Polarization, Orbital Magnetization and Topological Insulators*, Cambridge University Press (2018) ([doi:10.1017/9781316662205](https://doi.org/10.1017/9781316662205))

Review with emphasis on [mass terms](#MassTerms)

* {#Schnyder18} [[Andreas P. Schnyder]], *Accidental and symmetry-enforced band crossings in topological semimetals*, lecture notes (2018) $[$[pdf](https://www.fkf.mpg.de/6431357/topo_lecture_notes_schnyder_TMS18.pdf), [[Schnyder-Semimetals-2018.pdf:file]]$]$

* {#Schnyder20} [[Andreas P. Schnyder]], *Topological semimetals*, lecture notes (2020) $[$[pdf](https://www.fkf.mpg.de/7143520/topological_semimetals.pdf), [[Schnyder-Semimetals-2020.pdf:file]]$]$

For references on *[[graphene]]* see also there.

Discussion focusing on 3-dimensional semi-metals:

* A. A. Burkov, M. D. Hook, Leon Balents,  *Topological nodal semimetals*, Phys. Rev. B **84** (2011) 235126 ([arXiv:1110.1089](https://arxiv.org/abs/1110.1089), [doi:10.1103/PhysRevB.84.235126](https://doi.org/10.1103/PhysRevB.84.235126))

* [[Ari M. Turner]], [[Ashvin Vishwanath]], Part I of: *Beyond Band Insulators: Topology of Semi-metals and Interacting Phases*, in: *Topological Insulators*, Contemporary Concepts of Condensed Matter Science **6** (2013) 293-324 $[$[arXiv:1301.0330](https://arxiv.org/abs/1301.0330), [ISBN:978-0-444-63314-9](https://www.sciencedirect.com/bookseries/contemporary-concepts-of-condensed-matter-science/vol/6/suppl/C)$]$

* Bohm-Jung Yang, Naoto Nagaosa, *Classification of stable three-dimensional Dirac semimetals with nontrivial topology*, Nature Communications **5** (2014) 4898  ([doi:10.1038/ncomms5898](https://doi.org/10.1038/ncomms5898))

* N. P. Armitage, [[Eugene Mele]], [[Ashvin Vishwanath]], *Weyl and Dirac semimetals in three-dimensional solids*, Rev. Mod. Phys. **90** 015001 (2018) $[$[doi:10.1103/RevModPhys.90.015001](https://doi.org/10.1103/RevModPhys.90.015001)$]$

* Jiaheng Li, Zetao Zhang, Chong Wang, Huaqing Huang, Bing-Lin Gu, Wenhui Duan,  *Topological semimetals from the perspective of first-principles calculations*,
Journal of Applied Physics 128, 191101 (2020) ([doi:10.1063/5.0025396]( https://doi.org/10.1063/5.0025396))

* Faruk Abdulla, Ganpathy Murthy, Ankur Das, *Stable nodal line semimetals in the chiral classes in three dimensions*  [[arXiv:2401.02966](https://arxiv.org/abs/2401.02966)]

Discussion of the $\mathbb{Z}/2$-valued [[Berry phases]] around [[codimension]]=2 nodal loci in $P I$-symmetric (time-reversal +  inversion invariant)  semi-metals:

* Dan-Wei Zhang, Y. X. Zhao, Rui-Bin Liu, Zheng-Yuan Xue, Shi-Liang Zhu, Z. D. Wang, *Quantum simulation of exotic PT-invariant topological nodal loop bands with ultracold atoms in an optical lattice*, Phys. Rev. A **93** (2016) 043617  ([arXiv:1601.00371](https://arxiv.org/abs/1601.00371), [doi:10.1103/PhysRevA.93.043617](https://doi.org/10.1103/PhysRevA.93.043617))

  > (see sec II.A, these authors stand out as mentioning the relevant [[KO-theory]])

* {#FWDF16} Chen Fang, Hongming Weng, Xi Dai, Zhong Fang, *Topological nodal line semimetals*, Chinese Phys. B **25** (2016) 117106 ([arXiv:1609.05414](https://arxiv.org/abs/1609.05414), [doi:10.1088/1674-1056/25/11/117106](https://doi.org/10.1088/1674-1056/25/11/117106))

  > (see sec II.B)

* [Vanderbilt 18, 5.5.2](#Vanderbilt18)

More mathematical discussion of the case of [[Chern semi-metals]]:

* {#MathaiThiang17a} [[Varghese Mathai]], [[Guo Chuan Thiang]], *Global topology of Weyl semimetals and Fermi arcs*, J. Phys. A: Math. Theor. **50** (2017) 11LT01 [[arXiv:1607.02242](https://arxiv.org/abs/1607.02242),  [doi:10.1088/1751-8121/aa59b2](https://doi.org/10.1088/1751-8121/aa59b2)]

* {#MathaiThiang17b} [[Varghese Mathai]], [[Guo Chuan Thiang]], *Differential Topology of Semimetals*, Commun. Math. Phys. **355** (2017) 561–602 [[arXiv:1611.08961](https://arxiv.org/abs/1611.08961), [doi:10.1007/s00220-017-2965-z](https://doi.org/10.1007/s00220-017-2965-z)]

Discussion of (classification of) Dirac/Weyl mass terms:

