Goerss-Hopkins-Miller theorem

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

#Contents#
* table of contents
{:toc}

## Idea

The _Goerss-Hopkins-Miller theorem_ ([Goerss-Hopkins A](#GoerssHopkinsA), [Goerss-Hopkins B](#GoerssHopkinsB)) also called now the _Goerss-Hopkins-Miller-Lurie theorem_ ([[A Survey of Elliptic Cohomology|Lurie (Survey)]], [[Representability Theorems|Lurie (Representability)]]) as such establishes that the assignment to each [[elliptic curve]] of its [[elliptic cohomology|elliptic]] [[cohomology theory]] lifts in a [[coherence|coherent]] way to an assignment of the [[Brown representability theorem|representing]] [[spectra]] regarded as [[E-∞ rings]]. In summay this makes for a [[structure sheaf]] $\mathcal{O}_{\mathcal{M}^{der}}$ of the [[moduli stack of elliptic curves]] in [[derived algebraic geometry]], realizing that moduli stack  as a [[derived scheme]] $\mathcal{M}^{der}$ in [[E-∞ geometry]] ("spectral geometry") This is sketched in ([[A Survey of Elliptic Cohomology|Lurie]]) and discussed in detail in ([Behrens 13](#Behrens13)).

In detail the theorem utilizes and establishes more general tools concerning the [[obstruction]] theory of lifting [[formal group laws]] to [[cohomology theories]] and homotopy associative/commutative [[spectra]] to [[A-∞ rings]] and to [[E-∞ ring]] structure, respectively.

In the succinct and suggestive form of ([[A Survey of Elliptic Cohomology|Lurie, theorem 1.1]]) the Goerss-Hopkins-Miller theorem leads the existence of the lift $\mathcal{O}_{\mathcal{M}^{top}}$ in the [[diagram]]

$$
  \array{
    && E_\infty Rings
    \\
    &{}^{\mathllap{\mathcal{O}_{\mathcal{M}^{der}}}}\nearrow& \downarrow^{\mathrlap{Brown\;representation}}
    \\
    \mathcal{M}(R)
    &\longrightarrow&
    MCTs
  }
  \,,
$$

where the bottom map sends ([[derived elliptic curve|derived]]) [[elliptic curves]] over a [[ring]] $R$ to their associated [[elliptic cohomology|elliptic]] [[multiplicative cohomology theory]] ("MTC") and where the right vertical map sends a [[spectrum]] to the cohomology theory which it [[Brown representability theorem|represents]].

### Hopkins-Miller theorem

For $n \in \mathbb{N}$, consider the [[category]] whose [[objects]] are [[pairs]] consisting of a [[perfect field]] $k$ of finite [[characteristic]] and of a [[formal group]] of [[height of a formal group|height]] $n$, and whose [[morphisms]] are base change of fields (as rings).

The **Hopkins-Miller theorem** says that there is a functorial assignment $(k,\Gamma) \mapsto E_{k,\Gamma}$ to such data of an [[A-∞ ring]] [[spectrum]] $E_{k,\Gamma}$ such that

1. $\pi_2 E_{k,\Gamma}$ contains a [[unit]] and $\pi_{2n+1} E_{k,\Gamma} = 0$, hence in particular $E_{k,\Gamma}$ is [[complex oriented cohomology theory|complex orientable]];

1. the corresponding [[formal group law]] over $\pi_0 E_{k,\Gamma}$ is the [[Lubin-Tate formal group|universal deformation]] of $(k,\Gamma)$.

In this form this appears as ([Rezk 97, theorem 2.1](#Rezk97)).

### Goerss-Hopkins theorem

(...)

this appears reproduced as ([Behrens 13, theorem 4.2](#Behrens13))

### Lurie theorem
 {#LurieTheorem}

By ([[A Survey of Elliptic Cohomology|Lurie (Survey), theorem 4.1]]), the [[spectral Deligne-Mumford stack]] refinement $(\mathcal{M}_{ell}, \mathcal{O}^{top})$ of the [[moduli stack of elliptic curves]] is the moduli stack of [[derived elliptic curves]], in that there is a [[natural equivalence]] in [[E-∞ rings]] $A$ of the form

$$
  Hom(Spec(A), (\mathcal{M}_{ell},\mathcal{O}^{der}))
  \simeq
  E(A)
  \,,
$$

where on the left we have maps of [[structured (∞,1)-toposes]] and on the right the [[∞-groupoid]] of [[derived elliptic curves]] over $A$.

This is based on the [[Artin-Lurie representability theorem]] ([[A Survey of Elliptic Cohomology|Lurie (Survey), prop. 4.1]], [[Representability Theorems|Lurie (Representability)]]).

## Related concepts

[[!include moduli stack of curves -- table]]

## References

An introduction to and survey of the Goerss-Hopkins-Miller-Lurie theorem is in

* [[Paul Goerss]], _Topological modular forms (after Hopkins, Miller, and Lurie)_ Séminaire BOURBAKI Mars 2009 61ème année, 2008-2009, no 1005(2009)([arXiv:0910.5130](http://arxiv.org/abs/0910.5130))

which has grown out of

* _[[Topological Algebraic Geometry - A Workshop]]._

* [pdf](http://www.math.uni-bonn.de/people/grk1150/Past_Events/AG-programm-SS07.pdf)

Details on the Hopkins-Miller theorem are in

* {#Rezk97} [[Charles Rezk]], _Notes on the Hopkins-Miller theorem_, 1997 ([pdf](http://www.math.uiuc.edu/~rezk/hopkins-miller-thm.pdf))

and a survey is also in

* [[Akhil Mathew]], _[Hopkins-Miller theorem I](https://amathew.wordpress.com/tag/hopkins-miller-theorem/)_

