second lctvs diagram dot source

/*

[[!redirects lctvs lattice dot source]]

This is the graphviz source for an alternative version of the diagram at [[TVS relationships]].  The original intention was to see if it is possible to make this into a lattice.  However, that would have made the diagram too unwieldy.  Instead, the attempt is simply to expand and improve on the original diagram.

The current SVG can be seen at [[diagram of LCTVS properties]].

To recreate the SVG, the **whole** source of this page (or just the code below) should be run through `dot` (note that this part of the page is actually a comment in the dot source so doesn't get seen).  For example, assuming that the source has been saved as `lctvs.dot`, run:

    dot -Tsvg -o lctvs.svg lctvs.dot

~~~~
*/
digraph LCTVS {
splines=true;
overlap=orthoyx;
// style for properties relating to size
subgraph size {
style=none;
node [shape=box, style=filled, fillcolor="#ffffbb"];
FD [tooltip="Finite-Dimensional", href="http://ncatlab.org/nlab/show/Finite-Dimensional topological vector space"];
Hi [tooltip="Hilbert (technically, admits a Hilbertian structure)", href="http://ncatlab.org/nlab/show/Hilbert space"];
Nu [tooltip="Nuclear", href="http://ncatlab.org/nlab/show/Nuclear space"];
Ba [tooltip="Banach (technically, complete and normable)", href="http://ncatlab.org/nlab/show/Banach space"];
IP [tooltip="Topology from an inner-product", href="http://ncatlab.org/nlab/show/inner-product space"];
Mo [tooltip="Montel", href="http://ncatlab.org/nlab/show/Montel space"];
Sc [tooltip="Schwartz", href="http://ncatlab.org/nlab/show/Schwartz space"];
UB [tooltip="Ultrabornological", href="http://ncatlab.org/nlab/show/ultrabornological topological vector space"];
Fr [tooltip="Fréchet", href="http://ncatlab.org/nlab/show/Fréchet space"];
DF [tooltip="DF", href="http://ncatlab.org/nlab/show/DF topological vector space"];
No [tooltip="Normable space", href="http://ncatlab.org/nlab/show/normed vector space"];
Bo [tooltip="Bornological", href="http://ncatlab.org/nlab/show/bornological topological vector space"];
LF [tooltip="strict inductive sequence of Fréchet spaces", href="http://ncatlab.org/nlab/show/LF space"];
LB [tooltip="strict inductive sequence of Banach spaces", href="http://ncatlab.org/nlab/show/LB space"];
Me [tooltip="Metrisable", href="http://ncatlab.org/nlab/show/metrisable topological vector space"];
// no label
node [shape=circle, width=.2];
NuFr [label="", tooltip="Nuclear Fréchet space", href="http://ncatlab.org/nlab/show/nuclear topological vector space"];
}
// style for properties relating to completeness
subgraph complete {
node [shape=box, style=filled, fillcolor="#bbffff"];
LC [tooltip="Locally Complete", href="http://ncatlab.org/nlab/show/locally complete topological vector space"];
QC [tooltip="Quasi-Complete", href="http://ncatlab.org/nlab/show/quasi-complete topological vector space"];
Pt [tooltip="Ptak Space", href="http://ncatlab.org/nlab/show/Ptak space"];
BC [tooltip="Br Space", href="http://ncatlab.org/nlab/show/Ptak space"];
Sq [tooltip="Sequentially Complete", href="http://ncatlab.org/nlab/show/sequentially complete topological vector space"];
Cp [tooltip="Complete", href="http://ncatlab.org/nlab/show/complete topological vector space"];
}
// style for properties relating to duality
subgraph dual {
node [shape=box, style=filled, fillcolor="#ffbbff"];
Re [tooltip="Reflexive", href="http://ncatlab.org/nlab/show/reflexive topological vector space"];
SR [tooltip="Semi-Reflexive", href="http://ncatlab.org/nlab/show/semi-reflexive topological vector space"];
QB [tooltip="Quasi-Barrelled", href="http://ncatlab.org/nlab/show/quasi-barrelled topological vector space"];
Mk [tooltip="Mackey", href="http://ncatlab.org/nlab/show/Mackey topological vector space"];
Bl [tooltip="Barrelled", href="http://ncatlab.org/nlab/show/barrelled topological vector space"];
// no label
node [shape=circle, width=.2];
MkSR [label="", tooltip="Mackey and Semi-Reflexive", href="http://ncatlab.org/nlab/show/Mackey topological vector space"];
}
// Style for nodes with no label
node [shape=circle, width=.2, style=filled, fillcolor="#ffffbb"];
QCQB [label="", tooltip="Quasi-Complete and Quasi-Barrelled", href="http://ncatlab.org/nlab/show/barreled topological vector space"];
ReBa [label="", tooltip="Reflexive Banach space", href="http://ncatlab.org/nlab/show/reflexive topological vector space"];
ReFr [label="", tooltip="Reflexive Fréchet space", href="http://ncatlab.org/nlab/show/reflexive topological vector space"];

FD -> Hi;
Hi -> IP;
IP -> No;
No -> Me;
Hi -> ReBa;
ReBa -> Ba;
Ba -> LB;
Ba -> Fr;
Fr -> Me;
Fr -> LF;
LB -> LF;
Ba -> No;
FD -> NuFr;
Nu -> Sc;
NuFr -> Nu;
NuFr -> ReFr;
NuFr -> Mo;
No -> DF;
LB -> DF;

UB -> Bo;
Bo -> QB;
Bl -> QB;
QB -> Mk;
Re -> Bl;
Re -> MkSR;
QCQB -> Bl;
MkSR -> Mk;
MkSR -> SR;

Pt -> BC;
BC -> Cp;
Cp -> QC;
QC -> Sq;
Sq -> LC;

QCQB -> QC;
ReBa -> ReFr;
ReFr -> Re;
ReFr -> Fr;

SR -> QC;

Fr -> Pt;

Mo -> Re;

Me -> Bo;

LF -> UB;
UB -> QCQB;

/*
All the rest generates the key
*/

subgraph keys {
rank=same;
yFD  [label="",width=0,style=invis];
yDF [shape=box, style=solid, label="Key to symbols"];
yPt  [label="",width=0,style=invis];
}

subgraph cluster_key_col1 {
color=white;
node [label="",width=0,style=invis];
yHi;
yNu;
yBa;
yIP;
yMo;
ySc;
yUB;
yFr;
yZ1;
}

subgraph key1 {
node [shape=box, style=filled, fillcolor="#ffffbb"];
xFD [label="FD: Finite-Dimensional", href="http://ncatlab.org/nlab/show/Finite-Dimensional topological vector space"];
xHi [label="Hi: Hilbert (technically, admits a Hilbertian structure)", href="http://ncatlab.org/nlab/show/Hilbert space"];
xNu [label="Nu: Nuclear", href="http://ncatlab.org/nlab/show/Nuclear space"];
xBa [label="Ba: Banach (technically, complete and normable)", href="http://ncatlab.org/nlab/show/Banach space"];
xIP [label="IP: Topology from an inner-product", href="http://ncatlab.org/nlab/show/inner-product space"];
xMo [label="Mo: Montel", href="http://ncatlab.org/nlab/show/Montel space"];
xSc [label="Sc: Schwartz", href="http://ncatlab.org/nlab/show/Schwartz space"];
xUB [label="UB: Ultrabornological", href="http://ncatlab.org/nlab/show/ultrabornological topological vector space"];
xFr [label="Fr: Fréchet", href="http://ncatlab.org/nlab/show/Fréchet space"];
}

subgraph cluster_key_col2 {
color=white;
node [label="",width=0,style=invis];
yNo;
yBo;
yLF;
yLB;
yMe;
yLC;
yQC;
yZ2;
}

subgraph key2 {
node [shape=box, style=filled, fillcolor="#ffffbb"];
xDF [label="DF: DF", href="http://ncatlab.org/nlab/show/DF topological vector space"];
xNo [label="No: Normable space", href="http://ncatlab.org/nlab/show/normed vector space"];
xBo [label="Bo: Bornological", href="http://ncatlab.org/nlab/show/bornological topological vector space"];
xLF [label="LF: strict inductive sequence of Fréchet spaces", href="http://ncatlab.org/nlab/show/LF space"];
xLB [label="LB: strict inductive sequence of Banach spaces", href="http://ncatlab.org/nlab/show/LB space"];
xMe [label="Me: Metrisable", href="http://ncatlab.org/nlab/show/metrisable topological vector space"];
node [shape=box, style=filled, fillcolor="#bbffff"];
xLC [label="LC: Locally Complete", href="http://ncatlab.org/nlab/show/locally complete topological vector space"];
xQC [label="QC: Quasi-Complete", href="http://ncatlab.org/nlab/show/quasi-complete topological vector space"];
}

subgraph cluster_key_col3 {
color=white;
node [label="",width=0,style=invis];
yPt;
yBC;
ySq;
yCp;
yRe;
ySR;
yQB;
yMk;
yBl;
yZ3;
}

subgraph key2 {
node [shape=box, style=filled, fillcolor="#bbffff"];
xPt [label="Pt: Ptak Space", href="http://ncatlab.org/nlab/show/Ptak space"];
xBC [label="BC: Br Space", href="http://ncatlab.org/nlab/show/Ptak space"];
xSq [label="Sq: Sequentially Complete", href="http://ncatlab.org/nlab/show/sequentially complete topological vector space"];
xCp [label="Cp: Complete", href="http://ncatlab.org/nlab/show/complete topological vector space"];
node [shape=box, style=filled, fillcolor="#ffbbff"];
xRe [label="Re: Reflexive", href="http://ncatlab.org/nlab/show/reflexive topological vector space"];
xSR [label="SR: Semi-Reflexive", href="http://ncatlab.org/nlab/show/semi-reflexive topological vector space"];
xQB [label="QB: Quasi-Barrelled", href="http://ncatlab.org/nlab/show/quasi-barrelled topological vector space"];
xMk [label="Mk: Mackey", href="http://ncatlab.org/nlab/show/Mackey topological vector space"];
xBl [label="Bl: Barrelled", href="http://ncatlab.org/nlab/show/barrelled topological vector space"];
}

edge [style=invis];
yFD -> yHi;
yHi -> yNu;
yNu -> yBa;
yBa -> yIP;
yIP -> yMo;
yMo -> ySc;
ySc -> yUB;
yUB -> yFr;
yFr -> yZ1;

yDF -> yNo;
yNo -> yBo;
yBo -> yLF;
yLF -> yLB;
yLB -> yMe;
yMe -> yLC;
yLC -> yQC;
yQC -> yZ2;

yPt -> yBC;
yBC -> ySq;
ySq -> yCp;
yCp -> yRe;
yRe -> ySR;
ySR -> yQB;
yQB -> yMk;
yMk -> yBl;
yBl -> yZ3;

yFD -> xFD;
yHi -> xHi;
yNu -> xNu;
yBa -> xBa;
yIP -> xIP;
yMo -> xMo;
ySc -> xSc;
yUB -> xUB;
yFr -> xFr;
yDF -> xDF;
yNo -> xNo;
yBo -> xBo;
yLF -> xLF;
yLB -> xLB;
yMe -> xMe;
yLC -> xLC;
yQC -> xQC;
yPt -> xPt;
yBC -> xBC;
ySq -> xSq;
yCp -> xCp;
yRe -> xRe;
ySR -> xSR;
yQB -> xQB;
yMk -> xMk;
yBl -> xBl;

/*
FD -> Hi;
FD -> SC;
FD -> Nu;
FD -> Mo;
Hi -> Ba;
Hi -> IP;
Hi -> Re;
SC -> Me;
SC -> Se;
Mo -> Re;
Mo -> Pc;
Nu -> Sc;
Ba -> UB;
Ba -> Fr;
Ba -> DF;
Ba -> No;
IP -> No;
Re -> Bl;
Re -> SR;
UB -> Bo;
Fr -> Cn;
Fr -> Cp;
Fr -> Br;
Fr -> Me;
Cn -> Bo;
Cn -> LC;
Cp -> QC;
Br -> Bl;
Me -> Bo;
Me -> Pc;
SR -> QC;
Bo -> QB;
Bl -> QB;
QC -> Sq;
Pc -> Nm;
QB -> Mk;
Nm -> CP;
Sq -> LC;
LB -> LF;
LF -> Cp;
QCBo -> QC;
QCBo -> Bo;
QCBo -> Bl;
Hi -> AP;
Nu -> AP;
QCQB -> Bl;
QCQB -> QC;
Pt -> BC;
BC -> Cp;
Fr -> Pt;
Fr -> LF;
Ba -> LB;
*/
}

/*
~~~~
*/


second order algebraic theory

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

# Contents
* table of contents
{:toc}

## Idea

A _second-order algebraic theory_ is a generalization of the notion of [[Lawvere theory]] to describe algebraic structures with [[bound variable|variable binding]].

While single-sorted Lawvere theories are [[cartesian monoidal categories]] generated from a single object, single-sorted second-order algebraic theories are [[cartesian monoidal categories]] generated from a single [[exponentiable object]].

## Definition

A second-order algebraic theory is a category $T$ with cartesian products and an exponentiable object $o$ such that every object of $T$ is isomorphic to a finite cartesian product of objects of the form $o^n \Rightarrow o$.

A model of a theory $T$ is given by a functor $M : T \to \Set$ that preserves cartesian products (not necessarily exponentials). On objects, such a functor must give for each $o^n \Rightarrow o$ a set $M(n)$ and the functoriality determines a first-order Lawvere theory/cartesian multicategory structure on $M$. So models of second order algebraic theories are first order algebraic theories with additional structure.