* {#ChiuSchnyder14} [[Ching-Kai Chiu]], [[Andreas P. Schnyder]], Section A.2 of: *Classification of reflection-symmetry-protected topological semimetals and nodal superconductors*, Phys. Rev. B **90** 205136 (2014) $[$[doi:10.1103/PhysRevB.90.205136](https://doi.org/10.1103/PhysRevB.90.205136)$]$

following:

* {#MorimotoFurusaki13} Takahiro Morimoto and Akira Furusaki, Sec. V of: *Topological classification with additional symmetries from Clifford algebras*, Phys. Rev. B **88** (2013) 125129 ([arXiv:1306.2505](https://arxiv.org/abs/1306.2505),  [doi:10.1103/PhysRevB.88.125129](https://doi.org/10.1103/PhysRevB.88.125129))

* {#CTSR15} [[Ching-Kai Chiu]], [[Jeffrey C.Y. Teo]], [[Andreas P. Schnyder]], [[Shinsei Ryu]], Section III.C of *Classification of topological quantum matter with symmetries*, Rev. Mod. Phys. **88** (2016) 035005 ([arXiv:1505.03535](https://arxiv.org/abs/1505.03535), [doi:10.1103/RevModPhys.88.035005](https://doi.org/10.1103/RevModPhys.88.035005))

Example:

* {#DelftTopologyCourse} Delft Topology Course team, *[Haldane model, Berry curvature, and Chern number](https://topocondmat.org/w4_haldane/haldane_model.html#first-try)* - [First try](https://topocondmat.org/w4_haldane/haldane_model.html#first-try), [Second try](https://topocondmat.org/w4_haldane/haldane_model.html#second-try)

Related discussion of Dirac mass terms is in:

* {#FreedHopkins21} [[Daniel S. Freed]], [[Michael J. Hopkins]], *Reflection positivity and invertible topological phases*, Geom. Topol. **25** (2021) 1165-1330 $[$[arXiv:1604.06527](https://arxiv.org/abs/1604.06527), [doi:10.2140/gt.2021.25.1165](https://doi.org/10.2140/gt.2021.25.1165)$]$

Lists of actual semi-metals:

* Md. Rakibul Karim Akand. *Catalog of magnetic topological semimetals*, : AIP Advances **10** 095222 (2020) [[doi:10.1063/5.0020096](https://doi.org/10.1063/5.0020096)]

Simulation of Dirac semimetals with [[lattice field theory]]:

* *Lattice field theory simulations of Dirac semimetals*, Annals of Physics **391** (2018) 278-292 [[doi:10.1016/j.aop.2018.02.016](https://doi.org/10.1016/j.aop.2018.02.016)]

and other numerical calculuations:

* *The study of magnetic topological semimetals by first principles calculations*, npj Computational Materials **5** 96 (2019) [[doi:10.1038/s41524-019-0237-5](https://doi.org/10.1038/s41524-019-0237-5)]

* *Topological semimetals from the perspective of first-principles calculations*, Journal of Applied Physics **128** (2020) 191101 [[doi:10.1063/5.0025396](https://doi.org/10.1063/5.0025396)]

### Holographic description

On holographic descriptions of [[topological semimetals]] via the [[AdS-CMT correspondence]]:

* [[Karl Landsteiner]], [[Yan Liu]], *The holographic Weyl semi-metal*, Physics Letters B **753** (2016) 453-457 [[arXiv:1505.04772](https://arxiv.org/abs/1505.04772), [doi:10.1016/j.physletb.2015.12.052](https://doi.org/10.1016/j.physletb.2015.12.052)]

* [[Karl Landsteiner]], [[Yan Liu]], Ya-Wen Sun, *Quantum phase transition between a topological and a trivial semimetal from holography*, Phys. Rev. Lett. **116** 081602 (2016) [[arXiv:1511.05505](https://arxiv.org/abs/1511.05505), [doi:10.1103/PhysRevLett.116.081602](https://doi.org/10.1103/PhysRevLett.116.081602)]

* Ling-Long Gao, [[Yan Liu]], Hong-Da Lyu, *Black hole interiors in holographic topological semimetals* [[arXiv:2301.01468](https://arxiv.org/abs/2301.01468)]

Some of the above material is taken from

* {#SS22} [[Hisham Sati]], [[Urs Schreiber]]: *[[schreiber:Topological Quantum Computation in TED-K]]*, [PlanQC **2022**](https://icfp22.sigplan.org/home/planqc-2022) [**33**](https://planqc2022.hotcrp.com/paper/33), (15 Sep 2022) [[arXiv:2209.08331](https://arxiv.org/abs/2209.08331), slides: [pdf](/schreiber/files/TQCinTEDK-PlanQC-220905b.pdf), video: [YT](https://www.youtube.com/watch?v=hDwXQVcloB8&feature=emb_logo)]

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semi-Segal space

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#Contents#
* table of contents
{:toc}

## Idea

A _semi-Segal space_ is like a [[Segal space]] but without specified identities/degeneracies. It is to [[semicategories]] as Segal spaces are to [[categories]].

## Definition

Let $\mathcal{C} = $ [[sSet]], equipped with the standard [[model structure on simplicial sets]].

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

A **semi-Segal space** is a [[semi-simplicial object]] in $\mathcal{C}$ such that

1. it is a [[fibrant object]] in the [[Reedy model structure]] on $\mathcal{C}^{\Delta^{op}_+}$;

1. it satisfies the [[Segal conditions]] be [[weak equivalences]].