The Goerss-Hopkins theorem is laid out in

* {#GoerssHopkinsA} [[Paul Goerss]], [[Michael Hopkins]], _Moduli spaces of commutative ring spectra_, in _Structured ring spectra_, London
Math. Soc. Lecture Note Ser., vol. 315, Cambridge Univ. Press, Cambridge, 2004, pp. 151-200. ([pdf](http://www.math.northwestern.edu/~pgoerss/papers/sum.pdf), [doi:10.1017/CBO9780511529955.009](https://doi.org/10.1017/CBO9780511529955.009))

* {#GoerssHopkinsB} [[Paul Goerss]], [[Michael Hopkins]], _Moduli problems for structured ring spectra_ ([pdf](http://www.math.northwestern.edu/~pgoerss/spectra/obstruct.pdf))

The actual details of the application of the Goerss-Hopkins-Miller theorem to the construction of $\mathcal{O}_{\mathcal{M}^{der}}$ (and hence of [[tmf]]) is in

* {#Behrens13} [[Mark Behrens]], _Notes on the construction of $tmf$_, 2013 ([pdf](http://math.mit.edu/~mbehrens/papers/buildTMF.pdf))

An exposition with indications of further developments in [[(infinity,1)-category theory]] is in

* [[Aaron Mazel-Gee]], _Goerss-Hopkins obstruction theory for $\infty$-categories_, November 2013 ([pdf](http://math.berkeley.edu/~aaron/writing/thursday-cghost-beamer.pdf))

See also at _[survey of elliptic curves -- Gluing of curves to a spectrum](A+Survey+of+Elliptic+Cohomology#gluingallellipticcohomologytheoriestothetmfspectrum)_

A more abstract formulation of this is sketched in

*  [[Jacob Lurie]], _[[A Survey of Elliptic Cohomology]]_

with technical details in a sequence of articles, including

* [[Jacob Lurie]], _[[Formal Moduli Problems and DG-Lie Algebras|Moduli problems for ring spectra]]_ ([pdf](http://www.math.harvard.edu/~lurie/papers/moduli.pdf))

* [[Jacob Lurie]], _[[Representability Theorems]]_

[[!redirects Goerss-Hopkins-Miller-Lurie theorem]]

[[!redirects Hopkins-Miller theorem]]
[[!redirects Goerss-Hopkins theorem]]

[[!redirects Goerss-Hopkins-Miller-Lurie theorem]]


Goldbach conjecture

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

#Contents#
* table of contents
{:toc}

## Idea

In [[number theory]], the _Goldbach conjecture_ is the following [[conjecture]].

\begin{conjecture} Every [[even number]] greater than 2 is the [[sum]] of two [[prime numbers]]. \end{conjecture}

## Properties

### Relation to Bertrand's postulate

It was observed in [Ricardo 05](#Ricardo05) that the Goldbach conjecture implies a simple proof of [[Bertrand's postulate]] (which has been proven, independently of the Goldbach conjecture). See [there](Bertrand's+postulate#ProofFromGoldbach) for details.

## References

* {#Ricardo05} Henry J. Ricardo, _Goldbach's conjecture implies Bertrand's postulate_, Amer. Math. Monthly, 112, pg. 492, 2005

See also

* Wikipedia, _[Goldbach's conjecture](https://en.wikipedia.org/wiki/Goldbach%27s_conjecture)_

[[!redirects Goldbach's conjecture]]


Goldblatt-Thomason theorem

__Goldblatt-Thomason theorem__ is a theorem in [[modal logic]] which characterizes the elementary classes which are also modally definable classes. It says that if $K$ is an elementary class of Kripke frames, then $K$ is modally definable iff $K$ is closed under taking bounded morphic images, generated subframes, disjoint unions, and reflects ultrafilter extensions.

## Idea ##
The theorem is based on the duality between the category of [[Boolean algebras]] and Sets

$$
BA \stackrel{\overset{\Pi}{\leftarrow}}{\underset{\Sigma}{\to}} Set^{op}
$$

where $\Pi$ is the power set functor and $\Sigma$ assigns to a Boolean algebra the set of its [[ultrafilters]]. $\Sigma$ is left-adjoint to $\Pi$, but in general they do not form a dual equivalence of categories. This only holds in the finite case, where it is enough for $\Sigma$ to assign to a Boolean algebra the set of its atoms.

The idea of the proof is to translate between [[Kripke frames]] and Modal algebras, and obtain the above result by an application of [[Birkhoff's HSP theorem]].

## References ##

* _Brief summary of Goldblatt-Thomason theorem_, [pdf](https://users.soe.ucsc.edu/~btencate/esslli08course/esslli-fri1.pdf)

* _The Goldblatt-Thomason Theorem for Coalgebras_, [pdf](http://www.cs.le.ac.uk/people/akurz/Papers/CALCO-07/GTthm.pdf)

category: logic


golden ratio

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

#Contents#
* table of contents
{:toc}

## Idea

The [[irrational number]]

$$
  \phi \;\coloneqq\;
  \tfrac{1}{2}
  \left(
    1 + \sqrt{5}
  \right)
$$

is called the _golden ratio_ (sometimes the *golden mean*).

The golden ratio is an [[algebraic integer]] and generates the ring of algebraic integers in the quadratic subfield $\mathbb{Q}(\sqrt{5})$ of the [[cyclotomic field]] $\mathbb{Q}(\zeta)$ of fifth [[roots of unity]]. It also generates the [[group of units]] (modulo its [[torsion subgroup]]) of the ring of algebraic integers $\mathbb{Z}[\phi]$.