## Multicategory

Just as Lawvere theories can be identified with [[cartesian multicategories]], there should be a similar correspondence between second-order theories and some notion of second-order cartesian multicategory.

## Examples

* The [[untyped lambda calculus]] can be described as a second-order algebraic theory, and [[simply typed lambda calculus]] as a multi-sorted second-order theory.

* The calculus of [[derivatives]] or more generally [[partial derivatives]].

## References

* [[Marcelo Fiore]] and Chung-Kil Hur, _Term Equational Systems and Logics_, MFPS XXIV, [doi](https://doi.org/10.1016/j.entcs.2008.10.011)

* [[Marcelo Fiore]] and Ola Mahmoud, _Second-order Algebraic Theories_, MFCS 2010, [arxiv](https://arxiv.org/abs/1308.5409)

A formalization in [[Agda]]:

* Marcelo Fiore and Dmitrij Szamozvancev: [webpage](https://www.cl.cam.ac.uk/~ds709/agda-soas/) with an [accompanying talk](https://www.youtube.com/watch?v=nP2J9SJ9DVM).

Discussion via [[monads]]:

* [[Nathanael Arkor]], *Monadic and Higher-Order Structure*, PhD thesis, Cambridge (2022) [[doi:10.17863/CAM.86347](https://doi.org/10.17863/CAM.86347), [pdf](https://www.repository.cam.ac.uk/bitstreams/666be4fb-957b-4c2e-83fc-8124621f0c43/download), [[Arkor-MonadicAndHigherStructure.pdf:file]]]

[[!redirects second-order algebraic theory]]
[[!redirects second order algebraic theories]]
[[!redirects second-order algebraic theories]]

[[!redirects higher-order algebraic theory]]
[[!redirects higher-order algebraic theories]]


second quantization

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Quantum systems
+--{: .hide}
[[!include quantum systems -- contents]]
=--
#### Quantum field theory
+--{: .hide}
[[!include functorial quantum field theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The process of [[quantization]] is supposed to assign to a [[classical field theory]] ([[prequantum field theory]]) a corresponding [[quantum field theory]]. One tends to imagine that in some cases the resulting QFT can itself be regarded as a classical field theory, roughly, and the process of quantization be applied again. One then speaks of _second quantization_ .

In its simplest form the _free_ second quantization (in the sense of [[free field theory]], introducing no [[interaction]] between the second quantized quanta) is identified with the _[[Fock space]]_ construction on a given [[space of quantum states]].

For interacting second quantization see _[[sigma-model -- exposition of second quantization of sigma-models]]_.

[[!include second quantization -- table]]

## Properties

### Syntax in linear type theory

In [[linear type theory]], which may be regarded as [[quantum logic]] (by the discussion there), the [[Fock space]] construction may naturally be identified with the [[exponential modality]]. See there for more.

## Related concepts

* [[Fock space]]

* [[quantum field theory]]

* [[quantum many-body physics]]

* [[condensed matter physics]]

## References

* Wikipedia, _[Second quantization](http://en.wikipedia.org/wiki/Second_quantization)_

[[!redirects second quantizations]]


second quantization -- table

| [[worldvolume]] field theory | of fundamental [[branes]] | and their  [[second quantization]] | which in [[perturbation theory]] is given by |
|--|--|--|--|
| [[worldline formalism]] | [[particle]] |  [[quantum field theory]] | [[scattering amplitudes]] |
| [[2d CFT]] [[correlators]] | [[string]] | [[string field theory]] | [[string scattering amplitudes]] |


second stable homotopy group of spheres

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Stable Homotopy theory
+--{: .hide}
[[!include stable homotopy theory - contents]]
=--
#### Cobordism theory
+--{: .hide}
[[!include cobordism theory -- contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The second [[stable homotopy group of spheres]] (the second [[stable stem]]) is the [[cyclic group of order 2]]:

\[
  \label{ComplexHopfFibrationGeneratingpi1}
  \array{
    \pi_1^s &\simeq& \mathbb{Z}/2
  }
\]

## Related concepts

* [[first stable homotopy group of spheres]]

* [[third stable homotopy group of spheres]]

## References

[[Lev Pontryagin|Pontryagin]] had announced the computation in:

* {#Pontrjagin36} [[Lev Pontrjagin]], _Sur les transformations des sphères en sphères_ ([[PontrjaginSurLesTransformationDesSpheres.pdf:file]]) in: _Comptes Rendus du Congrès International des Mathématiques -- Oslo 1936_ ([pdf](https://www.mathunion.org/fileadmin/ICM/Proceedings/ICM1936.2/ICM1936.2.ocr.pdf))

The explanation of the proof strategy via [[Pontryagin's theorem]] in [[cobordism theory]] appears in

* {#Pontrjagin38b} [[Lev Pontrjagin]], _[[Classification of continuous maps of a complex into a sphere]]_, _Communication II_, Doklady Akademii Nauk SSSR 19(5) (1938), 361-363

but there was a mistake in the proof left, fixed in [Pontryagin 50](#Pontryagin50) and, independently and by different means, in [Whitehead 50](#Whitehead50):

* {#Pontryagin50} [[Lev Pontryagin]], _[[Homotopy classification of mappings of an (n+2)-dimensional sphere on an n-dimensional one]]_, Doklady Akad. Nauk SSSR (N.S.) 19 (1950), 957–959 ([pdf](https://www.maths.ed.ac.uk/~v1ranick/papers/pont3.pdf))

* {#Whitehead50} [[George Whitehead]], _The $(n+2)$nd Homotopy Group of the $n$-Sphere_, Annals of Mathematics Second Series, Vol. 52, No. 2 (Sep., 1950), pp. 245-247  ([jstor:1969466](https://www.jstor.org/stable/1969466))

A more comprehensive account of the computation and the [[cobordism theory]] behind it ([[Pontryagin's theorem]]) was then given in:

* {#Pontrjagin55} [[Lev Pontrjagin]], Section 15 of: _[[Smooth manifolds and their applications in homotopy theory]]_, Trudy Mat. Inst. im Steklov, No 45, Izdat. Akademii Nauk. USSR, Moscow, 1955 (AMS Translation Series 2, Vol. 11, 1959) ([doi:10.1142/9789812772107_0001](https://www.worldscientific.com/doi/abs/10.1142/9789812772107_0001))

These historical references are also listed, with brief commentary, in the first part of:

* [[Douglas Ravenel]], _[Early work on the stable 2-stem](https://people.math.rochester.edu/faculty/doug/AKpapers.html#2-stem)_

For more historical commentary see [p. 6](https://arxiv.org/pdf/math/0211216.pdf#page=6) of [[Michael Hopkins|Hopkins]]&[[Isadore Singer|Singer]]'s _[[Quadratic Functions in Geometry, Topology, and M-Theory]]_ [[Michael Hopkins|Hopkins]]'s talk at [Atiyah's 80th Birthday conference](https://www.maths.ed.ac.uk/~v1ranick/atiyah80.htm), [slide 8](https://www.maths.ed.ac.uk/~v1ranick/hopkins.pdf#page=8), [9:45](https://empg.maths.ed.ac.uk/Videos/Atiyah80/Hopkins.mov).

Review:

* {#WangXu10} [[Guozhen Wang]], [[Zhouli Xu]], Section 2.5 of: _A survey of computations of homotopy groups of Spheres and Cobordisms_, 2010 ([[WangXuHomotopyGroupsOfSpheres2010.pdf:file]])

* {#Putnam} [[Andrew Putman]], Section 10 of: _Homotopy groups of spheres and low-dimensional topology_ ([pdf](http://www.math.rice.edu/~andyp/notes/HomotopySpheresLowDimTop.pdf), [[PutmanHomotopyGroupsOfSpheres.pdf:file]])

See also:

* Mehmet Kirdar, *On the First, the Second and the Third Stems of the Stable Homotopy Groups of Spheres* [[arXiv:2107.06103](https://arxiv.org/abs/2107.06103)]

[[!redirects 2nd stable homotopy group of spheres]]

[[!redirects second stable stem]]
[[!redirects 2nd stable stem]]


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

#Contents#
* table of contents
{:toc}

## Statement

+-- {: .num_lemma #RegularSpacesWherOpenCoveringIsRefinedbyCountableLocallyFiniteConnectionsofOpensAreParacompact}
###### Lemma
**([[Michael's theorem]], [Michael 53, theorem 1](#Michael53))**

Let $X$ be a [[topological space]] such that

1. $X$ is [[regular topological space|regular]];

1. every [[open cover]] of $X$ has a [[refinement]] by a union of a [[countable set]] of [[locally finite sets of subsets|locally finite]] sets of open subsets (not necessarily covering).

Then $X$ is [[paracompact topological space]].

=--

Note that while in Michael's paper he assumes that paracompact spaces are Hausdorff, the assumption is not necessary, and the proof goes through for regular non-Hausdorff spaces.  See Kelley, p. 156.

+-- {: .num_prop }
###### Proposition
**(second-countable regular spaces are paracompact)**

Let $X$ be a [[topological space]] which is

1. [[second-countable topological space|second-countable]];

1. [[regular topological space|regular]].

Then $X$ is [[paracompact topological space]].

=--

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

Let $\{U_i \subset X\}_{i \in I}$ be an open cover. By [[Michael's theorem]] (lemma \ref{RegularSpacesWherOpenCoveringIsRefinedbyCountableLocallyFiniteConnectionsofOpensAreParacompact}) it is sufficient that we find a [[refinement]] by a [[countable cover]].

But second countability implies precisely that every open cover has a countable subcover:

Every open cover has a refinement by a cover consisting of [[base for the topology|base]] elements, and if there is only a countable set of these, then the resulting refinement necessarily contains at most this countable set of distinct open subsets.

=--

## Related statements

* [[locally compact and sigma-compact spaces are paracompact]]

* [[locally compact and second-countable spaces are sigma-compact]]

* [[CW-complexes are paracompact Hausdorff spaces]]

## References

* Kelley, _General Topology_, 1955.

* {#Michael53} [[Ernest Michael]], _A note on paracompact spaces_, Proc. Amer. Math. Soc. 1953 ([jstor](http://www.jstor.org/stable/2032419), [pdf](http://www.renyi.hu/~descript/papers/Michael_parakompakt.pdf))

* [[Akhil Mathew]], _[A theorem of Michael on paracompactness](https://amathew.wordpress.com/2010/08/19/a-theorem-of-michael-on-paracompactness/)_


second-countable space

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

# Second-countable spaces
* table of contents
{: toc}

## Idea

A [[space]] (such as a [[topological space]]) is _second-countable_ if, in a certain sense, there is only a [[countable set|countable]] amount of information globally in its topology.  (Change 'globally' to 'locally' to get a [[first-countable space]].)

## Definitions

+-- {: .num_defn #SecondContablTopologicalSpace}
###### Definition
**(second-countable topological space)**

A [[topological space]] is __second-countable__ if it has a [[base for a topology|base for its topology]] consisting of a [[countable set]] of [[subsets]].

=--

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

A [[locale]] is __second-countable__ if there is a [[countable set]] $B$ of [[open subspaces]] (elements of the [[frame of opens]]) such that every open $G$ is a [[join]] of some [[subset]] of $B$.  That is, we have
$$ G = \bigvee \{ U\colon B \;|\; U \subseteq G \} .$$
=--

## Generalisations

The __weight__ of a space is the [[minimum]] of the [[cardinalities]] of the possible bases $B$.  We are implicitly using the [[axiom of choice]] here, to suppose that this set of cardinalities (which really is a [[small set]] because bounded above by the number of open subspaces, and [[inhabited set|inhabited]] by this number as well) has a minimum.  But without Choice, we can still consider this collection of cardinalities.

Then a second-countable space is simply one with a countable weight.

## Examples

+-- {: .num_example #SecondCountableEuclideanSpace}
###### Example
**([[Euclidean space]] is second-countable)**

Let $n \in \mathbb{N}$. Consider the [[Euclidean space]] $\mathbb{R}^n$ with its [[Euclidean norm|Euclidean]] [[metric topology]]. Then $\mathbb{R}^n$ is second countable.

A [[countable set]] of [[topological base|base open subsets]] is given by the [[open balls]] $B^\circ_x(\epsilon)$ of [[rational number|rational]] [[radius]] $\epsilon \in \mathbb{Q}_{\geq 0} \subset \mathbb{R}_{\geq 0}$ and centered at points with [[rational number|rational]] [[coordinates]]: $x \in \mathbb{Q}^n \subset \mathbb{R}^n$.

=--

+-- {: .num_example }
###### Example

A [[compact space|compact]] [[metric space]] is second-countable.

=--

+-- {: .num_example }
###### Example

A [[separable space|separable]] [[metric space]], e.g., a [[Polish space]], is second-countable.

=--

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

It is *not* true that separable [[first-countable spaces]] are second-countable; a counterexample is the [[real line]] equipped with the half-open or lower limit topology that has as basis the collection of half-open intervals $[a, b)$.

=--

+-- {: .num_example }
###### Example

A [[Hausdorff space|Hausdorff]] [[locally Euclidean space]] is second-countable precisely  it is [[paracompact space|paracompact]] and has a [[countable set]] of  [[connected components]]. In this case it is called a _[[topological manifold]]_.

See at _[[topological space]]_ [this prop.](#RegularityConditionsForTopologicalManifoldsComparison).

=--

+-- {: .num_example}
###### Example

A [[countable set|countable]] [[coproduct]] ([[disjoint union space]]) of second-countable spaces is second-countable.

Countable [[products]] ([[product topological spaces]]) of second-countable spaces are second-countable.