=--

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

A [[semi-simplicial object]] $X_\bullet$ being Reedy fibrant means that for each $n \in \mathbb{N}$ the morphisms

$$
  \array{
   X_{n}
   \\
   \downarrow^{\mathrlap{(\partial_0, \cdots, \partial_{n})}}
   \\
    X^{\partial \Delta^n}
  }
$$

are [[fibrations]].

=--

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

Equivalently this says that it is a semi-simplicial object which satisfies the [[Segal conditions]] by [[homotopy pullbacks]]. This is just as for [[Segal spaces]], see there for details.

=--

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

A **complete semi-Segal space** is a semi-Segal space $X_\bullet$ such that the two morphisms

$$
  X_1^{inv} \hookrightarrow X_1 \stackrel{\partial_1, \partial_0}{\to} X_0
$$

are [[weak equivalences]].

=--

This is the completeness/[[univalence]] condition just as for [[complete Segal spaces]].

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

A semi-Segal space is **quasiunital** if (...)

=--

([Harpaz 12, p. 38](#Harpaz12))

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

A complete semi-Segal space, def. \ref{Completeness} is quasi-unital, def. \ref{QuasiUnitalness}.

=--

([Harpaz 12, Cor 4.1.11](#Harpaz12)).

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

A morphism of complete semi-Segal spaces $f_\bullet \colon X_\bullet \to Y_\bullet$ is quasi-unital if it preserves the weak equivalences, hence if

$$
  \array{
     X_1^{inv} &\hookrightarrow& X_1
     \\
     \downarrow^{\mathrlap{f_1|_{inv}}} && \downarrow^{\mathrlap{f_1}}
     \\
     Y_1^{inv} &\hookrightarrow & Y_1
  }
  \,.
$$

=--

## Related concepts

* [[semi-category]]

* [[semi-simplicial object]]

## References

The notion is mentioned in

* [[Jacob Lurie]], _[[On the Classification of Topological Field Theories]]_

More details are spelled out in

* {#Harpaz12} [[Yonatan Harpaz]], _Quasi-unital $\infty$-Categories_ ([arXiv:1210.0212](http://arxiv.org/abs/1210.0212))

See also

* {#Tanaka16} [[Hiro Lee Tanaka]], _Functors (between $\infty$-categories) that aren't strictly unital_ ([arXiv:1606.05669](http://arxiv.org/abs/1606.05669))

* Wolfgang Steimle, _Degeneracies in quasi-categories_, [arxiv:1702.08696](https://arxiv.org/abs/1702.08696)

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semi-simplicial object

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## Idea

A _semi-simplicial object_ is like a [[simplicial object]], but without degeneracy maps. In [[Set]] it is a [[semi-simplicial set]].

## Definition

### In terms of functors

For $\mathcal{C}$ a [[category]] or [[(∞,1)-category]], a semi-simplicial object $X$ in $\mathcal{C}$ is a [[functor]] or [[(∞,1)-functor]]
$$
  X \colon \Delta_+^{op}  \to \mathcal{C}
$$
from $\Delta_+$, the [[wide subcategory]] of the [[simplex category]] on the injective functions (the co-face maps).

### In homotopy type theory
 {#InHomotopyTypeTheory}

One may try to formulate semi-simplicial [[types]] in [[homotopy type theory]]. See *[[semi-simplicial types in homotopy type theory]]* and see the discussion at ([IAS](#IAS)).

## Properties

### Directedness

As opposed to the [[simplex category]] $\Delta$, the subcategory $\Delta_+$ is a [[direct category]].

## Examples
 {#Examples}

* in [[Sets]]: [[semisimplicial set]]:

* in [[TopologicalSpaces]]: [[semi-simplicial topological space]]

* in [[SimplicialSets]]: [[semi-Segal space]] (if extra conditions are met)

* in [[type theory]]: [[homotopytypetheory:semi-simplicial types]]

## Related concepts

* [[simplicial object]]

  * [[simplicial set]]

  * [[simplicial object in an (∞,1)-category]]

* **semi-simplicial object**

  * [[semisimplicial set]]

* [[semi-category]]

  * [[semi-Segal space]]

* [[Homotopy Type System]] ("HTS")

## References
 {#References}

> For more references see also at _[[semi-simplicial set]]_ and _[[semi-Segal space]]_.

### Semi-simplicial bundles

Discussion of semi-simplicial [[fiber bundles]] is in

* M. Barratt, V. Gugenheim and J. C. Moore, _On semisimplicial fibre bundles_, Amer. J. Math. 81 (1959), 639-657.