The golden ratio is of course an [[irrational number]], and is distinguished among irrational numbers by having the slowest [[convergence of a sequence|convergence]] rate of rational approximants obtained by truncating its [[continued fraction]] expansion

$$\phi = 1 + \frac1{1 + \frac1{1 + \frac1{1 + \ldots}}}$$

Its rational approximants $p/q$ are ratios of successive [[Fibonacci numbers]].

It is widely believed that this slowest rate of convergence explains observations of phyllotaxis in botany, i.e., the arrangement of leaves and plant structures such as the scales of pinecones and pineapples, in which nascent leaves or buds tend to form at the least crowded available spots on the growing plant, whereupon in the ensuing jostling the structures become arranged as "Fermat spirals" with a Fibonacci number of structures along each spiral.

## Related entries

* [[e]]

* [[pi]]

* [[binary icosahedral group]]

## References

See also

* Wikipedia, _[Golden ratio](https://en.wikipedia.org/wiki/Golden_ratio)_


Goldman bracket

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

#Contents#
* table of contents
{:toc}

## Idea

The _Goldman bracket_ of a [[compact space|compact]] [[closed manifold|closed]] [[surface]] $\Sigma$ is a [[Lie algebra]] structure on the [[free abelian group]] generated from the [[isotopy]] classes of based loops in $\Sigma$.

Equivalently, the Goldman bracket on $\Sigma$ is a structure on the 0th [[homology]] $H_0(L \Sigma)$ of the [[free loop space]] of $\Sigma$. It is in fact just the lowest degree of the [[string topology]] operations on $\Sigma$. See there for more details.

## Definition

Let $\Sigma$ be a compact closed and [[orientation|oriented]] surface ([[manifold]] of [[dimension]] 2). For $\gamma : S^1 \to \Sigma$ a [[continuous function]] from the based [[circle]], write $[\gamma]$ for the corresponding [[isotopy]] class.

For $[\gamma_1]$ and $[\gamma_2]$ two such classes, one can always find differentiable representatives $\gamma_1$ and $\gamma_2$ that intersect - if they intersect at some point $p$ - [[transversal map|transversally]]. Write $\gamma_1 \ast_p \gamma_2$ for the curve obtained by starting at the intersection point $p$, traversing along $\gamma_1$ back to that point and then along $\gamma_2$.

The **Goldman bracket** on the free abelian group on classes $[\gamma]$ is defined by

$$
  \left\{
    [\gamma_1], [\gamma_2]
  \right\}
  :=
  \sum_{p \in \gamma_1 \cap \gamma_2}
  sgn(p) [\gamma_1 \ast_p \gamma_2]
  \,,
$$

where $sgn(p)$ is +1 if $T_p \gamma_1, T_p \gamma_2$ is an oriented [[basis]] of the [[tangent space]] $T_p \Sigma$, and -1 otherwise.

## References

The original definition is due to

* W. Goldman, _Invariant functions on Lie groups and Hamiltonian flows of surface group representations_ ,
Invent. Math. (1986), no. 85, 263302.

The relation to [[string topology]] is due to

* [[Moira Chas]], [[Dennis Sullivan]], _String topology_, Ann. Math. [math.GT/9911159](http://arxiv.org/abs/math/9911159)


#Contents#
* table of contents
{:toc}

## Idea

Let $X$ be a smooth [[complex manifold]] with trivial [[canonical bundle]], and let $A^{0,*}(T_X)$ be the [[differential graded Lie algebra]] of smooth [[differential form]]s of type $(0,*)$ with coefficients in the [[tangent sheaf]] of $X$. The Goldman-Millson quasi-abelianity theorem [GM90](#citegoldman-millson) states that the [[differential graded Lie algebra|dgla]] $A^{0,*}(T_X)$ is quasi-abelian, i.e., it is [[quasi-isomorphism|quasi-isomorphic]] to an [[abelian differential graded Lie algebra|abelian dgla]].

Since the dgla $A^{0,*}(T_X)$ is a [[model]] for the [[derived functor|derived]] [[global section]]s $R\Gamma\mathcal{T}_X$ of the tangent sheaf $\mathcal{T}_X$ of $X$, the Goldman-Millson quasi-abelianity theorem can be stated by saying that if $X$ be a smooth complex manifold with trivial canonical bundle, then $R\Gamma\mathcal{T}_X$ is quasi-abelian. In this form the Goldman-Millson theorem can be generalized to smooth [[projective manifold]]s over an arbitrary [[characteristic]] zero [[algebraically closed
field]] $\mathbb{K}$, as done by Iacono and Manetti in [IM10](#citeiacono-manetti).

A description of the Goldman-Millson quasi-abelianity theorem within the framework of [[higher category theory]] is given in [FM10](#citefiorenza-martinengo).

## References ##

* {: #citegoldman-millson }
W. M. Goldman, J. J. Millson. _The [[nLab:homotopy]] invariance of the Kuranishi space._ Illinois J.
Math. 34 (1990) 337-367.

* {: #citeiacono-manetti }
D. Iacono, M. Manetti._An algebraic proof of Bogomolov-Tian-Todorov theorem_ Deformation Spaces. Vol. 39 (2010), p. 113-133; [arXiv:0902.0732](http://arxiv.org/abs/0902.0732)

* {: #citefiorenza-martinengo }
D. Fiorenza, E. Martinengo. _A short note on ∞-groupoids and the period map for projective manifolds_ Publications of the nLab. Vol. 2 (2012); [arXiv:0911.3845](http://arxiv.org/abs/0911.3845)


Goldstone boson

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Fields and quanta
+--{: .hide}
[[!include fields and quanta - table]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

(...)