[[subspace|Subspaces]] of second-countable spaces are second-countable.

=--

+-- {: .num_exampl}
###### Example

If $X$ is second-countable and there is an [[open map|open]] [[surjection]] $f \colon X \to Y$, then $Y$ is second-countable.

=--

+-- {: .num_example }
###### Example

For second-countable [[separation axiom|T_3]] spaces $X, Y$, if $X$ is [[locally compact topological space|locally compact]], then the [[mapping space]] $Y^X$ with the [[compact-open topology]] is second-countable.

Cf. [[Urysohn metrization theorem]] and [[Polish space]]. I ([[Todd Trimble]]) am uncertain to what extent the $T_3$ assumption can be removed.

=--

## Properties

* [[second-countable regular spaces are paracompact]]

* [[locally compact and second-countable spaces are sigma-compact]]

## Related countability properties

[[!include topology - countability axioms]]

[[!redirects second-countable space]]
[[!redirects second-countable spaces]]
[[!redirects second countable space]]
[[!redirects second countable spaces]]

[[!redirects second-countable topological space]]
[[!redirects second-countable topological spaces]]
[[!redirects second countable topological space]]
[[!redirects second countable topological spaces]]

[[!redirects second-countable locale]]
[[!redirects second-countable locales]]
[[!redirects second countable locale]]
[[!redirects second countable locales]]


second-countable spaces are Lindelöf

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

## Statement

\begin{theorem}
Using [[countable choice]], then:
Every [[second-countable space|second-countable]] [[topological space]] $X$ is [[Lindelöf space|Lindelöf]], i.e. any open cover admits a countable subcover.
\end{theorem}

\begin{proof}
Let $\{U_n\}$ be a countable base of the topology.
Given any open cover $\{V_\lambda\}$ of $X$, we can form the index set $I\subset \mathbb{N}$ of those $U_n$ that are contained in some $V_\lambda$.
By assumption $\bigcup_{i\in I} U_{i} = \bigcup_\lambda V_\lambda = X$.
The axiom of [[countable choice]] provides now a section of $\bigsqcup_{i\in I} \{\lambda \mid U_i \subset V_\lambda\}\to I$.
\end{proof}

## Related properties

[[!include topology - global countability axioms]]


second-order arithmetic

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

# Contents
* table of contents
{: toc}

## Idea

Second-order arithmetic is a [[theory]] of [[arithmetic]] dealing with [[natural numbers]] and, what makes it "second-order", [[sets]] of natural numbers[^fine]. SOA, or $Z_2$ as it is often denoted, is [[proof theory|proof-theoretically]] weak in comparison with [[ZFC]], but still strong enough to derive "almost all of undergraduate mathematics" (Friedman).

[^fine]: The logic that governs the language and theory of $Z_2$ is ordinary (first-order) predicate logic. The "second-order" aspect is really in the _models_, where one interprets the symbol $\in$ as membership in a background set theory, i.e., terms of type $P N$ are interpreted as subsets of the set that interprets the type $N$, and an extensionality axiom is in force. A _full model_ is where $P N$ is interpreted as the full power set of $N$. When one sees absoluteness assertions such as "there is only one (full) model of SOA up to isomorphism," it should be clear that this is meant with regard to a given background set theory. Cf. the fact that while there is, up to isomorphism, at most one [[natural numbers object]] $\mathbb{N}$ in a given [[topos]], the set of global elements $\Gamma(\mathbb{N})$ might contain "non-standard elements" as viewed against the background $Set$.

## Definition

The [[theory|language]] of SOA consists of two sorts, here denoted $N$ and $P N$, together with

* A constant $0$ of type $N$,

* An unary function symbol $s \colon N \to N$,

* Binary function symbols $+, \cdot \colon N \times N \to N$,

* A binary relation symbol $\lt$ of type $N \times N$.

* A binary relation symbol $\in$ of type $N \times P N$.

The axioms of SOA may be divided into two parts: the first part comprises the "first-order axioms" that deal only with the sort $N$, and defines which is known as _Robinson arithmetic_. The second part comprises [[induction]] and [[comprehension schemes]] that involve the symbol $\in$.

### "First-order" axioms

Omitting the sort $P N$ and the symbol $\in$ from the language, the [[theory|axioms]] in this section give [[Robinson arithmetic]]. The logic throughout is [[classical logic|classical]] [[first-order logic|first-order]] (predicate) logic with [[equality predicate|equality]].

For all the formulas in this section, it is tacitly understood that there are universal quantifiers at the heads of formulas, binding all variables which appear freely.

1. $s(m) = 0 \rightarrow \bot$.

2. $s(m) = s(n) \rightarrow m = n$.

3. $(0 = n) \vee (\exists_m s(m) = n)$.

4. $m + 0 = m$.

5. $m + s(n) = s(m+n)$.

6. $m \cdot 0 = 0$.

7. $m \cdot s(n) = (m\cdot n)+m]$.

8. $m \lt 0 \rightarrow \bot$.

9. $m \lt s(n) \leftrightarrow ((m \lt n) \vee (m=n))$.

10. $(0 = n) \vee (0 \lt n)$.

11. $((s(m) \lt n) \vee (s(m) = n)) \leftrightarrow m \lt n$.

### "Second-order" axioms

Note: according to convention, lower-case letters refer to terms of type $N$, and upper-case letters to terms of type $P N$.

1. [[comprehension scheme|Comprehension scheme]]: for any [[formula]] $\varphi$ in the language of SOA with [[free variables]] $\vec{m} = m_1, \ldots, m_j$ and $\vec{X} = X_1, \ldots, X_k$,
$$\forall \vec{m} \forall \vec{X} \exists Z \forall n (n\in Z \leftrightarrow \varphi(n))$$
provided that the variable $Z$ does not appear in $\varphi$. (Using [[alpha-equivalence]] to rename variables, it\'s enough that $Z$ does not appear free in $\varphi$.)

2. Full [[induction]] scheme: for $\varphi$ any formula with a free variable $n$ and possible remaining free variables $\vec{m} = m_1, \ldots, m_j$ and $\vec{X} = X_1, \ldots, X_k$,

$$
  \forall \vec{m} \forall \vec{X} ((\varphi(0) \wedge \forall n (\varphi(n) \rightarrow \varphi(s(n))) \rightarrow \forall n \varphi(n))
$$

The instance in the full induction scheme where $\varphi$ is the formula $n \in X$ is called simply the _induction axiom_. The induction axiom together with the comprehension scheme implies the full induction scheme.

## Subsystems of SOA

The theory described above gives full second-order arithmetic. However, in [[reverse mathematics]], one often studies subsystems of weaker proof-theoretic strength than SOA, by limiting in some way the comprehension scheme (often also beefing up the single induction axiom with more instances of the induction scheme, to offset the weakening). The main examples are given in [Wikipedia](http://en.wikipedia.org/wiki/Reverse_mathematics); a standard reference is [Simpson](#Sim).

Many important subsystems for SOA have been the subject of an [[ordinal analysis]].

Another important subsystem is _intuitionistic_ second-order arithmetic, or second-order [[Heyting arithmetic]].  Here, we use [[intuitionistic logic]] rather than classical logic as the underlying first-order logic.  In a subsystem without the induction axiom, we must add axioms stating that equality and order are [[decidable predicate|decidable]]:

* $(m = n) \vee ((m = n) \rightarrow \bot)$;

* $(m \lt n) \vee ((m \lt n) \rightarrow \bot)$.

Full (classical) SOA may be recovered from intuitionistic SOA by adding a similar axiom that set membership is decidable; in a subsystem without the comprehension scheme, this needs to be strengthened to an axiom scheme that every proposition (with any free variables) is true or false (effectively making the logic classical again).

## Related concepts

* [[foundations]]

* [[elementary function arithmetic]]

* [[descriptive set theory]]

## References

* Wikipedia _[Second order arithmetic](http://en.wikipedia.org/wiki/Second-order_arithmetic)_

* Harvey Friedman's [Home Page](http://www.math.osu.edu/~friedman.8/)

* {#Sim} Stephen G. Simpson, _Subsystems of second order arithmetic_, Perspectives in Logic (2nd ed.), Cambridge University Press (2009) ([pdf](http://www.personal.psu.edu/t20/sosoa/chapter1.pdf))

[[!redirects SOA]]
[[!redirects second order arithmetic]]
[[!redirects second-order arithmetic]]


second-order logic

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Foundations
+-- {: .hide}
[[!include foundations - contents]]
=--
#### $(0,1)$-Category theory
+--{: .hide}
[[!include (0,1)-category theory - contents]]
=--
#### Type theory
+-- {: .hide}
[[!include type theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

_Second-order logic_(SOL) is an extension of [[first-order logic]] with [[quantifiers]] and [[variables]] that range over [[subsets]] of the [[universe]] of discourse, and hence is a [[higher-order logic]] of stage 2.

An important fragment is _Monadic Second-order logic_  (MSOL), where second-order quantification is restricted to second-order unary relations between subsets i.e. MSOL quantifies only over set predicates e.g. $\forall X.\varphi(X)$ but not $\forall X\forall Y .\varphi(X,Y)$.

Second-order logic could be characterised as a [[first order theory]] with [[dependent types]]. There is a [[type]] $V$ called the [[domain of discourse]], and for each term $x:V$, a dependent type $\mathcal{P}(x)$ whose terms $P(x):\mathcal{P}(x)$ are propositions depending on $x$.

##Remark
As SOL permits characterisation of mathematical structures up to isomorphism, it is sometimes promoted as a contender to [[set theory]] for the [[foundations of mathematics]] (cf. references). However, characterising structures up to isomorphism is insufficient for [[category theory]] and [[higher category theory]], as [[weak category|weak categories]] could only be characterised up to equivalence of categories, and weak [[∞-groupoid|∞-groupoids]] could only be characterised up to [[homotopy equivalence]].

## Related concepts

* [[propositional logic]] (0th order)

* [[predicate logic]] (1st order)

* **second-order logic**

* [[higher-order logic]]

* [[dependent type theory]]

* [[System F]]

## References

Textbook account in view of the [[Curry-Howard correspondence]]:

* Morten Heine Sørensen, Pawel Urzyczyn, *Lectures on the Curry-Howard isomorphism*, Studies in Logic **149**, Elsevier (2006) [[ISBN:9780444520777](https://www.elsevier.com/books/lectures-on-the-curry-howard-isomorphism/sorensen/978-0-444-52077-7), [pdf](https://disi.unitn.it/~bernardi/RSISE11/Papers/curry-howard.pdf)]

See also:

* Wikipedia, _[Second-order logic](http://en.wikipedia.org/wiki/Second-order_logic)_

* Stewart Shapiro, _Do Not Claim Too Much: Second-Order Logic and First-Order Logic_ , Phil. Math. **3** no. 7 (1999) pp.42-64 . ([pdf](http://www.accionfilosofica.com/misc/1214145300crs.pdf))

* [[Jouko Väänänen]], _Second-order Logic and Foundations of Mathematics_ , BSL **7** no. 4 (2001) pp.504-520. ([ps](http://www.math.ucla.edu/~asl/bsl/0704/0704-003.ps))

* [[Jouko Väänänen]], _Second-order logic, set theory, and the foundations of mathematics_ , Ms. ([pdf](http://www.math.helsinki.fi/logic/people/jouko.vaananen/second_order_or_set_theory.pdf))

[[!redirects second order logic]]
[[!redirects SOL]]


secondary characteristic class

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Cohomology
+--{: .hide}
[[!include cohomology - contents]]
=--
#### Differential cohomology
+--{: .hide}
[[!include differential cohomology - contents]]
=--
#### $\infty$-Chern-Weil theory
+--{: .hide}
[[!include infinity-Chern-Weil theory - contents]]
=--
=--
=--

# Contents
* table of contents
{:toc}

## Idea

In its most refined form, a _secondary characteristic class_ is a [[characteristic class]] in [[ordinary differential cohomology]]. The term "secondary" refers to the fact that such a differential cohomology class in degree $n$ not only encodes a degree-$n$ class in [[integral cohomology]], but in addition [[connection on an infinity-bundle|higher connection data]] in degree $(n+1)$: the data of a [[circle n-bundle with connection]].

The refined [[Chern-Weil homomorphism]] takes values in such "secondary characteristic classes".

But the precise meaning of the term _secondary characteristic class_ varies a little in the literature, as follows. Historically it was first understood in more restricted senses.

1. In the strict sense of the word, a _secondary characteristic class_ is a characteristic of a situation where an ordinary [[characteristic class]] vanishes ([PetersonStein1962](#PetersonStein)).

1. More specifically, a special case of this situation in [[differential geometry]] arises where the [[characteristic class]] is represented in [[de Rham cohomology]] by a [[curvature characteristic form]]. If that curvature form happens to vanish, the corresponding [[Chern-Simons form]] itself becomes closed, and now itself represents a cohomology class, in one degree lower. This is often called the corresponding _Chern-Simons secondary characteristic class_ . Sometimes the term "secondary geometric invariants" is used for [[Chern-Simons form]]s (see for instance the review ([FreedII](#FreedCSII))).

1. Using refined [[Chern-Weil theory]] the notions of [[curvature characteristic form]]s and their [[Chern-Simons form]]s are unified into the notion of cocycles in [[ordinary differential cohomology]]. The notion of [[Cheeger-Simons differential character]] was introduced to describe this unification, and it is has become tradition to call these differential characters themselves _secondary characteristic classes_ independently of whether the corresponding ordinary [[characteristic class]]/[[curvature characteristic form]] vanishes or not (for instance ([DupontKamber](#Kamber), [Karlsson](#Karlsson)). More descriptively, this case is maybe better referred to as a _[[differential characteristic class]]_ . See there for more details.