* S. Weingram, _The realization of a semisimplicial bundle map is a $k$-bundle map_ ([pdf](http://www.ams.org/journals/tran/1967-127-03/S0002-9947-1967-0231382-7/S0002-9947-1967-0231382-7.pdf))

### In homotopy type theory
 {#ReferencesInHomotopyTypeTheory}

Discussion of formulation of semsiplicial [[types]] in the context of [[homotopy type theory]] (for use as discussed at _[[category object in an (infinity,1)-category]]_) is in

* {#IAS} [[UF-IAS-2012]], _[Semi-simplicial types](https://web.archive.org/web/20180728221737/http://uf-ias-2012.wikispaces.com/Semi-simplicial+types)_

[[Coq]]-code for semi-simplicial types in [[homotopy type theory]] had been proposed in

* [[Vladimir Voevodsky]], _[[Voevodsky-semisimplicial.txt:file]]_

but its execution requires augmenting [[homotopy type theory]] with an auxilirary [[extensional type theory|extensional]] [[identity type]], discussed in

* [[Vladimir Voevodsky]], _A type system with two kinds of identity types_ (Feb. 2013) [[[Voevodsky-HTS.pdf:file]]]

See at _[[Homotopy Type System]]_ ("HTS") for more on this.

More along these lines is in

* [[Hugo Herbelin]], _A dependently-typed construction of semi-simplicial types_ (March 2013) [[[Herbelin-SemiSimplicial.pdf:file]]]

* [[Bruno Barras]], [[Thierry Coquand]], [[Simon Huber]], _A generalization of Takeuti-Gandy Interpretation_  [[[BarrasCoquandHuber-TakeutiGandyInterpretation.pdf:file]]]

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semi-simplicial set

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# Semi-simplicial sets
* table of contents
{:toc}

## Idea

A *semi-simplicial set* is like a [[simplicial set]], but without the degeneracy maps: it is a sequence $\{X_n\}_{n \in \mathbb{N}}$ of [[sets]] together with [[functions]] called _face maps_ between them which encode that an element in $X_{n+1}$ has $(n+1)$ "faces" (boundary segments) which are elements in $X_{n}$.

The semi-simplicial set version of a [[simplicial complex]] is also called a _[[Delta set]]_.

## Definition

Let $\Delta$ denote the [[simplex category]], which is a [[skeleton]] of the [[category]] of [[inhabited set|inhabited]] [[finite set|finite]] [[totally ordered sets]].   Let $\Delta_+$ denote the [[wide subcategory]] of $\Delta$ containing only the [[injective functions]] (it is sometimes written $\Delta_{inj}$, $\Delta_i$, $\widehat{\Delta}$, $\Delta'$ or $\overline{\Delta}$).   Thus, $\Delta_+$ is equivalent to the category of inhabited finite totally ordered sets and order-preserving injections.

Recall that a [[simplicial set]] is a [[presheaf]] $X\colon \Delta^{op}\to Set$.  Similarly, a **semi-simplicial set** is a presheaf $X\colon \Delta_+^{op} \to Set$.

More generally, for $\mathcal{C}$ any other category, a functor $\Delta_+^{op} \to \mathcal{C}$ is a _[[semi-simplicial object]]_ in $\mathcal{C}$.

## Properties

### Adjunction with simplicial sets
 {#AdjunctionWithSimplicialSets}

The [[forgetful functor]] from [[SimplicialSets]] to the [[category]] of semi-simplicial sets is given by precomposition with the [[opposite functor]] of the non-[[full subcategory|full]] [[wide subcategory]] inclusion

$$
  \Delta_{inj} \xrightarrow{\;j\;} \Delta
$$

into the [[simplex category]] and hence has both a [[left adjoint]] as well as a [[right adjoint]] given by left/right [[Kan extension]], respectively:

\begin{tikzcd}[column sep=small]
  \mathrm{SemiSimplSets}
  \ar[
    rr,
    shift left=18pt,
    "j_!"{above}
  ]
  \ar[
    rr,
    shift right=18pt,
    "j_\ast"{below}
  ]
  &&
  \mathrm{SimplSets}
  \ar[
    ll,
    "j^\ast"{description}
  ]
  \ar[ll, shift left=10pt, phantom, "\scalebox{.7}{$\bot$}" ]
  \ar[ll, shift right=10pt, phantom, "\scalebox{.7}{$\bot$}" ]
\end{tikzcd}

The [[adjunction unit]] of the left [[adjoint pair]]

$$
  X \xrightarrow{ \;\eta_X\; } j^\ast j_! X
$$

is a [[weak homotopy equivalence]] in the sense that its [[geometric realization]] is so ([Rourke & Sanderson 71, Rem. 5.8](#RourkeSanderson71)).

Notice that

1. for $X \in $ [[SimplicialSets]], the [[geometric realization]] of the underlying semi-simplicial set $j^\ast X$ is the *[[fat geometric realization]]* of $X$ (see [there](geometric+realization+of+simplicial+topological+spaces#FatGeometricRealization)):

   $$
     \left\Vert X \right\Vert
     \;\coloneqq\;
     \left\vert j^\ast X \right \vert
   $$

1. There is (by [this Prop.](https://ncatlab.org/nlab/source/geometric+realization+of+simplicial+topological+spaces#FatRealizationOfGoodSimplicialSpaces)) a natural [[weak homotopy equivalence]] from the fat to the ordinay geometric realization of a simplicial set (which is always "[[good simplicial topological space|good]]" when regarded as a simplicial space):

   $$
     \left\vert j^\ast X \right \vert
     \;\simeq_{whe}\;
     \left\vert X \right \vert
     \,.
   $$

It follows (see also [MO:a/75101](https://mathoverflow.net/a/75101/381)) that also the [[adjunction counit]]

$$
  j_! j^\ast X \xrightarrow{\; \epsilon_X \;} X
$$

is a [[simplicial weak equivalence]].