## Examples

* [[mesons]] are the [[Goldstone bosons]] of [[chiral symmetry breaking]] in the [[effective field theory|effective]] [[chiral perturbation theory]] of [[QCD]]

## Related concepts

* [[spontaneous symmetry breaking]]

* [[hidden local symmetry]]

## References

Named after [[Jeffrey Goldstone]].

See also

* Wikipedia, _[Goldstone boson](https://en.wikipedia.org/wiki/Goldstone_boson)_

[[!redirects Goldstone bosons]]


Goncalo Tabuada

**Gonçalo Tabuada** is a mathematician with main results in the [[homotopy theory]] of [[dg-categories]], [[noncommutative motives]], and [[K-theory]]. His Ph.D. advisor was [[Bernhard Keller]].

* [web](http://math.mit.edu/~tabuada/)

##Selected publications and preprints

* (with [[Ivo Dell'Ambrogio]]), _A Quillen model structure for classical Morita theory and a tensor categorification of the Brauer group_ , arXiv:1211.2309 (2012). ([pdf](http://arxiv.org/pdf/1211.2309v1))

* (with [[Ivo Dell'Ambrogio]]), Morita homotopy theory of $C^*$-categories, Journal of Algebra, 398 (2014), 162-199

On [[noncommutative motives]]:

* [[Goncalo Tabuada]], _A guided tour through the garden of noncommutative motives_, in Guillermo Cortinas, _Topics in Noncommutative Geometry_ Clay Mathematics Proceedings Vol 16, 2012 ([arxiv1108.3787](http://arxiv.org/abs/1108.3787));

in relation to [[algebraic K-theory]]:

* {#BlumbergGepnerTabuada10} [[Andrew Blumberg]], [[David Gepner]], [[Gonçalo Tabuada]], _A universal characterization of higher algebraic K-theory_,  Geometry and Topology 17 (2013) 733–838 ([arXiv:1001.2282](http://arxiv.org/abs/1001.2282))

category: people

[[!redirects Gonçalo Tabuada]]
[[!redirects G. Tabuada]]
[[!redirects Tabuada]]


Gonzalo E. Reyes

[[!redirects G. E. Reyes]]
[[!redirects Gonzalo Reyes]]

* [blog](https://reyes-reyes.com/)

## Selected writings

On [[categorical logic]] and [[forcing]] via [[classifying toposes]]:

* (with [[Michael Makkai]]), _First Order Categorical Logic_ , LNM 611, Springer Berlin 1977.

On [[synthetic differential geometry]]:

* (with [[Ieke Moerdijk]]), _[[Models for Smooth Infinitesimal Analysis]]_ , Springer Berlin 1991.

  > (on [[synthetic differential geometry]])

An introduction to [[topos theory]] with emphasis on [[presheaf toposes]]:

* [[Marie La Palme Reyes]], [[Gonzalo E. Reyes]], [[Houman Zolfaghari]], _Generic figures and their glueings.  A constructive approach to functor categories_, Polimetrica (2008) [[pdf](https://marieetgonzalo.files.wordpress.com/2004/06/generic-figures.pdf), ISBN:8876990046]

On the history of [[categorical logic]]:

* {#MarquisReyes09} [[Jean-Pierre Marquis]], [[Gonzalo Reyes]], (2009) _The History of Categorical Logic 1963-1977 ([pdf](https://www.webdepot.umontreal.ca/Usagers/marquisj/MonDepotPublic/HistofCatLog.pdf), [[MarquisReyes_CategoricalLogic.pdf:file]])

On the early history of [[topos theory]]:

* [[Gonzalo Reyes]], _Topos Theory in Montréal in the 1970s: My Personal Involvement_,  History and Philosophy of Logic Volume 40, 2019 - Issue 4 (2019) ([doi:10.1080/01445340.2018.1554471](https://doi.org/10.1080/01445340.2018.1554471))

## Related $n$Lab entries

* [[Marie La Palme Reyes]]

* [[Models for Smooth Infinitesimal Analysis]]

* [[syllogism]]

category: people


good monoidal (∞,1)-category

A monoidal (∞, 1)-category $S$ is **good** if $S$ admits small sifted colimits, and the tensor product functor $\otimes: S \times S \to S$ preserves small sifted colimits.

This is ([Lurie 09, def. 4.1.7](#Lurie09)). The terminology is explained in [[Higher Topos Theory]].

## References

* {#Lurie09} [[Jacob Lurie]], section 4.1 of _[[On the Classification of Topological Field Theories]]_ ([arXiv:0905.0465](http://arxiv.org/abs/0905.0465))


good open cover

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

#Contents#
* table of contents
{:toc}

## Definition

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

A [[cover]] $\{U_i \to X\}$ of a [[topological space]] $X$ is called a **good cover** -- or **good open cover** if it is

1. an [[open cover]];

1. such that all the $U_i$ and all their inhabited finite intersections are [[contractible topological spaces]].

=--

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

For $X$ a [[topological manifold]] one often requires that the inhabited finite intersections are [[homeomorphism|homeomorphic]] to an [[open ball]]. Similarly, for $X$ a [[smooth manifold]] one often requires that the finite inhabited intersections are [[diffeomorphism|diffeomorphic]] to an [[open ball]].

In the literature this is traditionally glossed over, but this is in fact a subtle point, see the discussion at [[open ball]] and see below at _[Existence on paracompact smooth manifolds](#ExistenceOnParacompactManifolds)_.