## Higher order invariants and boundary field theory
 {#HigherOrderInvariantsAndBoundaryFieldTheory}

The statement that secondary invariants are indeed secondary to primary invariants can be formalized in the language of [[prequantum field theory|prequantum]] [[boundary field theory]] by saying that (higher) [[topological Yang-Mills theory]] (which is controled by differential [[Chern classes]]) has as [[boundary field theory]] ([[higher dimensional Chern-Simons theory|higher]]) [[Chern-Simons theory]].

As explained at _[[boundary field theory]]_, this statement is reflected by the existence of a universal [[correspondence]] in the [[slice (∞,1)-topos]] $\mathbf{H}_{/\flat \mathbf{B}^{n+1}U(1)}$, where $\mathbf{H}$ is [[Smooth∞Grpd]], $\flat$ is the [[flat modality]] and $\mathbf{B}^{n+1}U(1)$ is the [[circle n-group|circle (n+2)-group]]:

$$
  \array{
    && X
    \\
    & \swarrow &\downarrow^\nabla& \searrow^{\mathrlap{F_{(\nabla)}}}
    \\
    \ast &\leftarrow& \mathbf{B}^n U(1)_{conn} &\stackrel{F_{(-)}}{\to}& \Omega^{n+1}_{cl}
    \\
    & \searrow &\swArrow& \swarrow_{\mathrlap{}}
    \\
    && \flat \mathbf{B}^{n+1}U(1)
  }
  \,,
$$

where the bottom right map is the canonical one in the context of [[differential cohesion]], the outer diagram is any specified one and the factorization through the

This exhibits a secondary invariant $\nabla$ (a [[cocycle]] in [[ordinary differential cohomology]]) of degree (n+1) as a boundary condition for the theory of closed characteristic forms.

In this picture, one obtains further "higher order invariants" by successively further [[transgression|transgrssing]] and forming higher order boundaries.

If we view here, as we may $\nabla$ as the [[local Lagrangian]] of an [[schreiber:infinity-Chern-Simons theory]], then the ternary invariants are _[[WZW terms]]_; the quaternary invariants are _[[Wilson loop]]_ terms.
([lpqft](#lpqft))

## Related concepts

* [[Chern-Simons invariant]]

* [[Cheeger-Simons class]], [[complex volume]]

## References

The notion in its general cohomological sense appears in

* {#PetersonStein} Franklin P. Peterson, Norman Stein, _Secondary characteristic classes_, Annals of mathematics, Vol 76, No. 3 (1962)

The notion of [[Chern-Simons forms]] originates in

* [[Shiing-shen Chern]], [[James Simons]], _[[Characteristic forms and geometric invariants]]_, The Annals of Mathematics, Second Series, Vol. 99, No. 1 (1974) ([jstor](http://www.jstor.org/stable/1971013)).

The special meaning in the context of [[Chern-Weil theory]] in [[differential geometry]] was established by the introduction of [[Cheeger-Simons differential character]] in

* {#CheegerSimons} [[Jeff Cheeger]],  [[James Simons]], _Differential characters and geometric invariants_, pages 50–80 in: _Geometry and Topology_, Lecture Notes in Mathematics, vol 1167. Springer (1985)  ([pdf](http://numr.wdfiles.com/local--files/differential-cohomology/cheeger-simons.pdf), [doi:10.1007/BFb0075212](https://doi.org/10.1007/BFb0075212))

Review:

* {#FreedCSII} [[Dan Freed]], _Classical Chern-Simons theory II_ ([pdf](http://www.ma.utexas.edu/users/dafr/cs2.pdf))

* {#Kamber} [[Johan Dupont]], Franz Kamber, _Gerbes, Simplicial forms and Invariants for Families of Foliated Bundles_ ([pdf](http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.152.1375&rep=rep1&type=pdf))

* {#Karlsson} Michelle Karlsson, _Characteristic classes and bounded cohomology_ ([pdf](http://www.math.ethz.ch/u/burger/karlsson.pdf))

Lecture notes on secondary cohomology classes in [[differential cohomology]] for flat connections is presented in

* {#Bunke12} [[Ulrich Bunke]], _Differential cohomology_ ([arXiv:1208.3961](https://arxiv.org/abs/1208.3961))

Discussion in complex [[algebraic geometry]] is in

* {#BlochEsnault97} [[Spencer Bloch]], [[Hélène Esnault]], _Algebraic Chern-Simons theory_, American Journal of Mathematics, Volume 119, Number 4, August 1997, pp. 903-952 ([pdf](http://www.mi.fu-berlin.de/users/esnault/preprints/helene/39-algebraic-chern-simons.pdf))

Discussion of secondary classes being [[torsion elements]]:

* [[Alexander Reznikov]], _All regulators of flat bundles are torsion_, Annals of Mathematics Second Series, Vol. 141, No. 2 (Mar., 1995), pp. 373-386 (14 pages) ([arXiv:dg-ga/9407006](https://arxiv.org/abs/dg-ga/9407006), [jstor:2118525](https://www.jstor.org/stable/2118525))

* [[Alexander Reznikov]], _Rationality of secondary classes_, J. Differential Geom., Volume 43, Number 3 (1996), 674-692. ([arXiv:dg-ga/9407007](https://arxiv.org/abs/dg-ga/9407007), [euclid:jdg/1214458328](https://projecteuclid.org/euclid.jdg/1214458328))

* Jaya N. Iyer, [[Carlos Simpson]], _Regulators of canonical extensions are torsion: the smooth divisor case_ ([arXiv:0707.0372](https://arxiv.org/abs/0707.0372))

* [[Hélène Esnault]],  _Algebraic Differential Characters of Flat Connections with Nilpotent Residues_,  In: Baas N., Friedlander E., Jahren B., Østvær P. (eds.) _Algebraic Topology_, Abel Symposia, vol 4. Springer 2009 ([doi:10.1007/978-3-642-01200-6_5](https://doi.org/10.1007/978-3-642-01200-6_5))

Discussion of secondary characteristic classes in [[string cobordism cohomology theory]] and in [[tmf]]:

* [[Ulrich Bunke]], [[Niko Naumann]], _Secondary Invariants for String Bordism and tmf_, Bulletin des Sciences Mathématiques, Volume 138, Issue 8, 2014, Pages 912-970
([arXiv:0912.4875](https://arxiv.org/abs/0912.4875),
[doi:10.1016/j.bulsci.2014.05.002](https://doi.org/10.1016/j.bulsci.2014.05.002))

Discussion of secondary and higher order invariant as higher order [[boundary field theories]] to higher [[topological Yang-Mills theory|topological Yang-Mills]] prequantum field theory is in

* {#lpqft} [[Urs Schreiber]] et al., _[[schreiber:Local prequantum field theory]]_

Specifically the secondary Chern-Simons and quaternary WZW invariants in this context are discussed in

* [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]], _[[schreiber:A higher stacky perspective on Chern-Simons theory]]_, in Damien Calaque et al. (eds.)
_Mathematical Aspects of Quantum Field Theories_, Springer 2014

[[!redirects secondary characteristic class]]
[[!redirects secondary characteristic classes]]

[[!redirects secondary characteristic form]]
[[!redirects secondary characteristic forms]]

[[!redirects secondary invariant]]
[[!redirects secondary invariants]]


secondary differential calculus

[[!redirects secondary calculus]]

## Related concepts

* [[differential calculus]]

* [[variational calculus]]

* [[variational bicomplex]]

* [[D-geometry]]

## References

* Luca Vitagliano, _Secondary calculus_, 2007 ([pdf](http://www.impan.pl/~jagrab/vitagliano.pdf))

* Wikipedia, _[Secondary calculus and cohomological physics](http://en.wikipedia.org/wiki/Secondary_calculus_and_cohomological_physics)_


section

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Category theory
+-- {: .hide}
[[!include category theory - contents]]
=--
#### Mapping space
+--{: .hide}
[[!include mapping space - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition

### Explicitly

A **section** of a [[morphism]] $f : A \to B$ in some [[category]] is a **right-[[inverse]]**: a [[morphism]] $\sigma : B \to A$ such that
$$
  f \circ \sigma : B \stackrel{\sigma}{\to} A \stackrel{f}{\to} B
$$
equals the [[identity morphism]] on $B$.

Typically $A \stackrel{f}{\to} B$ is thought of as a [[bundle]] and then one speaks of sections of bundles. For topological bundles one considers _continuous sections_, for smooth bundles _smooth sections_, etc.

### In terms of dependent product
 {#InTermsOfDependentProduct}

In a [[locally cartesian closed category]] $\mathcal{C}$, regard the morphism $f\colon A \to B$ as an object $[f] \in \mathcal{C}_{/B}$ in the [[slice category]] over $B$. Then there is the [[dependent product]]

$$
  \underset{B}{\prod} [f] \in \mathcal{C}
  \,.
$$

This is the [[space of sections]] of $f$. A single section $\sigma$ is a [[global element]] in here

$$
  \sigma \colon \ast \to \underset{B}{\prod} [f]
  \,.
$$

See at _[dependent product -- In terms of spaces of sections](dependent%20product#RelationToSpacesOfSections)_ for more on this.

## Split idempotents

In the case that $f$ has a section $\sigma$, $f$ may also be called a [[retraction]] or **cosection** of $\sigma$, $A$ may be called a [[retract]] of $B$, and the entire situation is said to split the [[idempotent]]
$$
  A \stackrel{f}{\to} B \stackrel{\sigma}{\to} A
  \,.
$$

A **[[split epimorphism]]** is a morphism that *has* a section; a **[[split monomorphism]]** is a morphism that *is* a section.  A [[split coequalizer]] is a particular kind of split epimorphism.

## Sections of bundles and sheaves

If one thinks of $f : A \to B$ as a [[bundle]] then its sections are sometimes called [[global sections]]. This leads to a notion of global sections of [[sheaves]] and further of objects in a general [[topos]]. See

* [[global section]]

for more on this.

## Related concepts

* [[local section]], [[global section]]

* [[flat section]]

* [[space of sections]]

* [[ex-space]]

* [[field history]]

[[!redirects sections]]
[[!redirects cross section]]
[[!redirects cross-section]]
[[!redirects right inverse]]
[[!redirects right-inverse]]
[[!redirects cross sections]]
[[!redirects cross-sections]]
[[!redirects right inverses]]
[[!redirects right-inverses]]

[[!redirects smooth section]]
[[!redirects smooth sections]]

[[!redirects continuous section]]
[[!redirects continuous sections]]


section conjecture

#Contents#
* table of contents
{:toc}

## Idea

Let $k$ be a [[field]] of [[characteristic zero]] with [[algebraic closure]] $\overline{k}$ and [[absolute Galois group]]
$\Gamma_k := Gal (\overline{k}/k)$. Let $X$ be a geometrically connected [[variety]] over $k$. Fix a [[geometric point]], $\overline{x}\in X (\overline{k})$ and let $\pi_1 (X ) \coloneqq \pi_1 (X,  \overline{x})$ be the  [[Galois theory|étale fundamental group]] of $X$. Set $\overline{X} = X \times_k  \overline{k}$
and denote by $\pi_1 (\overline{X}) := \pi_1 (\overline{X} ,  \overline{x})$ the  étale fundamental group of $\overline{X}$.

Grothendieck's
fundamental [[short exact sequence]] of [[profinite groups]] ([[SGA1]]), is:

$$
  1  \to\pi_1 (\overline{X})\to\pi_1 (X) \to \Gamma_k \to 1
  \,.
$$

By [[functor|functoriality]] of $\pi_1$, the existence of a $k$-point on $X$ implies that the above exact sequence has a [[section]] (is a [[split exact sequence]]).

This suggests that the converse statement is true whenever $X$ is a proper hyperbolic [[algebraic curve]] over a [[number field]]. This [[conjecture]] is the _section conjecture_ ([Grothendieck 97](#Grothendieck97)).

> (Needs more from Galois Theory and Diophantine geometry below)

##Blog discussions

* [n-Category café](http://golem.ph.utexas.edu/category/2008/03/kim_on_fundamental_groups_in_n.html)

* [Motivic stuff: Cohomology, homotopy theory, and arithmetic geometry](http://homotopical.wordpress.com/tag/section-conjecture/) (Andreas Holmstrom on October 4, 2009)

##Related entries

* [[anabelian geometry]]

##References

* {#Grothendieck97} [[Alexander Grothendieck]],  letter to [[Gerd Faltings]], London Math. Soc. Lecture Note Ser., 242, Geometric Galois actions, 1, 49–58, Cambridge Univ. Press, Cambridge, 1997.

* [[Mohamed Saidi]], _Around the Grothendieck Anabelian Section Conjecture_, [ArXiv:1010.1314v2](http://arxiv.org/pdf/1010.1314v2.pdf).

* [[Minhyong Kim]], _Galois Theory and Diophantine geometry, 2009 ([pdf](http://www.ucl.ac.uk/~ucahmki/cambridgews.pdf))

* [[Jakob Stix]], Evidence for the section conjecture in the theory of arithmetic fundamental groups Habilitationsschrift, School of Mathematics and Computer Science at the Ruprecht-Karls-Universität Heidelberg, January 2011, x+190 pages, to appear as
Rational Points and Arithmetic of Fundamental Groups, Evidence for the Section Conjecture.
Springer Lecture Notes in Mathematics 2054, xx+pp.249, Springer, 2013.