### Relation to semi-categories

The [[nerve]] of a [[semicategory]] is a semi-simplicial set (satisfying the [[Segal conditions]]) just as the nerve of a [[category]] is a [[simplicial set]].

### Model category structure

There is a _[[model structure on semi-simplicial sets]]_, [[transferred model structure|transferred]] along the [[right adjoint]] to the [[forgetful functor]] from the [[model structure on simplicial sets]].

## Related concepts

* [[simplicial object]]

  * [[simplicial set]]

  * [[simplicial object in an (∞,1)-category]]

* [[semi-simplicial object]]

* [[semicategory]]

* [[semi-Segal space]]

## Historical and terminological remarks
 {#History}

The original paper [Eilenberg & Zilber 50](#EilenbergZilber50) defined both (what we now call) semi-simplicial sets, under the name *semi-simplicial complexes*, and (what we now call) simplicial sets, under the name *complete semi-simplicial complexes*.  The motivation for the name "semi-simplicial" was that a semi-simplicial set is like a [[simplicial complex]], but lacks the property that a [[simplex]] is uniquely determined by its vertices.  Then they added the degeneracies and a corresponding adjective "complete."

Over time it became clear that "complete semi-simplicial complexes" were much more important and useful than the non-complete ones.  This seems to have led first to the omission of the adjective "complete," and then the omission of the prefix "semi" (and at some point the replacement of "complex" by "set"), resulting in the current name [[simplicial sets]].

The concept is essentially the same as that of $\Delta$-set, as used by [Rourke & Sanderson 71](#RourkeSanderson71). Their motivation was from geometric topology.

On the other hand, in other contexts the prefix "semi-" is used to denote absence of identities (such as a [[semigroup]] (which is, admittedly, missing more than identities relative to a group) or a [[semicategory]]),  thus if we start from the modern name "simplicial sets" it makes independent sense to refer to their degeneracy-less variant as "semi-simplicial sets."  This is coincidentally in line with the original terminology of Eilenberg and Zilber, but not of course with the intermediate usage of "semi-simplicial set" for what we now call a "simplicial set."

Note also the existence of an alternative terminology "presimplicial sets", or "pre-simplicial sets", which can be traced back at least to the textbook "Cellular Structures in Topology" by Fritsch and Piccinini in 1990. This terminology is commonly used e.g. in the context of simplicial models for concurrent programs or higher-dimensional [[automata]] (see e.g. "First introduction to simplicial sets" by [[Sina Hazratpour]]).

Similarly, the subcategory of injective functions of the simplex category was written $\Delta$ at some time of the history (e.g. in [Rourke & Sanderson 71](#RourkeSanderson71)) but this is now the standard notation for the [[simplex category]]. In the more recent history, different notations can be found but none seems to be widely adopted. $\Delta_+$ emphasizes that it is the subcategory of $\Delta$ that raise the degree when $\Delta$ is seen as a [[Reedy category]] but the $+$ may also ambiguously suggests that it adds something to $\Delta$. The notation $\Delta_{inj}$  emphasizes that it is the subcategory of injective morphisms of $\Delta$. Similarly for $\Delta_i$ though less explicitly. The notation $\widehat{\Delta}$ (e.g. in [Friedman](#Friedman12)) has the risk of introducing a confusion for readers used with the hat notation for presheaves. The notation $\Delta'$ and $\overline{\Delta}$ (e.g. in [[Sina Hazratpour]]) express that it is a variant of $\Delta$ but without giving precisions.

## Related concepts

* [[semi-simplicial object]]

* [[semi-simplicial topological space]]

## References

* {#EilenbergZilber50} [[Samuel Eilenberg]], [[Joseph Zilber]], *Semi-Simplicial Complexes and Singular Homology*, *Annals of Mathematics* vol. 51 no. 3 (1950), 499--513 ([jstor:1969364](https://www.jstor.org/stable/1969364))

* [[Daniel Kan]], *Is an ss complex a css complex?*, Advances in Mathematics
Volume 4, Issue 2, April 1970, Pages 170-171 (doi:10.1016/0001-8708(70)90021-6)

*  {#RourkeSanderson71} [[Colin Rourke]],  [[Brian Sanderson]], _Δ-Sets I: Homotopy Theory_. The Quarterly Journal of Mathematics 22: 321–338 (1971) ([doi:10.1093/qmath/22.3.321](https://doi.org/10.1093/qmath/22.3.321), [pdf](http://www.maths.ed.ac.uk/~aar/papers/deltars.pdf))

* S. Buoncristiano, [[Colin Rourke]], [[Brian Sanderson]], _A geometric approach to Homology Theory_, LMS Lect. Notes 18, (1976)

* [[Peter Hilton]], _On a generalization of nilpotency to semi-simplicial complexes_ ([pdf](http://plms.oxfordjournals.org/content/s3-10/1/604.full.pdf))

* [[Alex Heller]], _Homotopy resolutions of semi-simplicial complexes_, Transactions of the American Mathematical Society Vol. 80, No. 2 (Nov., 1955), pp. 299-344 ([JSTOR](http://www.jstor.org/stable/1992992))

* [[James McClure]], On semisimplicial sets satisfying the Kan condition ([arXiv:1210.5650](http://arxiv.org/abs/1210.5650)).