=--

Due to this subtlety, it is instructive to make explicit the following definition:

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

Given a [[smooth manifold]] $X$ a _differentiably good open cover_ ([FSS 2010, Def. 6.3.9](#FSS10)) is a good open cover one all whose finite non-empty intersections are in fact [[diffeomorphism|diffeomorphic]] to an [[open ball]], hence to a [[Cartesian space]].

=--

## Properties {#Properties}

### Existence on paracompact smooth manifolds
 {#ExistenceOnParacompactManifolds}

\begin{remark}
\label{LiteratureOnExistenceOfDifferentiablyGoodOpenCovers}
The existence result (Prop. \ref{ExistenceOfDifferentiablyGoodOpenCovers} below) of differentiably good open covers (Def. \ref{DifferentiablyGoodOpenCover}) on smooth manifolds seems to have been a [[folk theorem]], based in turn on the [[folk theorem]] that every [[geodesic convexity|geodesically convex]] [[open neighbourhood]] in a [[Riemannian manifold]] is *[[diffeomorphic]]* (as opposed to just [[homeomorphic]]) to an [[open ball]], hence to a [[Cartesian space]].

For instance, the latter fact is asserted without proof or reference inside the proof of Theorem 5.1 of [Bott & Tu 1982](#BottTu82).

Actual proofs that [[star-shaped subsets]] of [[Euclidean space]] are [[diffeomorphism|diffeomorphic]] to [[open balls]]/[[Cartesian spaces]] have then been made explicit in [Gonnord & Tosel 1998, p. 60](#GonnordTosel98) and  [Ferus 2007, Thm. 237](#Ferus07).

(These proofs had remained obscure: In the textbooks [Conlon 2008](ball#Conlon08) and [Lee 2009](ball#Lee09) the statement is still referred to as lacking a citable proof: Conlon writes that a proof is "extremely difficult", while Lee writes that "references to a complete proof are hard to find", see also [this remark](ball#LiteratureOnStarShapedOpenDiffeoToOpenBall) at _[[open  ball]]_.)

That this argument completes the existence proof of differentiably good open covers has  made explicit in [FSS 2010, Prop. A.1](#FSS10). In [Guillemin & Haine 19 this is Thm. 5.3.2](GuilleminHaine19), where the crucial step is left as Exercise 5.3, also App. C. See also [Demailly 2012, Lem. IV 6.9](#Demailly12).
\end{remark}

\begin{proposition}
\label{ExistenceOfDifferentiablyGoodOpenCovers}
**(existence of differentiably good open covers)**
\linebreak
Every [[paracompact manifold|paracompact]] [[smooth manifold]] admits a good open cover, def. \ref{GoodOpenCover}, in fact
a differentiable good open cover, def. \ref{DifferentiablyGoodOpenCover}
\end{proposition}

The following proof of Prop. \ref{ExistenceOfDifferentiablyGoodOpenCovers} follows that given in [FSS 2010, Prop. A.1](#FSS10):

\begin{proof}
By ([Greene](#Greene)) every paracompact smooth manifold admits a [[Riemannian metric]] with positive [[convexity radius]] $r_{\mathrm{conv}} \in \mathbb{R}$. Choose such a metric and choose an [[open cover]] consisting for each point $p\in X$ of the geodesically convex open subset $U_p := B_p(r_{conv})$ given by the geodesic $r_{conv}$-ball at $p$. Since the [[injectivity radius]] of any metric is at least $2r_{\mathrm{conv}}$ it follows from the minimality of the geodesics in a geodesically convex region that inside every finite nonempty intersection $U_{p_1} \cap \cdots \cap U_{p_n}$ the [[geodesic flow]] around any point $u$ is of radius less than or equal the injectivity radius and is therefore a diffeomorphism onto its image.

Moreover, the [[preimage]] of the intersection region under the geometric flow is a [[star-shaped]] region in the [[tangent space]] $T_u X$: because the intersection of geodesically convex regions is itself geodesically convex, so that for any $v \in T_u X$ with $\exp(v) \in U_{p_1} \cap \cdots \cap U_{p_n}$ the whole geodesic segment $t \mapsto \exp(t v)$ for $t \in [0,1]$ is also in the region.

So we have that every finite non-empty intersection of the $U_p$ is diffeomorphic to a star-shaped region in a vector space. By the results cited at [[ball]] (e.g. theorem 237 of ([Ferus 2007](#Ferus07))) this star-shaped region is diffeomorphic to an $\mathbb{R}^n$.
\end{proof}

Here is another proof of Prop. \ref{ExistenceOfDifferentiablyGoodOpenCovers}:

\begin{proof}
Every [[paracompact manifold|paracompact]] [[smooth manifold]] admits a [[Riemannian metric]], and for any point in a [[Riemannian manifold]] there is a [[geodesically convex]] [[neighborhood]] (any two points in the neighborhood are connected by a unique geodesic in the neighborhood, one whose length is the distance between the points; see for example the remark after ([Milnor, lemma 10.3 on page 59](#Milnor)), or ([do Carmo, Proposition 4.2](#doCarmo))).
A nonempty intersection of finitely many such geodesically convex neighborhoods is also geodesically convex.
The inverse of the [[exponential map]] based at any
interior point of a geodesically convex open subset
gives a [[diffeomorphism]] from this subset to a star-shaped
open subset of $\mathbf{R}^n$.
Indeed, the [[Gauss lemma]] shows that the tangent
map of the exponential map is invertible.
By definition of geodesic convexity the exponential
map is injective, hence a diffeomorphism.
By [this theorem](open+ball#StarShapedOpenDiffeomorphicToOpenBall), star-shaped open subsets
of $\mathbf{R}^n$ are diffeomorphic to $\mathbf{R}^n$,
which completes the proof.
\end{proof}

### Coverages of good open covers

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

The [[category]] $ParaSmMfd$ of [[paracompact topological space|paracompact]] [[smooth manifolds]] admits a [[coverage]] whose covering families are good open covers.