[[!redirects Grothendieck's section conjecture]]
[[!redirects anabelian section conjecture]]


sectional curvature

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###Context###
#### Riemannian geometry
+--{: .hide}
[[!include Riemannian geometry - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A measure of [[curvature]] in [[Riemannian geometry]].

[[Riemannian manifolds]]

* of [[constant function|constant]] [[positive number]] sectional curvature are the topic of [[elliptic geometry]];

* of [[constant function|constant]] [[zero]] sectional curvature are the topic of [[Euclidean geometry]];

* of [[constant function|constant]] [[negative number]] sectional curvature are the topic of [[hyperbolic geometry]];

## Related concepts

[[!include curvature in Riemannian geometry -- table]]

* [[Clifford-Klein space form]]

  * [[spherical space form]]

## References

See also

* Wikipedia, _[Sectional curvature](https://en.wikipedia.org/wiki/Sectional_curvature)_

[[!redirects sectional curvatures]]


sedenion

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#### Algebra
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[[!include higher algebra - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The __sedenions__ are the [[non-associative algebra]] $\mathbb{S}$ over the [[real numbers]] obtained by applying the [[Cayley–Dickson construction]] to the [[octonions]].

## Properties

Although every sedenion has a multiplicative [[inverse element|inverse]], $\mathbb{S}$ is *not* a [[division algebra]], since it has [[zero divisors]].

## References

Wikipedia, _[Sedenion](https://en.wikipedia.org/wiki/Sedenion)_

[[!redirects sedenion]]
[[!redirects sedenions]]


Seely isomorphism

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### Context
#### Foundations
+-- {: .hide}
[[!include foundations - contents]]
=--
#### $(0,1)$-Category theory
+-- {: .hide}
[[!include (0,1)-category theory - contents]]
=--
#### Monoidal categories
+-- {: .hide}
[[!include monoidal categories - contents]]
=--
=--
=--

# Contents
* table of contents
{:toc}

## Idea

In [[linear logic]] and its categorical models, the [[exponential modality]] verifies $!(A \& B) \cong !A \otimes !B$. This is called the Seely isomorphism, named after [[Robert Seely]].

## In sequent calculus

... to come ...

## In category theory

... to come ...

## Related concepts

- [[graded Seely isomorphism]]


Seerp Roald Koudenburg

* [institute page](https://www.ncc.metu.edu.tr/seerp-roald-koudenburg)

* [Mathematics Genealogy page](https://www.genealogy.math.ndsu.nodak.edu/id.php?id=174115)

## Selected writings

On [[formal category theory]] via [[double categories]]:

* {#Koudenburg15} [[Seerp Roald Koudenburg]], *A double-dimensional approach to formal category theory* ([arXiv:1511.04070](http://arxiv.org/abs/1511.04070)) 2015.

On [[augmented virtual double categories]]:

* {#Koudenburg19} [[Seerp Roald Koudenburg]], *Augmented virtual double categories*, Theory and Applications of Categories, Vol. 35, 2020, No. 10, pp 261-325 ([arXiv:1910.11189](https://arxiv.org/abs/1910.11189), [tac:35-10](http://www.tac.mta.ca/tac/volumes/35/10/35-10abs.html))

  > (streamlined and expanded version of Sec. 1, 2 and 3 of [Koudenburg 15](#Koudenburg15) )

category: people

[[!redirects Seerp Koudenburg]]
[[!redirects Seerp R. Koudenburg]]


seesaw mechanism

## Related concepts

* [[GUT]]

* [[neutrino]]

## References

* [[Fabien Besnard]], _A remark on the mathematics of the seesaw mechanism_ ([arXiv:1611.08591](https://arxiv.org/abs/1611.08591))

* Wikipedia, _[Seesaw mechanism](https://en.wikipedia.org/wiki/Seesaw_mechanism)_.

* Robert D. Klauber, _[The Seesaw Mechanism](http://www.quantumfieldtheory.info/TheSeesawMechanism.htm)_

[[!redirects seesaw mechanisms]]

[[!redirects Seesaw Mechanism]]
[[!redirects Seesaw Mechanisms]]

[[!redirects Seesaw mechanism]]
[[!redirects Seesaw mechanisms]]


Sefi Ladkani

Sefi Ladkani is a mathematician at the University of Haifa with areas of interest including [[representation theory]] and [[homological algebra]]. PhD defended in 2009 at the University of Yerusalem, under supervision of [[David Kazhdan]].

* [page](https://cris.haifa.ac.il/en/persons/sefi-ladkani) at Haifa
* [scholar](https://scholar.google.com/citations?user=xUO5mgQAAAAJ).google
* [orcid](https://orcid.org/0000-0002-9135-7827)
* [mathgenealogy](https://www.mathgenealogy.org/id.php?id=295343).org
* older page at [uni-bonn](https://www.math.uni-bonn.de/people/sefil)

## Selected papers

On [[Coxeter transformation]]s and [[Euler bilinear form]]s.

* _On the periodicity of Coxeter transformations and the non-negativity of their Euler forms_, [pdf](https://www.math.uni-bonn.de/people/sefil/Papers/coxhered.pdf)
* _Categorification of a linear algebra identity and factorization of Serre functors_,  Mathematische Zeitschrift __285__ (2017) 879--896 [doi](https://doi.org/10.1007/s00209-016-1731-9)

category: people


Segal category

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#### Enriched category theory
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[[!include enriched category theory contents]]
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#Contents#
* table of contents
{:toc}

## Idea

The notion of _Segal category_ is one of the models for that of _[[(∞,1)-category]]_, given by regarding an $(\infty,1)$-category as an [[∞Grpd]]-[[enriched (∞,1)-category]].

So the notion can be understood as modelling the notion of an [[sSet]]-[[enriched category|enrichment]] _up to [[coherence|coherent]] [[homotopy]]_, i.e. a _weak_ enrichment. As such it is closely related to the notion of [[complete Segal space]], which models the notion of an [[internal category in an (∞,1)-category|internal category]] in [[sSet]].

Indeed, Segal categories may be considered with enrichment not just over [[sSet]], but over other suitable [[model categories]]. In particular, an iterated enrichment over itself gives rise to the notion of _[[Segal n-category]]_ which is a model for _[[(∞,n)-categories]]_.

Since the major difference between ([[small category|small]])
$\mathcal{V}$-[[enriched categories]] and $\mathcal{V}$-[[internal categories]] is that in the first case the [[objects]] (as opposed to all the [[hom objects]]) form an ordinary [[set]], while in the second these form an object of $\mathcal{V}$, too, accordingly the definition of _Segal category_ is like that of _([[complete Segal space|complete]]) [[Segal space]]_, only that the simplicial set of objects is required to be an ordinary set (a [[discrete object|discrete]] simplicial set).

## Definition

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

A **Segal category** is

* a [[simplicial object|simplicial]] [[simplicial set]] ($\simeq$ [[bisimplicial set]]) $X \in [\Delta^{op}, sSet]$,

  where we call $X_0$ the _simplicial set of [[objects]]_, $X_1$ the _simplicial set of [[morphisms]]_;  and $X_k$ for $k \geq 2$ the _simplicial set of sequences of composable morphisms of length $k$_;

* such that $X_0$ is a [[discrete object|discrete]] (= constant) simplicial set;

* and such that the [[Segal maps]]

  $$
    X^{Sp[k] \hookrightarrow \Delta[k]}
     :
     X_k
       \stackrel{\simeq}{\to}
     X_1 \times_{X_0} \cdots \times_{X_0} X_1 \;\;(k factors)
  $$

(With the defintion of X_k, shouldn't X_k be equal to X_1 \times_{X_0}... X_1?)

  induced by the [[spine]] inclusions $Sp[k] \hookrightarrow \Delta[k]$ are [[model structure on simplicial sets|weak equivalences of simplicial sets]] for $k \geq 2$.

=--

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

There is _no_ condition that a Segal category be [[fibrant object|fibrant]] with respect to the [[Reedy model structure]] on bisimplicial sets.

=--

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

For $X$ a Segal category, the [[fiber product]] simplicial set $X_1 \times_{X_0} X_1$ is manifestly the space of pairs of composable 1-[[morphism]]s in $X$, and the [[weak equivalence]]

$$
  (d_0,d_2)
    :
  X_2 \stackrel{\simeq}{\to} X_1 \times_{X_0} X_1
$$

given by the above definition together with the remaining face map $d_1 : X_2 \to X_1$ constitutes an [[∞-anafunctor]]

$$
  \circ : X_1 \times_{X_0} X_1 ⇸ X_1
$$

given by the [[span]]

$$
  \array{
    X_2 &\stackrel{d_1}{\to}& X_1
    \\
    {}^{\mathllap{\simeq}}\downarrow
    \\
    X_1 \times_{X_0} X_1
  }
  \,.
$$

This encodes the [[composition]] operation in the Segal category $X$.

Accordingly, the analogous spans out of $X_k$ for $k \geq 3$ encode the [[associativity]] of this composition as well as all its [[coherence|coherences]].

=--

## Properties

### Model category structure

The category of [[bisimplicial sets]] carries a [[model category]] structure whose [[fibrant objects]] are the [[Reedy model structure|Reedy fibrant]] Segal categories. This _[[model structure for Segal categories]]_ is a [[presentable (∞,1)-category|presentation]] of the [[(∞,1)-category of (∞,1)-categories]].

### Operadic version

The [[operad|operadic]] generalization of Segal category is that of _[[Segal operad]]_. Segal categories are precisely those Segal operads whose only [[inhabited set|inhabited]] operations-spaces are those of unary operations.

## Examples

### Inclusion of ordinary categories

Let $C$ be an ordinary [[small category]] and write $N(C) \in sSet$ for its [[nerve]]. Regard this as a bisimplicial set under the inclusion $sSet \simeq [\Delta^{op}, Set] \hookrightarrow [\Delta^{op}, sSet]$.

Then $N(C)$ is a Segal category. Each simplicial set $N(C)_k$ is discrete, for all $k \in \mathbb{N}$, and all the morphisms

$$
  N(C)_k \to Mor(C) \times_{Obj(C)} \cdots \times_{Obj(C)} Mor(C)
$$

are in fact [[isomorphisms]] / [[bijections]] of sets. This property of the [[nerve]] of an ordinary category goes by the name _Segal condition_ and is what gave Segal categories its name.

One may also form the $n$-fold [[comma object]]-fiber product of a choice of base points $\pi_0(C) \to C$ with itself. This yields a Segal category incarnation of $C$ where in degree 1 we have the [[groupoid]] [[core]] of the [[arrow category]] of $C$. For more on this see at _[Segal space -- Examples - From a category](Segal+space#ConstructionFromACategory)_.

## Related concepts

* [[complete Segal space]]

* [[Gamma space]]

* [[Segal operad]]

* [[table - models for (infinity,1)-operads]]

## References

The idea of Segal categories goes back (implicitly) to

* [[Graeme Segal]], _Categories and cohomology theories_, Topology **13** 3 (1974) 293-312 [doi:10.1016/0040-9383(74)90022-6]

and is explicit in

* {#DwyerKanSmith89} [[William Dwyer]], [[Daniel Kan]], [[Jeff Smith]], *Homotopy-commutative diagrams and their realizations*, Journal of Pure and Applied Algebra **57** 1 (1989) 5-24 [doi:10.1016/0022-4049(89)90023-6]

as kind of bisimplicial sets "which are 'special' in the sense that they satisfy a slight variation on a condition of Segal" (But the terminology "*Segal category*" is not used in [Dwyer, Kan & Smith 1989](#DwyerKanSmith89).)

Discussion in the broader context of *[[Segal n-categories|Segal $n$-categories]]*:

* [[André Hirschowitz]], [[Carlos Simpson]], *Les $n$-catégories de Segal*, §2 in: _Descente pour les $n$-champs (Descent for $n$-stacks)_ [[arXiv:9807049](http://arxiv.org/abs/math/9807049)]

Discussion with emphasis on the comparison of the various [[model category]] presentations of [[(infinity,1)-categories|$(\infty,1)$-categories]]:

* [[Julie Bergner]], *Segal Categories*, §5 in: _A survey of $(\infty,1)$-categories_, In: [[John Baez]], [[Peter May]] (eds.),  _[[Towards Higher Categories]]_, The IMA Volumes in Mathematics and its Applications **152**, Springer (2007) [[arXiv:math/0610239](http://arxiv.org/abs/math/0610239), [doi:10.1007/978-1-4419-1524-5_2](https://doi.org/10.1007/978-1-4419-1524-5_2)]

* {#Joyal08} [[André Joyal]], pp 164-169 in: *[[The Theory of Quasi-Categories and its Applications]]*, lectures at *[Advanced Course on Simplicial Methods in Higher Categories](https://lists.lehigh.edu/pipermail/algtop-l/2007q4/000017.html)*, CRM (2008) [[pdf](http://mat.uab.cat/~kock/crm/hocat/advanced-course/Quadern45-2.pdf), [[JoyalTheoryOfQuasiCategories.pdf:file]]]

Discussion in the context of [[enriched (∞,1)-categories]]:

* [[Carlos Simpson]], _[[Homotopy Theory of Higher Categories]]_ ([arXiv:1001.4071](http://arxiv.org/abs/1001.4071))

* {#Lurie} [[Jacob Lurie]], section 2 of: _(Infinity,2)-Categories and the Goodwillie Calculus I_ ([arXiv:0905.0462](http://arxiv.org/abs/0905.0462))

[[!redirects Segal categories]]

[[!redirects Segal groupoid]]
[[!redirects Segal groupoids]]