* {#Friedman12} [[Greg Friedman]], _An elementary illustrated introduction to simplicial sets_, Rocky Mountain J. Math. 42(2): 353-423 (2012) ([arXiv:0809.4221](http://arxiv.org/abs/0809.4221)).

* [[Andrew Ranicki]], _Algebraic L-Theory and Topological Manifolds_, Cambridge University Press 2002.

See also the references at _[[semi-simplicial object]]_ and:

* Wikipedia, _[Delta sets](http://en.wikipedia.org/wiki/Delta_set)_

* MO, _[Semi-simplicial versus simplicial sets (and simplicial categories)](http://mathoverflow.net/questions/75094/semi-simplicial-versus-simplicial-sets-and-simplicial-categories)_
    _[Degeneracies for semi-simplicial Kan complexes](http://mathoverflow.net/questions/57653/degeneracies-for-semi-simplicial-kan-complexes/)_

In [[homotopy type theory]]:

* [[Thorsten Altenkirch]], [[Paolo Capriotti]], [[Nicolai Kraus]], _Extending Homotopy Type Theory with Strict Equality_, CSL 2016, [arXiv](http://arxiv.org/abs/1604.03799)

On the [[model structure on semi-simplicial sets]]:

* [[Benno van den Berg]], _A note on semisimplicial sets_, 2013 ([[vandenBerg_SemisimplicialSets.pdf:file]])

as a [[weak model category]]:

* {#Henry18} [[Simon Henry]], Theorem 5.5.6 of: *Weak model categories in classical and constructive mathematics*, Theory and Applications of Categories, Vol. 35, 2020, No. 24, pp 875-958.  ([arXiv:1807.02650](https://arxiv.org/abs/1807.02650), [tac:35-24](http://www.tac.mta.ca/tac/volumes/35/24/35-24abs.html))

as a [[semi-model category]]:

* {#Rooduijn2018} Jan Rooduijn, *A right semimodel structure on semisimplicial sets*, Amsterdam 2018 ([pdf](https://eprints.illc.uva.nl/id/eprint/1663/1/MoL-2018-34.text.pdf), [mol:4787](https://msclogic.illc.uva.nl/theses/archive/publication/4787/A-right-semimodel-structure-on-semisimplicial-sets))

as a [[fibration category]] and [[cofibration category]]:

* {#Sattler18} [[Christian Sattler]], *Constructive homotopy theory of marked semisimplicial sets* ([arXiv:1809.11168](https://arxiv.org/abs/1809.11168))

[[!redirects semi-simplicial sets]]
[[!redirects semisimplicial set]]
[[!redirects semisimplicial sets]]

[[!redirects Delta set]]
[[!redirects Delta sets]]


semi-simplicial topological space

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{:toc}

## Idea

A *semi-simplicial topological space* (often just "semi-simplicial space") is a [[semi-simplicial object]] [[internalization|internal]] to [[TopologicalSpaces]].

This is like a [[simplicial topological space]], but without degeneracy maps.

## Related concepts

* [[semi-simplicial set]]

* [[geometric realization of simplicial topological spaces]]

## References

* [[Johannes Ebert]], [[Oscar Randal-Williams]], *Semi-simplicial spaces*, Algebr. Geom. Topol. 19 (2019) 2099-2150 ([arXiv:1705.03774](https://arxiv.org/abs/1705.03774))

[[!redirects semi-simplicial topological spaces]]

[[!redirects semisimplicial topological space]]
[[!redirects semisimplicial topological spaces]]

[[!redirects semi-simplicial space]]
[[!redirects semi-simplicial spaces]]

[[!redirects semisimplicial space]]
[[!redirects semisimplicial spaces]]


semi-simplicial types in homotopy type theory

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#Contents#
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{:toc}

## Idea

Note: This page is an update of a page initially posted on the wiki of the special year on Univalent Foundations at IAS in 2012-13.

One interesting open problem (considered by [[Vladimir Voevodsky]] and others): define _[[semi-simplicial object|semi-simplicial]]_ types in Homotopy Type Theory.

Classically, a _[[semi-simplicial object]]_ in some [[category]] is much like a [[simplicial object]], but without any [[degeneracy maps]]; i.e. it is a [[contravariant functor]] from the category $\Delta_i$, of finite nonempty ordinals and just [[injections]] between them as [[morphisms]].

Can we define these internally to [[type theory]], that is as a [[functor]] from $\Delta_i$ to a [[type universe]] $U$ in [[homotopy type theory]], in some way?

The initial idea, proposed at IAS, was to represent semi-simplicial types not as a contravariant functor but, iterating the [[categorical semantics of dependent types|correspondence between fibrations and families]], as a family of families, what could be called an "indexed" presentation of semi-simplicial types based on "iterated dependencies".

## Semi-simplicial types as families of families of $n$-semi-simplices

### Small semi-simplicial types

To illustrate the idea of semi-simplicial types as families of families of $n$-semi-simplices, let us consider the finite-dimensional parts of such semi-simplicial type:

Let $U$ be a [[type universe]]. Then,

* A $U$-small 0-semi-simplicial type is just a $U$-small type $X_0:U$.