The same holds true for [[full subcategories]] such as

* [[CartSp]] -- [[Cartesian spaces]].

=--

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

It is sufficient to check this in $ParaSmMfd$.
We need to check that for $\{U_i \to U\}$ a good open cover and $f : V \to U$ any morphism, we get commuting squares

$$
  \array{
     V_j &\to& U_{i(j)}
     \\
     \downarrow && \downarrow
     \\
     V &\stackrel{f}{\to}& U
  }
$$

such that the $\{V_i \to V\}$ form a good open cover of $V$.

Now, while $ParaSmMfd$ does not have all [[pullbacks]], the pullback  of an [[open cover]] does exist, and since $f$ is necessarily a [[continuous function]] this is an [[open cover]] $\{f^* U_i \to V\}$. The $f^* U_i$ need not be contractible, but being open subsets of a paracompact manifold, they are themselves paracompact manifolds and hence admit themselves good open covers $\{W_{i,j} \to f^* U_i\}$.

Then the family of composites $\{W_{i,j} \to f^* U_i \to V\}$ is clearly a good open cover of $V$.

=--

### Existence on CW complexes

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

Every finite [[CW complex]] admits a good open cover.

=--

Hopefully someone can find a clear reference to a proof. The assertion for finite CW complexes is found for example [here](https://books.google.com/books?id=dL8FCAAAQBAJ&pg=PA37&lpg=PA37&dq=%22good+cover%22+%22CW+complex%22&source=bl&ots=GrZW-ZGSpU&sig=eLwTeBNnBYJuEYBJ-6JCPTPvw1g&hl=en&sa=X&ved=0CDgQ6AEwBGoVChMIlJCJzIOGyAIVCmg-Ch2FQwlj#v=onepage&q=%22good%20cover%22%20%22CW%20complex%22&f=false) (Topology of Tiling Spaces by Sadun, p. 37). It is not immediately clear from the remarks there what obstructions would exist to generalizing the assertion to all CW complexes.

As indicated at [[CW complex]], every CW complex is *homotopy equivalent* to a simplicial complex, and simplicial complexes certainly admit good covers by taking open stars.

### (Non-)Existence for topological manifolds

For a (paracompact) [[topological manifold]] the construction via [[Riemannian metrics]] or similar smooth constructions in general does not work.

In ([Osborne-Stern 69](#OsborneStern69)) the following discussion for sufficient conditions getting "close" to good open covers is discussed:

Let $X$ be a [[n-connected space|k-connected]] [[topological manifold]] of [[dimension]] $n$ (without [[boundary]]), and define

$$
  q \coloneqq
  min(k,n-3)
  \,.
$$

For $p \in \mathbb{N}$ such that $p(q+1) \gt n$ then $X$ admits a cover by $p$ open balls and such that all nonempty intersections of the covering cells are [[n-connected space|(q−1)-connected]].

### Refining covers

A cover $\{U_i\to X\}_{i\in I}$ refines another cover $\{V_j\to X\}_{j\in J}$ if each map $V_j\to X$ is some $U_i\to X$.

Each differentially good cover has a unique smallest refinement to a differentially good cover that is closed under intersection.

## $n$POV

The following [[nPOV]] perspective on good open covers gives a useful general "explanation" for their relevance, which also explains the role of good covers in [[Cech cohomology]] generally and [[abelian sheaf cohomology]]  in particular.

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

Let $sPSh(CartSp)_{proj}$ be the category of [[simplicial presheaves]] on the category [[CartSp]] equipped with the projective [[model structure on simplicial presheaves]].

Let $X$ be a [[smooth manifold]], regarded as a 0-[[truncated]] object of $sPSh(C)$.

Let $\{U_i \to X\}$ be a good open cover by [[open ball]]s in the strong sense: such that every finite non-empty intersection is diffeomorphic to an $\mathbb{R}^d$.

Then: the [[Cech nerve]] $C(\{U\}) \in sPSh(C)$ is a cofibrant resolution of $X$ in the [[local model structure on simplicial presheaves]].

=--

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

By assumption we have that $C(U)$ is degreewise a [[coproduct]] of [[representable functor|representables]]. It is also evidently a [[split hypercover]].

This implies the statement by the [characterization of cofibrant objects in the projective structure](http://ncatlab.org/nlab/show/model+structure+on+simplicial+presheaves#CofibrantObjects).

=--

This has a useful application in the [[nerve theorem]].

Notice that the [[descent]] condition for simplicial presheaves on [[CartSp]] at (good) covers is very weak, since all [[Cartesian space]]s are topologically contractible, so it is easy to find the fibrant objects $A \in sPSh(C)_{proj, loc}$ in the [[topological localization]] of $sPSh(C)_{proj}$ for the canonical [[coverage]] of [[CartSp]]. The above observation says that for computing the $A$-valued [[cohomology]] of a [[diffeological space]] $X$, it is sufficient to evaluate $A$ on (the Cech nerve of) a good cover of $X$.

We can turn this around and speak for any [[site]] $C$ of a covering family $\{U_i \to X\}$ as being _good_ if the corresponding [[Cech nerve]] is degreewise a coproduct of representables. In the projective model structure on simplicial presheaves on $C$ such good covers will enjoy the central properties of good covers of topological spaces.

## Related concepts

* [[equivariant good open cover]]

* [good covers by Stein manifolds](Stein+manifold#GoodCoversBySteinManifolds) - note that this is a different concept, with vanishing Dolbeault cohomology replacing contractibility.