Segal completion -- table

| ([[equivariant cohomology|equivariant]]) [[generalized (Eilenberg-Steenrod) cohomology|cohomology]] | [[Brown representability theorem|representing]]
 [[spectrum]] | [[equivariant cohomology]]
 of the [[point]] $\ast$ |  [[generalized (Eilenberg-Steenrod) cohomology|cohomology]]
 of [[classifying space]] $B G$ |
|---|----|---|---|
| ([[equivariant ordinary cohomology|equivariant]])
 [[ordinary cohomology]] | [[Eilenberg-MacLane spectrum|HZ]] |  [[Borel equivariant cohomology|Borel equivariance]]
 $H^\bullet_G(\ast) \simeq H^\bullet(B G, \mathbb{Z})$ |   |
| ([[equivariant K-theory|equivariant]])
 [[complex K-theory]] | [[KU]] |  [[representation ring]]
 $KU_G(\ast) \simeq R_{\mathbb{C}}(G)$ | [[Atiyah-Segal completion theorem]]
 $R(G) \simeq  KU_G(\ast)  \overset{ \text{compl.} }{\longrightarrow} \widehat {KU_G(\ast)}  \simeq KU(B G)$  |
| ([[equivariant complex cobordism cohomology theory|equivariant]])
 [[complex cobordism cohomology]] | [[MU]] | $MU_G(\ast)$ | [[completion theorem for complex cobordism cohomology]]
 $MU_G(\ast)  \overset{ \text{compl.} }{\longrightarrow} \widehat {MU_G(\ast)}  \simeq MU(B G)$ |
| ([[equivariant algebraic K-theory|equivariant]])
 [[algebraic K-theory]] | $K \mathbb{F}_p$ | [[representation ring]]
 $(K \mathbb{F}_p)_G(\ast) \simeq R_p(G)$ | [[Rector completion theorem]]
 $R_{\mathbb{F}_p}(G) \simeq  K (\mathbb{F}_p)_G(\ast)  \overset{ \text{compl.} }{\longrightarrow} \widehat {(K \mathbb{F}_p)_G(\ast)} \!\! \overset{\text{Rector 73}}{\simeq} \!\!\!\!\!\! K \mathbb{F}_p(B G) $
 |
| ([[equivariant stable cohomotopy|equivariant]])
 [[stable cohomotopy]] | $K \mathbb{F}_1 \overset{\text{Segal 74}}{\simeq} $ [[equivariant sphere spectrum|S]]  | [[Burnside ring]]
 $\mathbb{S}_G(\ast) \simeq A(G)$ |  [[Segal-Carlsson completion theorem]]
 $A(G) \overset{\text{Segal 71}}{\simeq}  \mathbb{S}_G(\ast)  \overset{ \text{compl.} }{\longrightarrow} \widehat {\mathbb{S}_G(\ast)} \!\! \overset{\text{Carlsson 84}}{\simeq} \!\!\!\!\!\! \mathbb{S}(B G) $
  |


Segal condition

+-- {: .rightHandSide}
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### Context
#### Category theory
+-- {: .hide}
[[!include category theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The _Segal condition_ is a condition on a [[simplicial object]] $X_\bullet \in \mathcal{C}^{\Delta^{op}}$ which says that each component $X_{n \geq 2} \in \mathcal{C}$ is obtained from $X_1 \stackrel{\overset{\partial_0}{\to}}{\underset{\partial_1}{\to}} X_0$ by the $n$-fold [[fiber product]] of $X_1$ over $X_0$ which "glues" $n$ copies of $X_1$ end-to-end.

Hence if one thinks of $X_0$ as a collection of [[objects]] and of $X_1$ as a collection of [[morphisms]], then $X_\bullet$ satisfies the Segal condition precisely if each $X_{n \geq 2}$ can be interpreted as the collection of sequences of [[composition|composable]] morphisms of length $n$. The precise formulation is below in _[Definition -- For simplicial objects](#ForSimplicialObjects)_.

Accordingly, if $\mathcal{C} = $ [[Set]] is the category of [[sets]], then the Segal condition characterizes precisely those [[simplicial sets]] which are the [[nerve]] of a ([[strict category|strict]]) [[small category]], theorem \ref{NerveTheorem} below. This is the observation due to [Grothendieck 1961](#Grothendieck61) as recalled in [Segal 1968](#Segal68)), which today gives the Segal condition its name. Sometimes this statement also called the _nerve theorem_ (no relation to what is called _[[nerve theorem]]_ in [[homotopy theory]]).

It is useful to decompose this statement into its constituents as follows:

A [[small category]] $C$ may be thought of as a [[directed graph]] $U(C)$ equipped with a [[unit law|unital]] [[associativity|associative]] [[composition]] operation. This corresponds to a sequence of inclusions of [[sites]]

$$
  (1 \stackrel{\overset{d_0}{\leftarrow}}{\underset{d_1}{\leftarrow}} 0) \hookrightarrow \Delta_0 \to \Delta
$$

into the [[simplex category]], where $\Delta_0$ is the category of finite non-empty directed linear graphs:

* a [[directed graph]] is equivalently a [[presheaf]] on $(1 \stackrel{\overset{d_0}{\leftarrow}}{\underset{d_1}{\leftarrow}} 0)$;

* a [[presheaf]] on $\Delta$, hence a [[simplicial set]] encodes via its face and degeneracy maps a kind of associative and unital composition -- but not necessarily "of composable morphisms" if $X_{n\geq 2}$ is not given in the above fashion.

In terms of this we can say that equipping a directed graph with the structure of a category is equivalent to asking for its pushforward along $(1 \stackrel{\overset{d_0}{\leftarrow}}{\underset{d_1}{\leftarrow}} 0) \hookrightarrow \Delta_0$ (which encodes all the collections of sequences of composable edges) to be equipped with a lift to a simplicial object through the pullback along $\Delta_0 \to \Delta$. Conversely, the simplicial objects obtained as such lifts are precisely the simplicial objects that satisfy the Segal condition.

Formulated in this way one sees that the Segal condition has a large variety of generalizations to structures with richer kinds of composition operations, such as [[globular operads]]. This is made precise below in _[Definition -- For cellular objects](#ForCellularObjects)_.

## Definition

### For simplicial objects
 {#ForSimplicialObjects}

Let $\mathcal{C}$ be a [[category]] with [[pullbacks]].

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

A [[simplicial object]]

$$
  X : \Delta^{op} \to \mathcal{C}
$$

is said to satisfy the **Segal condition** if it sends the [[colimits]] in the [[simplex category]] to [[limits]], hence if the [[Segal maps]] exhibit [[equivalences]]

$$
  X_n
   \stackrel{\simeq}{\longrightarrow}
  \underbrace{
    X_1 \times_{X_0} \cdots \times_{X_0} X_1
  }_{n\; factors}
$$

for all $n \in \mathbb{N}$.

=--

More in detail:

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

For all $n \in \mathbb{N}$, consider $\Delta[n]$ naturally as a [[cocone]] in the [[simplex category]] under the [[diagram]]

$$
  \array{
     \Delta[0] && && \Delta[0] && && \Delta[0] &&& \cdots
      \\
      & {}_{\mathllap{d_1}}\searrow && {}^{\mathllap{d_0}}\swarrow && \searrow^{\mathrlap{d_1}} && {}_{d_0}\swarrow && \cdots
      \\
      && \Delta[1] &&&& \Delta[1] && \cdots
  }
$$

with $n$ copies of $\Delta[1]$ at the bottom, such that the cocone injection of the $k$th copy is $\Delta[1] \simeq (k-1,k) \hookrightarrow (0,1,2, \cdots, n) \simeq \Delta[n]$.

A [[simplicial object]] $X \colon \Delta^{op} \to \mathcal{C}$ satisfies the **Segal conditions** if it sends these cocones to [[limit]] [[cones]] in $\mathcal{C}$.

=--

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

This definition immediately generalizes to [[(∞,1)-category theory]] where $\mathcal{C}$ is an [[(∞,1)-category]] and $X$ is a [[simplicial object in an (∞,1)-category]]. Then for $X$ to satisfy the Segal conditions means that it sends the cocones of def. \ref{SegalCones} to [[(∞,1)-limit]] cones in $\mathcal{C}$.

Such a simplicial object is also called a **[[pre-category object in an (∞,1)-category]]** in $\mathcal{C}$.

=--

### For cellular objects
 {#ForCellularObjects}

A _[[globular theory]]_ is a [[wide subcategory]] inclusion

$$
  i_A \colon \Theta_0 \to \Theta_A
$$

of the [[globular site]] $\Theta_0$. There is an [[equivalence of categories]]

$$
  \omega Grph \simeq Sh(\Theta_0)
  \,.
$$

of [[∞-graphs]] and sheaves on the [[globular site]]. In particular for $\Theta_A = \Theta$ the [[cell category]] ([[Theta category]]) a [[presheaf]] on $\Theta$ is a [[cellular object]].

The **Segal condition** on a cellular object $X \colon \Theta^{op} \to \mathcal{C}$ is that the restriction $i^* X \colon \Theta_0^{op} \to \Theta^{op} \to \mathcal{C}$ to the [[cellular site]] is a [[sheaf]] there.

The cellular objects that satisfy the Segal condition are precisely the [[∞-category]] objects ([Berger](#Berger)).

The cellular spaces/ cellular simplicial sets/cellular [[∞-groupoids]] that satisfy the Segal condition as a [[weak homotopy equivalence]]/ [[equivalence of ∞-groupoids]] is a _[[Theta_n-space]]_ an [[(∞,n)-category]].

## Properties

### Characterization of nerves of (higher) categories

#### Of simplicial nerves of small categories

The archetypical role of the Segal condition is to make the following statement true.

+-- {: .num_theorem #NerveTheorem}
###### Theorem
**(nerve theorem)**

A [[simplicial set]] is the [[nerve]] of a [[small category]] precisely if it satsfies the Segal conditions.

=--

This is due to [Grothendieck 61](#Grothendieck61), recalled in [Segal 1968](#Segal68).

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

There is an entirely unrelated theorem in [[homotopy theory]] also often called "the" _[[nerve theorem]]_. See there for more. Not to be confused with the discussion here.

=--

#### Of complete Segal spaces

By refining the above result from sets to $\infty$-groupoids, one obtains the [[pre-category object in an (infinity,1)-category]].

#### Of cellular nerves of strict $\omega$-categories

Similarly, a [[cellular set]] is the cellular nerve of a [[strict omega-category]] precisely if it satisfies the cellular Segal condition. ([Berger](#Berger)).

#### Of cellular models of $(\infty,n)$-categories

See at _[[Theta-space]]_.

### In terms of sheaf conditions
 {#InTermsOfSheafConditions}

We discuss an equivalent formulation of the Segal condition in terms of notions in [[topos theory]]/[[(∞,1)-topos theory]]. This perspective for instance lends itself more to a formulation of Segal conditions in terms of the [[internal language]] of toposes.

#### For simplicial objects and category objects
 {#InTermsOfSheafConditionForSimplicialObjects}

We characterize below in prop. \ref{CatAsPullback} the category of categories as the pullback of the [[topos]] of [[simplicial set]] along the inclusion of the topos of graphs into that of presheaves on finite linear graphs.

First we state some preliminaries.

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

The condition in def. \ref{SegalCones} superficially looks like a [[sheaf]] condition for [[coverings]] of $\Delta[n]$ by $n$ subsequent copies of $\Delta[1]$. However, these coverings do _not_ form a [[coverage]] on the [[simplex category]] $\Delta$: the refinement-of-covers-axiom is not satisfied:

For instance for $d_1 \colon \Delta[1] \to \Delta[2]$ the map that sends the single edge of $\Delta[1]$ to the composite edge in $\Delta[2]$ there is no way to "pull back" the cover

$$
  \{\Delta[1] \coprod \Delta[1] \stackrel{((0,1),(1,2))}{\to} \Delta[2]\}
$$

along this morphism, not even in the weak sense of _[[coverage]]_.

However, as this example also makes clear, the problem is precisely only with the morphisms in $\Delta$ that are no injective on generating edges.

=--

Therefore consider instead the following:

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

Let

$$
  j \colon \Delta_0 \hookrightarrow Graph
$$

be the [[full subcategory]] of that of [[directed graphs]] on the linear graphs $\{0 \to 1 \to \cdots \to n\}$ for $n \in \mathbb{N}$.

=--

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

Morphisms in $\Delta_0$ have to send elementary edges to elementary edges. So there are

* precisely $n$ morphisms $\Delta_0[1] \to \Delta_0[n]$

* precisely $n$ morphisms $\Delta_0[2] \to \Delta_0[n+1]$

* precisely $n$ morphisms $\Delta_0[3] \to \Delta_0[n+2]$

* etc.

=--

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

Write

$$
  i
    \colon
  (1 \stackrel{\leftarrow}{\leftarrow} 0) \hookrightarrow \Delta_0
$$

for the [[full subcategory]] on the linear graphs with no edge and with one edge.

=--

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

The [[category]] of [[directed graphs]] is [[equivalence of categories|equivalently]] the [[category of presheaves]] over $(1 \stackrel{\leftarrow}{\leftarrow} 0)$, def. \ref{ParallelMorphismCategory}:

$$
  Graph(\mathcal{C}) \simeq PSh((1 \Leftarrow 0), \mathcal{C})
  =
  \mathcal{C}^{(1 \Rightarrow 0)}
  \,.
$$

=--

+-- {: .num_prop #AdjointTripleFromGraphs}
###### Definition

Write

$$
  (i_! \dashv i^* \dashv i_*)
  \colon
  \mathcal{C}^{1 \Rightarrow 0}
  \stackrel{\overset{i_!}{\to}}{\stackrel{\overset{i^*}{\leftarrow}}{\underset{i_*}{\to}}}
  \mathcal{C}^{\Delta_0^{op}}
$$

for the [[adjoint triple]] induced on [[categories of presheaves]] by the inclusion $i$ of def. \ref{ParallelMorphismCategory}: $i^*$ is given by precomposition with $i$, $i_!$ is left and $i_*$ is right [[Kan extension]] along $i$.