* A $U$-small 1-semi-simplicial type is $U$-small 0-semi-simplicial type $X_0$ with a family of $U$-small types
$$x_0:X_0, x_1:X_0 \vdash X_1(x_0, x_1):U$$

* A $U$-small 2-semi-simplicial type is a $U$-small 1-semi-simplicial type $(X_0, X_1)$ with a family of $U$-small types
$$x:X_0, x_1:X_0, x_2:X_0, x_{0, 1}:X_1(x_0, x_1), x_{1, 2}:X_1(x_1, x_2), x_{0, 2}:X_1(x_0, x_2) \vdash X_2(x_0, x_1, x_2, x_{0, 1}, x_{1, 2}, x_{0, 2}):U$$
representing triangular configurations from $X_0$ and $X_1$.

* A $U$-small 3-semi-simplicial type is a $U$-small 2-semi-simplicial type $(X_0, X_1, X_2)$ with a family of $U$-small types
$$\begin{array}{l}
x:X_0, x_1:X_0, x_2:X_0, x_3:X_0, \\
x_{0, 1}:X_1(x_0, x_1), x_{1, 2}:X_1(x_1, x_2), x_{2, 3}:X_1(x_2, x_3), \\
x_{0, 2}:X_1(x_0, x_2), x_{1, 3}:X_1(x_1, x_3), x_{0, 3}:X_1(x_0, x_3), \\
x_{0, 1, 2}:X_2(x_0, x_1, x_2, x_{0, 1}, x_{1, 2}, x_{0, 2}), x_{0, 1, 3}:X_2(x_0, x_1, x_3, x_{0, 1}, x_{1, 3}, x_{0, 3}), \\
x_{0, 2, 3}:X_2(x_0, x_2, x_3, x_{0, 2}, x_{2, 3}, x_{0, 3}), x_{1, 2, 3}:X_2(x_1, x_2, x_3, x_{1, 2}, x_{2, 3}, x_{1, 3}), \\
\vdash X_2(x_0, x_1, x_2, x_3, x_{0, 1}, x_{1, 2}, x_{2, 3}, x_{0, 2}, x_{1, 3}, x_{0, 3}, x_{0, 1, 2}, x_{0, 1, 3}, x_{0, 2, 3}, x_{1, 2, 3}):U
\end{array}$$
representing tetrahedral configurations from $X_0$, $X_1$ and $X_2$.

And so on.

Each of these can be packed up in single type through [[product types]] and [[dependent function types]]. Thus, given a type universe $U$,

* Can we define a function "$\text{Semi-simplicial}:\mathbb{N} \to U$", such that for $n = 0,1,2,3$, the $U$-small type $\text{Semi-simplicial}(n)$ is (equivalent to) the objects explicitly defined here?

* Can we define a type of $U$-small semi-simplicial types (i.e. infinity-semi-simplicial types) with $n$-simplices for all $n$?

A first sketch of a definition was given in by [Voevodsky (2012)](#Voevodsky12), suggesting that defining such families requires to define restrictions, then to prove that restrictions of restrictions commute, but then also to prove higher-dimensional coherence conditions on restrictions.

[[Hugo Herbelin]] worked at the IAS on a formalization in [[Coq]] based on the presentation of $\Delta_i$ with face maps $d_i$. In [Herbelin (2014)](#Herbelin14) he describes the construction in [[Coq]] extended with the principle of [[uniqueness of identity proofs]] (UIP). UIP was used to cut down the need for higher-dimensional [[coherence]], so morally, it meant that types were actually [[h-sets]]. What the construction shows is how to define in all details such alternative presentation of semi-simplicial sets based on iterated dependencies, what can also be seen as an effective and generic construction of [[Reedy model structure|matching object]] representatives in a semi-simplicial set.

To cut the need for higher-dimensional coherences, several extensions of homotopy type theory were considered:

* [[Vladimir Voevodsky]] proposed *[[Homotopy Type System]]* (HTS), a homotopy type theory extended with non-[[fibrant types]] and with an extra equality which is extensional in the sense of [[extensional type theory]];

* [Part & Luo (2015)](#PartLuo15) proposed a "Logic-enriched Homotopy Type Theory", in particular, they formalized a construction of semi-simplicial types in this logic using the Plastic proof assistant;

* [Altenkirch, Capriotti & Kraus (2016)](#AltenkirchCapriottiKraus16) proposed a two-level type theory generalizing HTT, leading to 2LTT by [[Dannil Annenkov]], [[Paolo Capriotti]], [[Nicolai Kraus]] and [[Christian Sattler]], providing another construction of semi-simplicial types in an extended type theory;

* According to [Kraus (2018)](#Kraus18), semi-simplicial types can also be constructed in Angiuli-Harper-Wilson's computational higher-dimensional type theory.

## Other approaches

The "indexed" approach as families of families with iterated dependencies is only one possible approach! Other approaches to the problem are also possible, and may be better.

An alternative idea is something like "in the simplicial model, or more generally in other homotopy-theoretic model, they should be equivalent to coherent (possibly: [[Reedy model structure|Reedy-fibrant]]) semi-simplicial objects".

Why not imitate the classical definition, using internal functors from the internally-defined category $\Delta{}_i$? Giving a reasonable definition of $\Delta{}_i$ is not too hard: the objects [n] are very well-behaved. But defining functors out of it is problematic, because there are coherence issues. One would have to specify the functoriality laws using equations, i.e. inhabitants of equality types, which homotopically we treat as paths. But specified paths in homotopy theory have to be coherent to define a useful notion of functor; thus we need equations between our equations (associativity pentagons, etc), then higher equations between those, and so on forever. Thus we end up with the same problem we had before of specifying infinitely much data using a finite description in type theory.