* [[quasicompact quasiseparated scheme]]: schemes that admit a finite cover by affine opens such that the intersection of any two elements is itself covered by finitely many affine opens.

## References
 {#References}

Proof of existence of differentiably good open covers (Def. \ref{DifferentiablyGoodOpenCover}) of smooth manifolds:

* {#BottTu82} [[Raoul Bott]], [[Loring Tu]], Thm. 5.1 in: *[[Differential Forms in Algebraic Topology]]*, Graduate Texts in Mathematics 82 Springer 1982 ([doi:10.1007/978-1-4757-3951-0](https://link.springer.com/book/10.1007/978-1-4757-3951-0), [pdf](http://www.maths.ed.ac.uk/~aar/papers/botttu.pdf))

* {#FSS10} [[Domenico Fiorenza]], [[Urs Schreiber]], [[Jim Stasheff]], Prop. A.1 of:  *[[schreiber:Cech cocycles for differential characteristic classes]]*, Advances in Theoretical and Mathematical Physics, **16** 1 (2012) 149-250 ([arXiv:1011.4735](http://arxiv.org/abs/1011.4735), [euclid:1358950853](http://projecteuclid.org/euclid.atmp/1358950853), [doi:10.1007/BF02104916](https://doi.org/10.1007/BF02104916))

* {#Demailly12} [[Jean-Pierre Demailly]], Lemma IV.6.9 of: _Complex Analytic and Differential Geometry_, 2012 ([pdf](https://www-fourier.ujf-grenoble.fr/~demailly/manuscripts/agbook.pdf))

* {#GuilleminHaine19} [[Victor Guillemin]], [[Peter Haine]], Thm. 5.3.2 and Appendix C of: _Differential Forms_, World Scientific (2019) ([doi:10.1142/11058](https://doi.org/10.1142/11058))

These proofs require one to show that [[star-shaped subsets]] of [[Cartesian space]]/[[Euclidean space]] $\mathbf{R}^n$ are [[diffeomorphism|diffeomorphic]] to $\mathbf{R}^n$ (see the article *[[ball]]* for details, and see the references [there](ball#ReferencesStarShapedReasonDiffeomorphicToOpenBall)).

Such proofs have been given in:

* {#GonnordTosel98} Stéphane Gonnord, Nicolas Tosel, page 60 of: _Calcul Différentiel_, ellipses (1998) (English translation: [MO:a/212595](http://mathoverflow.net/a/212595), [[Aubry_reproducing_GonnordAndTosel.pdf:file]])

* {#Ferus07} [[Dirk Ferus]], [theorem 237](http://www.math.tu-berlin.de/~ferus/ANA/Ana3.pdf#page=154) in: _Analysis III_ (2007) ([pdf](http://www.math.tu-berlin.de/~ferus/ANA/Ana3.pdf), [[Ferus_AnalysisIII.pdf:file]])

Other references on good open covers:

* {#doCarmo} [[Manfredo do Carmo]], _Riemannian geometry_ (trans. Francis Flaherty), Birkhäuser (1992)

* {#Milnor} [[John Milnor]], _Morse theory_ , Princeton University Press (1963)

* {#Greene} R. Greene, _Complete metrics of bounded curvature on noncompact manifolds_  Archiv der Mathematik Volume 31, Number 1 (1978)

* {#OsborneStern69} RP Osborne and JL Stern. _Covering Manifolds with Cells_, Pacific Journal of Mathematics, Vol 30, No. 1, 1969.

* MathOverflow, _[Proving the existence of good covers](http://mathoverflow.net/q/102161/381)_

[[!redirects good cover]]
[[!redirects good covers]]
[[!redirects good open covers]]

[[!redirects good covering]]
[[!redirects good coverings]]

[[!redirects differentiably good open cover]]
[[!redirects differentiably good open covers]]


good orbifold

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

#Contents#
* table of contents
{:toc}

## Definition

An [[orbifold]] $\mathcal{X}$ is called _good_ ([Thurston 92, Ch. 13, Def. 13.2.3](#Thurston92)) or _developable_ if it is [[isomorphism|isomorphic]] to a [[global quotient orbifold|global quotient]] of a [[smooth manifold]] $M$ by the [[action]] of a [[discrete group]] $\flat G$ (not necessarily [[finite group|finite]]):

$$
  \mathcal{X} \,\simeq\, M \sslash \flat G
$$

Otherwise $\mathcal{X}$ is called _bad_.

If $\mathcal{X}$ is even the [[global quotient orbifold|global quotient]] of a [[smooth manifold]] by a [[finite group]] [[group action|action]] one says that it is _very good_.

In the other direction, an orbifold that is the [[global quotient orbifold|global quotient]] of a [[smooth manifold]] by some ([[compact Lie group|compact]]) [[Lie group]]  is called a *[[presentable orbifold]]*.

## References

* {#Thurston92} [[William Thurston]], _Three-dimensional geometry and topology,_ preliminary draft, University of Minnesota, Minnesota, (1992). [1979 version](https://archive.org/details/ThurstonTheGeometryAndTopologyOfThreeManifolds/mode/2up), [scan of 1991 copy](https://math.berkeley.edu/~ianagol/276.S10/bookdraft.pdf) (note the pages are upside down and need to be rotated after downloading), [pdf of 2002 version](https://www.math.unl.edu/~jkettinger2/thurston.pdf).

The first three chapters are published in expanded form as

* _The Geometry and Topology of Three-Manifolds_ (1997) [Princeton University Press page](https://press.princeton.edu/books/hardcover/9780691083049/three-dimensional-geometry-and-topology-volume-1)

The book has its own [Wikipedia page](https://en.wikipedia.org/wiki/The_geometry_and_topology_of_three-manifolds).