=--

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

The functor $i_* \colon Graph(\mathcal{C}) \to \mathcal{C}^{\Delta_0^{op}}$
of def. \ref{AdjointTripleFromGraphs} sends a graph $X_1 \stackrel{\overset{\partial_1}{\to}}{\underset{\partial_0}{\to}} X_0$ to the presheaf $i_*(X)$ which on $n \in \mathbb{N}$ is given by th iterated [[pullback]]

$$
  i_*(X) \colon n \mapsto \underbrace{X_1 \times_{X_0} \times \cdots \times_{X_0} X_1}_{n \; factors}
$$

and which sends an inclusion $\Delta[k] \simeq (j, \cdots, j+k) \hookrightarrow (0,\cdots, n)\simeq \Delta[n] $ to the corresponding [[projection]] map out of the pullback.

We may call $i_*(X)$ the **nerve of the graph** $X$.

=--

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

Using the [[Yoneda lemma]] and the defining [[hom functor|hom]]-[[isomorphisms]] of the [[adjunction]] as well as the fact that the [[hom functor]] sends [[colimits]] in the first argument to [[limits]], we have

$$
  \begin{aligned}
    i_*(X)(n) & \simeq Hom(\Delta[n], i_*(X))
    \\
    & \simeq Hom(i^* \Delta[n], X)
    \\
    & \simeq Hom( \underbrace{\Delta[1] \coprod_{\Delta[0]} \cdots \coprod_{\Delta[0]} \Delta[1]}, X )
   \\
   & \simeq
   \underbrace{
     Hom(\Delta[1], X)
       \underset{Hom(\Delta[0], X)}{\times}
       \cdots
       \underset{Hom(\Delta[0], X)}{\times}
     Hom(\Delta[1], X)
   }_{n\;factors}
   \\
   & \simeq
   \underbrace{
     X \times_{X_0} \cdots \times_{X_0} X_1
   }_{n\; factors}
  \end{aligned}
  \,.
$$

=--

+-- {: .num_prop #CoverageOnDelta0}
###### Proposition

For $n \in \mathbb{N}$ declare a unique [[covering]] family of $\Delta[n] \in \Delta_0$ to be

$$
  \left\{
    \Delta\left[1\right] \simeq \left(k,k+1\right)
    \hookrightarrow \Delta\left[n\right] \right\}_{k = 0}^{n-1}
  \,.
$$

Then this is a [[coverage]] on $\Delta_0$.

=--

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

A [[presheaf]] $X \colon \Delta_0^{op} \to \mathcal{C}$ is a [[sheaf]] with respect to the coverage of def. \ref{CoverageOnDelta0} precisely if it is in the [[essential image]] of the graph-nerve functor

$$
  i_*
  \colon
  Graph(\mathcal{C})
   \simeq
  \mathcal{C}^{(1 \stackrel{\to}{\to} 0)}
    \stackrel{}{\to}
  \mathcal{C}^{\Delta_0^{op}}
$$

of prop. \ref{NerveOfGraph}. This yields an [[equivalence of categories]]

$$
  Graph(\mathcal{C})
  \simeq
  Sh(\Delta_0)
$$

with the [[category of sheaves]] on $\Delta_0$. The graph-nerve functor is a [[full and faithful functor]]

$$
  i_* \colon Graph \simeq Sh(\Delta_0)
  \hookrightarrow
  PSh(\Delta_0)
  \,.
$$

=--

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

Write

$$
  (j_! \dashv j^* \dashv j_*)
  \colon
  \mathcal{C}^{\Delta_0^{op}}
  \stackrel{\overset{j_!}{\to}}{\stackrel{\overset{j^*}{\leftarrow}}{\underset{j_*}{\to}}}
  \mathcal{C}^{\Delta^{op}}
$$

for the [[adjoint triple]] induced on [[categories of presheaves]] by the inclusion $j$ of def. \ref{FiniteLinearGraphsInAllGraphs}: $j^*$ is given by precomposition with $j$, $j_!$ is left and $j_*$ is right [[Kan extension]] along $j$.

=--

In terms of all this the nerve theorem \ref{NerveTheorem} says the following:

We have [[geometric morphisms]] of [[toposes]]

$$
  Grph \simeq Sh(\Delta_0)
  \stackrel{\overset{i_!}{\to}}{
  \stackrel{\overset{i^*}{\leftarrow}}{\underset{i_*}{\hookrightarrow}}}
  PSh(\Delta_0)
   \stackrel{\overset{j_!}{\to}}{\stackrel{\overset{j^*}{\leftarrow}}{\underset{j_*}{\to}}}
  PSh(\Delta)
  \simeq sSet
$$

which capture the Segal condition as follows.

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

The [[commuting diagram]] of [[1-categories]]

$$
  \array{
    Cat &\stackrel{N}{\hookrightarrow}& PSh(\Delta) \simeq sSet
    \\
    \downarrow^{\mathrlap{U}} && \downarrow^{\mathrlap{j^*}}
    \\
    Graph \simeq Sh(\Delta_0) &\stackrel{i_*}{\hookrightarrow}& PSh(\Delta_0)
  }
  \,,
$$

where

* $N$ forms the [[nerve of a category]];

* $U$ is the [[forgetful functor]] that sends a category to its underlying graph

is a [[pullback]].

=--

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

The morphisms in the commuting diagram of prop. \ref{CatAsPullback} participate in further [[adjunctions]], and in terms of these the Segal condition may further be reformulated as a restriction condition on [[algebras over an operad]]:

First of all the nerve has a [[left adjoint]] $\tau \colon PSh(\Delta) \to Cat$. With this the left adjoint $j_!$ of $j^*$ induces a left adjoint

$$
  F \simeq \tau j_! i_*
$$

of $U$, which is the [[free category]] functor.

Moreover, with $U$ also $j^*$ is a [[monadic functor]] and the [[monad]] $U F \colon Grph \to Grph$ of which [[Cat]] is the category of [[algebras over a monad|algebras]] is the restriction of the monad $j^* \circ j_!$:

$$
  i_* U F \simeq j^* j_! i_*
  \,.
$$

(All this is discussed in ([Berger, p. 13](#Berger)), and actually in the further generality of cellular sets that we get to [below](#InTermsOfPullBackOfCatOfSheavesForCellularObjects).)

In summary we have a diagram of adjoint pairs of functors of the form

$$
  \array{
    Cat &\stackrel{\overset{\tau}{\leftarrow}}{\underoverset{N}{\bottom}{\hookrightarrow}}& PSh(\Delta) \simeq sSet
    \\
    {}^{\mathllap{U}}\downarrow \vdash \uparrow^{\mathrlap{F}} && {}^{\mathllap{j^*}}\downarrow \vdash \uparrow^{\mathrlap{j_!}}
    \\
    Graph \simeq Sh(\Delta_0) &\stackrel{\overset{i^*}{\leftarrow}}{\underoverset{i_*}{\bottom}{\hookrightarrow}}& PSh(\Delta_0)
  }
$$

where several (however not all) subdiagrams of functors commute, as discussed above.
In terms of this the reformulation of the Segal condition as in prop. \ref{CatAsPullback} is now further reformulated as:

_A category is equivalently an algebra over the monad $j^* j_!$ on $PSh(\Delta_0)$ which satisfies the Segal condition in that it is in the essential image of the functor $i_*$ of prop. \ref{NerveOfGraph}._

=--

#### For cellular objects and $\omega$-category objects
 {#InTermsOfPullBackOfCatOfSheavesForCellularObjects}

The immediate generalization of prop. \ref{CatAsPullback} from simplicial objects to [[cellular objects]] is the following.

Let

$$
  j \colon \Theta_0 \to \Theta
$$

be the defining inclusion of the [[cellular site]] into the [[cell category]].

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

The category $Str\omega Cat$ of [[strict ∞-categories]] is the [[pullback]]

$$
  \array{
    Str\omega Cat
     &\underoverset{\simeq}{N}{\to}& Mod_\Theta &\hookrightarrow&
PSh(\Theta)
    \\
    \downarrow^{\mathrlap{U}} && \downarrow^{} && \downarrow^{\mathrlap{j^*}}
   \\
   \omega Graph &\stackrel{\simeq}{\to}& Sh(\Theta_0) &\hookrightarrow& PSh(\Theta_0)
  }
  \,.
$$

=--

See at _[[globular theory]]_ for more.

## Related concepts

* [[Segal space]], [[complete Segal space]], [[n-fold complete Segal space]]

## References

The "Segal conditions" are originally due to:

* {#Grothendieck61} [[Alexander Grothendieck]], Prop. 4.1 of: [[FGA]] _Techniques de construction et théorèmes d'existence en géométrie algébrique III : préschémas quotients_, Séminaire Bourbaki : années 1960/61, exposés 205-222, Séminaire Bourbaki, no. 6 (1961), Exposé no. 212,   ([numdam:SB_1960-1961__6__99_0](http://www.numdam.org/item/?id=SB_1960-1961__6__99_0), [pdf](http://www.numdam.org/item/SB_1960-1961__6__99_0.pdf), [English transl. pdf](https://labs.thosgood.com/builds/fga-3-III.pdf))

and named after their mentioning in

* {#Segal68} [[Graeme Segal]], Section 2 of: _Classifying spaces and spectral sequences_,  Publications Mathématiques de l'IHÉS, Volume 34  (1968), p. 105-112  ([numdam:PMIHES_1968__34__105_0](http://www.numdam.org/item/PMIHES_1968__34__105_0/))

The interpretation of the Segal condition as a [[sheaf]] condition is reviewed for instance in section 2 of

* [[Joachim Kock]], _Polynomial functors and trees_ ([pdf](http://mat.uab.es/~kock/cat/poly-trees.pdf))

and discussed for [[strict infinity-categories]] in

* [[Clemens Berger]], _A cellular nerve for higher categories_, Advances in Mathematics 169, 118-175 (2002) ([pdf](http://math1.unice.fr/~cberger/nerve.pdf))
 {#Berger}

Based on that, an iterative and homotopy-theoretic formulation of the cellular Segal conditions is in section 5 of

* {#Rezk} [[Charles Rezk]], _A cartesian presentation of weak $n$-categories_ Geom. Topol. 14 (2010), no. 1, 521–571 ([arXiv:0901.3602](http://arxiv.org/abs/0901.3602))

[[!redirects Segal conditions]]


Segal map

#Contents#
* table of contents
{:toc}

## Idea

Segal maps, named after [[Graeme Segal]], appear in the definition of the _[[Segal conditions]]_ on a [[simplicial object]].

In the context of [[higher category theory]] they
were used by [[Charles Rezk]] to define [[Segal categories]] and [[complete Segal spaces]].

## Definition

Let $X$ be a [[bisimplicial set]].  Assume for simplicity that $X$ is [[fibrant]] with respect to the [[Reedy model structure]] on the [[functor category]] $[\Delta^{op}, SSet]$.  The **$n$-th Segal map** of $X$ is the canonical [[morphism]] of [[simplicial sets]]
  $$ X_n \to (X_1)^{\times_{X_0} (n)} = X_1 \times_{X_0} \cdots \times_{X_0} X_1. $$
Here the right-hand side is the [[limit]] of the diagram
  $$ X_1 \stackrel{d_1}{\to} X_0 \stackrel{d_0}{\leftarrow} X_1 \stackrel{d_1}{\to} \cdots \stackrel{d_1}{\to} X_0 \stackrel{d_0}{\leftarrow} X_1 $$
where there are $n$ copies of $X_1$.

More explicitly, this morphism is induced by the morphisms $a_i: X_n \to X_1$ ($0 \le i \le n-1$), which are induced by the morphisms $\alpha^i: [1] \to [n]$ in $\Delta$, the [[simplex category]], which map $0 \mapsto i$ and $1 \mapsto i+1$.

If $X$ is not [[Reedy model structure|Reedy fibrant]], then one must replace the [[limit]] above with a [[homotopy limit]].

## References

* [[Julie Bergner]], _Three models for the homotopy theory of homotopy theories_, Topology 46 (2007), 397-436, [arXiv:math/0504334](http://arxiv.org/abs/math/0504334).

[[!redirects Segal maps]]

[[!redirects Segal morphism]]
[[!redirects Segal morphisms]]


Segal n-category

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Higher category theory
+--{: .hide}
[[!include higher category theory - contents]]
=--
#### Enriched category theory
+--{: .hide}
[[!include enriched category theory contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The notion of _Segal $n$-category_ is a model for [[(∞,n)-categories]] via $n$-fold iterated [[enriched (∞,1)-category|(∞,1)-enrichment]]. It is the [[higher category theory|higher]] analog of the notion of [[Segal categories]],

## References

The model structure for Segal $n$-categories was introduced in

* {#HS} [[André Hirschowitz]], [[Carlos Simpson]], _Descente pour les $n$-champs (Descent for $n$-stacks)_ [[arXiv:9807049](http://arxiv.org/abs/math/9807049)]

Further developments are in

* [[Carlos Simpson]], _[[Homotopy Theory of Higher Categories]]_ ([arXiv:1001.4071](http://arxiv.org/abs/1001.4071))

and in section 2 of

* [[Jacob Lurie]], _(Infinity,2)-Categories and the Goodwillie Calculus I_ ([arXiv:0905.0462](http://arxiv.org/abs/0905.0462))
 {#Lurie}

[[!redirects Segal n-categories]]


Segal space

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Internal $(\infty,1)$-Categories
+--{: .hide}
[[!include internal infinity-categories contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A _Segal space_ is a [[precategory object in an (∞,1)-category|pre-category object]] in [[∞Grpd]].