If one restricts the types involved to [[h-sets]], then this problem go away; but then one has again only defined semi-simplicial [[h-sets]], which are (presumably) strictly less general.

Another approach is to extend type theory with streams typed by streams of types as in [a HoTTEST talk by Astra Kolomatskaia](#Kolomatskaia22).

## Why semi-simplicial, not simplicial?

With semi-simplicial sets, the “iterated dependency” approach gives us at least a candidate approach for tackling coherence issues. With simplicial sets, it’s hard to see how one might tackle or avoid them. (Comparing it to the semi-simplicial approach, one requires degeneracy maps and equations between them, which lets loose the spectre of coherence again.) Furthermore, reasoning about Kan simplicial sets seems to insist on classical logic. For example, the classical result stating the homotopy equivalence of fibers of a Kan fibration cannot be proved constructively ([pdf](https://ncatlab.org/ufias2012/files/countermodel.pdf)).

In homotopy-theoretic terms, this is because $\Delta{}_i$ is a [[direct category]], while the [[simplex category]] $\Delta$ is not.

## About the original page on the wiki of the Univalent Foundations year

The original page about semi-simplicial types on the wiki of the Univalent Foundations year contained also material related to semi-simplicial sets in general though not to semi-simplicial types properly speaking. It is kept here for historical reasons.

**Update 4/12**: [A note on semisimplicial sets](https://ncatlab.org/ufias2012/files/semisimplicialsets.pdf) by Benno van den Berg.

**Update 4/15**: [A note on a presheaf model for simplicial sets](https://ncatlab.org/ufias2012/files/countermodel.pdf) by Marc Bezem and Thierry Coquand.

**Update 6/24**: Accompanying files to Update 4/15:

* [CL.pl](https://ncatlab.org/ufias2012/files/CL.pl), the prover for coherent logic (requires [SWI-Prolog](http://www.swi-prolog.org/))
* [X.in](https://ncatlab.org/ufias2012/files/X.in), a formalization of the model in coherent logic
* [X.model](https://ncatlab.org/ufias2012/files/X.model), the edges, fill1 and fill2, day by day, generated by the prover from input X.in.
* [X.v](https://ncatlab.org/ufias2012/files/X.v), a Coq script verifying the proof that no homotopy equivalence between the fibers can exist in the model.

## See also

* [[open problems in homotopy type theory]]

## References

* {#Voevodsky12} [[Vladimir Voevodsky]], sketch of a definition (2012) [Coq [code](https://ncatlab.org/ufias2012/files/semisimplicial.v)]

* {#Herbelin14} [[Hugo Herbelin]], *A dependently-typed construction of semi-simplicial types*, Mathematical Structures in Computer Science, Vol 25 (special issue 05), 2015 [[pdf](http://pauillac.inria.fr/~herbelin/articles/mscs-Her14-semisimplicial.pdf)] [Coq [code](http://pauillac.inria.fr/~herbelin/articles/semisimplicial.v)]

* {#PartLuo15} [[Fedor Part]], [[Zhaohui Luo]]:  *Semi-simplicial Types in Logic-enriched Homotopy Type Theory* [[arXiv:1506.04998](http://arxiv.org/abs/1506.04998)] (Submitted on 16 Jun 2015) [Plastic [code](https://github.com/part-xx/hott-plastic/tree/master/lib/Univalence/SimplicialTypes)]

* {#AltenkirchCapriottiKraus16} [[Thorsten Altenkirch]], [[Paolo Capriotti]], [[Nicolai Kraus]]: *Extending Homotopy Type Theory with Strict Equality* [[arXiv:1604.03799](https://arxiv.org/abs/1604.03799)] (Submitted on Apr 2016) [Agda [code](https://github.com/nicolaikraus/HoTT-Agda/blob/master/nicolai/SemiSimp/SStypes.agda) simulating semi-simplicial types in HoTT with strict equality]

* [[Danil Annenkov]], [[Paolo Capriotti]], [[Nicolai Kraus]], [[Christian Sattler]]: *Two-Level Type Theory and Applications* [[arXiv:1705.03307](https://arxiv.org/abs/1705.03307)] (Submitted on May 2017)

* [[Nicolai Kraus]], [[Christian Sattler]]: *Space-valued diagrams, type-theoretically (extended abstract)* [[arXiv:1704.04543](https://arxiv.org/abs/1704.04543)] (Submitted on Apr 2017)

* {#Kraus18} [[Nicolai Kraus]]: *On the Role of Semisimplicial Types*, 2018 [[pdf](https://www.cs.nott.ac.uk/~psznk/docs/on_semisimplicial_types.pdf)]

* {#Kolomatskaia22} [[Astra Kolomatskaia]], *Semi-Simplicial Types*, [[Homotopy Type Theory Electronic Seminar Talks]], 15 December 2022 ([slides](https://www.uwo.ca/math/faculty/kapulkin/seminars/hottestfiles/Kolomaskaia-2022-12-15-HoTTEST.pdf), [video](https://www.youtube.com/watch?v=fQv2FpeFxew))