A genuine [[category object in an (∞,1)-category|category object]] in [[∞Grpd]] is a _[[complete Segal space]]_.  This is a way of speaking of [[(∞,1)-categories]].

## Definition

A **Segal space** $X_\bullet$
  is a [[simplicial topological space]] or [[bisimplicial set]]
  $X_\bullet : \Delta^{op} \to Top$ which satisfies the [[Segal conditions]]:

for all $m,n \in \mathbb{N}$ the square

$$
  \array{
    X_{m+n} &\stackrel{
      p^*_{0,\cdots, m}
    }{\to}& X_m
    \\
    {}^{\mathllap{p^*_{m, \cdots, m+n}}}\downarrow
    && \downarrow^{\mathrlap{p^*_m}}
    \\
    X_n &\stackrel{p^*_0}{\to}& X_0
  }
$$

is a [[homotopy pullback]] square.

A Segal space for which $X_0$ is a [[discrete space]] is called a _[[Segal category]]_. See there for more dicussion.

## Examples
 {#Examples}

### In $Set$
 {#InSetByNervesOfCategories}

For $\mathcal{C}$ a ([[small category|small]]) [[category]] we may regard its ordinary [[nerve]] [[simplicial set]] $N(\mathcal{C}) \in Set^{\Delta^{op}}$ as a Segal space, under the canonical inclusion $Set \hookrightarrow \infty Grpd$,

$$
  N(\mathcal{C}) \in Set^{\Delta^{op}} \hookrightarrow \infty Grpd^{\Delta^{op}}
  \,.
$$

In fact, the classical "nerve theorem" about the [[Segal conditions]] says that a simplicial set is the nerve of a category precisely if it is a Segal space.

Notice that $Equiv(N(\mathcal{C})_1) \hookrightarrow N(\mathcal{C})_1$ is precisely the [[subset]] of [[isomorphisms]] in all [[morphisms]] of $\mathcal{C}$.

Therefore under this identification, $N(\mathcal{C})$ is a [[complete Segal space]] precisely if $\mathcal{C}$ is a [[gaunt category]], hence precisely if the only isomorphisms in $\mathcal{C}$ are the [[identities]].

In particular if $\mathcal{C}$ is a [[(0,1)-category]], hence a [[preordered set]], then $N(\mathcal{C})$ is complete Segal precisely if $\mathcal{C}$ is in fact an [[partially ordered set]].

### Construction in $1Grpd$ from a category
 {#ConstructionFromACategory}

Let $\mathcal{C}$ be an ordinary category. We discuss how Segal spaces are associated with this.

Let $\mathcal{K}$ be a [[groupoid]] and $p \colon \mathcal{K} \to \mathcal{C}$ a [[functor]] which is [[essentially surjective functor|essentially surjective]].

Then let $X_1 \coloneqq (p/p)\in Grpd$ be the "lax fiber product" of $p$ with itself, or rather the [[comma object]] of $p$ with itself, hence the [[comma category]], sitting in the universal square

$$
  \array{
    X_1 &\stackrel{\partial_1}{\to}& \mathcal{K}
    \\
    {}^{\mathllap{\partial_0}}\downarrow &\swArrow& \downarrow^{\mathrlap{p}}
    \\
    \mathcal{K} &\underset{p}{\to}& \mathcal{C}
  }
  \,.
$$

Next let $X_2 = (p/p/p)$ be the "3-fold comma category", hence the comma category in

$$
  \array{
    X_2 &\stackrel{\partial_{0}}{\to}& \mathcal{K}
    \\
    {}^{\mathllap{}}\downarrow &\swArrow& \downarrow^{\mathrlap{p}}
    \\
    X_1 &\underset{\partial_1}{\to}& \mathcal{C}
  }
$$

and so forth: $X_n \coloneqq p^{/^{n+1}}$.

This way an [[object]] of $X_n$ is an $(n+1)$-[[tuple]] of objects $(x_0,x_1, \cdots, x_n) \in \mathcal{K}$ together with a sequence of $n$ composable morphisms $p(x_0) \to p(x_1) \to \cdots \to p(x_n)$, and a [[morphism]] is an $(n+1)$-tuple of morphisms $(f_0,f_1, \cdots, f_n) \in \mathcal{K}$ and a [[pasting]] [[commuting diagram]]

$$
  \array{
    p(x_0) &\to& p(x_1) &\to&  \cdots &\to& p(x_1)
    \\
    {}^{\mathllap{p(f_0)}}\downarrow && \downarrow^{\mathrlap{p(f_1)}} && \cdots   && \downarrow^{\mathrlap{p(f_1)}}
    \\
    p(x'_0) &\to& p(x_1) &\to& \cdots &\to& p(x'_1)
  }
$$

in $\mathcal{C}$.

By direct inspection, the maps $X_n \to X^{\partial \Delta^n}$ obtained this way are [[isofibrations]], hence [[fibrations]] in the [[canonical model structure]] on [[Grpd]] and so the [[homotopy pullbacks]] that enter the [[Segal conditions]] for $X_\bullet$ are given by ordinary [[fiber products]]. These clearly satisfy the Segal conditions, hence

$$
  X_\bullet \in Grpd^{\Delta} \hookrightarrow \infty Grpd^{\Delta^{op}}
$$

constructed this way is a Segal space.

Two special case of the functor $p$ are important:

* if $\mathcal{K} \simeq core(\mathcal{C})$ is the [[core]] of $\mathcal{C}$ and $p$ is the canonical core inclusion, one finds that $Equiv(X_1) \hookrightarrow X_1$ by the above construction is $Equiv(X_1) = Core(\mathcal{C})^{\Delta^1}$, the [[arrow category]] of the [[core]] of $\mathcal{C}$. This is [[equivalence of categories|equivalent]] to $\mathcal{C}$ by, for instance, the source or  restriction map. Hence for $p$ the core inclusion, the above construction gives the [[complete Segal space]] corresponding to the category $\mathcal{C}$.

* if $p \colon \pi_0(\mathcal{C}) \to \mathcal{C}$ is a choice of basepoints in each [[isomorphism class]] of $\mathcal{C}$, then $X_\bullet$ is the [[Segal category]] incarnation of the category $\mathcal{C}$.

### In $1Grpd$
 {#ExamplesInIGrpd}

We consider the situation of _[From a category](#ConstructionFromACategory)_, but now conversely, starting with a Segal space in groupoids and then extracting a category from it.

Consider a Segal space that is degreewise just a [[1-groupoid]], hence a simplicial object in the inclusion

$$
  X_\bullet \in 1Grpd^{\Delta^{op}} \hookrightarrow \infty Grpd^{\Delta^{op}}
  \,.
$$

Choosing this to be [[Reedy model structure|Reedy]] [[fibrant object|fibrant]], the map $(\partial_0,\partial_1) \colon X_1 \to X_0 \times X_0$ is an [[isofibration]].

We may write an object $K \in X_1$ as a horizontal morphism

$$
  \partial_1(K) \stackrel{K}{\to} \partial_2(K)
$$

and a morphism $\lambda \colon L \to K$ in $X_1$ as a vertical [[double category]] arrow:

$$
  \array{
     \partial_1(L) & \stackrel{L}{\to} & \partial_0(L)
     \\
     {}^{\mathllap{\partial_1(\lambda)}}\downarrow &\Downarrow^{\mathrlap{\lambda}}& \downarrow^{\mathrlap{\partial_0(\lambda)}}
     \\
     \partial_1(K) &\stackrel{K}{\to}& \partial_0(K)
  }
  \,.
$$

Then the fact that $(\partial_1,\partial_0)$ is an [[isofibration]] means that for every "niche"

$$
  \array{
     y_0 & & y_1
     \\
     {}^{\mathllap{f_0}}\downarrow
     &&
    \downarrow^{\mathrlap{f_1}}
     \\
     x_0 &\stackrel{K}{\to}& x_1
  }
  \,,
$$

namely for every pair of morphisms $f_0, f_1$ in $X_0$ and lift of its codomain to an object $K \in X_1$, there is a "niche filler"

$$
  \array{
     y_0 & \stackrel{L}{\to} & Y_1
     \\
     {}^{\mathllap{f_0}}\downarrow
     &\Downarrow^{\mathrlap{\lambda}}&
    \downarrow^{\mathrlap{f_1}}
     \\
     x_0 &\stackrel{K}{\to}& x_1
  }
  \,,
$$

namely a lift of the whole pair $(f_0,f_1)$ to a morphism $\lambda$ in $X_1$, and this is necessarily universal in that any other such lift uniquely factors through this one (because $X_1$ is a groupoid).

Comparison with the definition of a [[2-category equipped with proarrows]] in the incarnation _[as a double category](2-category+equipped+with+proarrows#DefinitionAsDoubleCategory)_ shows that this is the beginning of the construction of a [[pseudo double category]] whose vertical category is $X_0$ and whose weak horizontal composition is that induced by the Segal maps.

Assume next that $X_3 \to X^{\partial \Delta^2}$ is a [[1-monomorphism]], as are all the higher $X^n \to X^{\partial \Delta^n }$, for $n \geq 3$, hence that $X_\bullet$ is [[coskeleton|2-coskeletal]] as a simplicial object. This means that the horizontal composition in this pseudo double category has unique composites, hence that the horizontal category is an ordinary category. If then furthermore the composite $Equiv(X_1) \to X_1 \stackrel{\partial_0}{\to} X_0$ is an equivalence, hence is the Segal space is a [[complete Segal space]] this means that $X_\bullet$ arises from this horizontal category by the construction [above](#ConstructionFromACategory).

## Related notions

* [[reduced Segal space]], [[Segal category]],

* [[semi-Segal space]]

* [[complete Segal space]], [[model structure for complete Segal spaces]]

* [[higher Segal space]]

* [[Segal type]]

## References

See generally the references at _[[complete Segal space]]_.

The "[[Segal conditions]]" are first discussed in

* [[Graeme Segal]], _Classifying spaces and spectral sequences_, Inst. Hautes Études Sci. Publ. Math., vol. 34, pp. 105–112 (1968),

where it is attributed to [[Alexander Grothendieck]].

The term "Segal space" is due to

* [[Charles Rezk]], ...

The _invertible_ case of Segal spaces, hence models for [[groupoid objects in an (infinity,1)-category]] are discussed in section 3 of

* [[Julia Bergner]], _Adding inverses to diagrams II: Invertible homotopy theories are spaces_, Homology, Homotopy and Applications, Vol. 10 (2008), No. 2, pp.175-193. ([web](http://www.intlpress.com/hha/v10/n2/a9/), [arXiv:0710.2254](http://arxiv.org/abs/0710.2254))

[[!redirects Segal spaces]]


Segal type

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

\tableofcontents

## Idea

Segal types are the (infinity,1)-categorical version of [[precategories]] in [[simplicial type theory]]

## Definition

In [[simplicial type theory]], a **Segal type** is a type $A$ such that given elements $x:A$, $y:A$, and $z:A$ and morphisms $f:\mathrm{hom}_A(x, y)$ and $g:\mathrm{hom}_A(y, z)$, the type
$$\sum_{h:\mathrm{hom}_A(x, z)} \left\langle \Delta^2 \to A \vert_{[x, y, z, f, g, h]}^{\partial \Delta^2} \right\rangle$$
is a [[contractible type]], where $\Delta^2$ is the $2$-simplex probe shape and $\partial \Delta^2$ is its boundary.

## Related concepts

* [[hom type]]
* [[discrete Segal type]]
* [[Rezk type]]
* [[isomorphism in a Segal type]]
* [[Segal space]]
* [[op modality]]

## References

* {#RiehlShulman17} [[Emily Riehl]], [[Michael Shulman]], *A type theory for synthetic $\infty$-categories* $[$[arXiv:1705.07442](https://arxiv.org/abs/1705.07442)$]$


Segal's category

> This entry is about _the_ [[small category]] named after [[Graeme Segal]]. Not to be confused with the model in [[higher category theory]] called _[[Segal categories]]_.

# Contents
* table of contents
{:toc}

## Definition

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

**Segal's category**, denoted $\Gamma$ is the category opposite to the [[skeleton]] of the [[category]] $FinSet^{*/}$ of [[pointed object|pointed]] [[finite sets]]:

$$
  \Gamma^{op} \simeq FinSet^{*/}
  \,.
$$

=--

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

The category $\Gamma$ is related to [[(infinity,1)-operads]] in a way similar to how the [[simplex category]] (non-empty and _linearly ordered_  finite sets) is related to [[(∞,1)-categories]].

=--

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

A morphism $f \colon \{*\} \coprod S \to \{*\} \coprod T$ in $\Gamma$ may be thought of as a [[partially defined function]] $\tilde f \colon S \to T$ which is undefined on all elements of $S$ that $f$ sends to the point.

=--

## Related concepts

* [[Gamma space]]

## References

For instance notation 2.0.0.2 in

* [[Jacob Lurie]], _[[Higher Algebra]]_
  {#Lurie}

[[!redirects Segal's category]]
[[!redirects Segal's category]]
[[!redirects Segal's category]]
[[!redirects Segal's category]]
[[!redirects Segal\'s category]]
[[!redirects Gamma category]]
[[!redirects Gamma-category]]
[[!redirects Segal Gamma-category]]