[[!redirects good orbifolds]]

[[!redirects bad orbifold]]
[[!redirects bad orbifolds]]

[[!redirects very good orbifold]]
[[!redirects very good orbifolds]]


Goodwillie calculus

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### $(\infty,1)$-Category theory
+--{: .hide}
[[!include quasi-category theory contents]]
=--
#### Stable Homotopy theory
+--{: .hide}
[[!include stable homotopy theory - contents]]
=--
#### Goodwillie calculus
+--{: .hide}
[[!include Goodwillie calculus - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

Goodwillie calculus is an [[(∞,1)-category theory|(∞,1)-]][[categorification]] of [[differential calculus]], where [[smooth functions]] between [[Cartesian spaces]] are replaced by [[topological functors]] between [[categories]] of [[pointed topological spaces]].

### As an approximation to stabilization of a functor

The operation of [[stabilization]] that sends an [[(∞,1)-category]] $C$ to the [[stable (∞,1)-category]] $Stab(C)$ does not in general extend to a functor.

We may think of this operation as the analog of _linearizing_ a space. Turning an [[(∞,1)-functor]] $F \colon C \to D$ into a functor $Stab(C) \to Stab(D)$ is not unlike performing a first order [[Taylor expansion]] of a function, whence one speaks here of [[Goodwillie-Taylor towers]].

This is what Goodwillie calculus studies.

Let $F: \mathcal{C} \to \mathcal{D}$ (where $\mathcal{C}$ and $\mathcal{D}$ are each either $Top_*$, the category of [[pointed topological spaces]], or $Spec$, the category of [[spectra]]) be a pointed homotopy functor. Associate with $F$ a sequence of spectra, called the derivatives of $F$, denoted by $\partial_1 F, \partial_2 F,\cdots, \partial_n F, \cdots$, or, collectively, by $\partial_* F$. For each $n$ the spectrum $\partial_n F$ has a natural [[action]] of the [[symmetric group]] $\Sigma_n$. Thus, $\partial_\bullet F$ is a [[symmetric sequence]] of spectra.

The Goodwillie-derivatives of $F$ contain substantial information about the homotopy type of $F$. We can form a sequence of 'approximations' to $F$ together with natural transformations forming a *[[Goodwillie-Taylor tower]]*. This tower takes the form

$$
F \to \cdots \to P_n F \to P_{n-1} F \to \cdots\to P_0 F
$$

with $P_n F$ being the universal *n-excisive* approximation to $F$. (A functor is [[n-excisive (∞,1)-functor|n-excisive]] if it takes any $n + 1$-dimensional cube with homotopy pushout squares for faces to a homotopy cartesian cube.) For '[[analytic (∞,1)-functors]]' $F$, this tower converges for sufficiently highly connected $X$, that is

$$
F(X) \simeq \underset{n}{holim} P_n F(X).
$$

The fibre $D_n F$ of the map $P_n F \to P_{n-1} F$ is an *n-homogeneous* functor in an appropriate sense, and is determined by $\partial_n F$, via the following formula. If $F$ takes values in $Spec$ then

$$
D_n F(X) \simeq (\partial_n F \wedge X^{\wedge n})_{h \Sigma_n}.
$$

If $F$ takes values in $Top_*$ then one needs to prefix the right hand side with $\Omega^{\infty}$. (Arone & Ching)

### Analogy between homotopy theory and calculus {#Analogy}

Here is an overview of the relation between [[homotopy theory]]/[[∞-groupoid]] theory and [[algebraic approaches to differential calculus|differential calculus]] that is the starting point for Goodwillie calculus.

> based on a message by [[André Joyal]] to the Category Theory Mailing list, May 12, 2010

Write $k[ [x] ]$ for the [[ring]] of [[formal power series]] in one
variable over a [[field]] $k$. The ring $k[ [x] ]$ bears some resemblances
with the category of [[pointed object|pointed]] [[homotopy type]]s (= pointed spaces up to [[weak homotopy equivalence]]s). The category of pointed
homotopy types is a ring (the product is the [[smash product]]
and the sum is the [[wedge sum]]).

The following dictionary indicates what the correspondence between the two subjects is.

* $k$ $\stackrel{\text{corresponds to}}{\mapsto}$ the category of [[pointed set]]s;

* $k[ [x] ]$
  $\mapsto$
  the category of pointed [[homotopy type]]s;

* $x$ $\mapsto$ the pointed circle;

* the augmentation $(k[ [x] ] \to k)$
  $\mapsto$
  the [[homotopy group|connected components functor]]
  $\pi_0$ :  pointed homotopy types $\to$ pointed sets

* the augmentation ideal $J$
  $\mapsto$
  the [[subcategory]] of pointed [[connected]] spaces;

* the $n+1$ power of the augmentation ideal
  $J^{n+1}$
  $\mapsto$
  the subcategory of pointed $n$-[[connected]] spaces;

* the product of an element in $J^{n+1}$
  with an element of $J^{m+1}$ is an element of
  $J^{n+m+2}$
  $\mapsto$
  the [[smash product]] of an $n$-[[connected]] space
  with a $m$-connected space is $(n+m+1)$-connected;

* multiplication by $x$ $\mapsto$ the [[suspension]] functor.

* division by $x$ $\mapsto$ the [[loop space]] functor;

  Notice here the difference: the loop functor is
  [[right adjoint]] to the suspension functor, not its [[inverse]].
  Moreover, the loop space of a space has a special structure
  (it is a [[group object in an (infinity,1)-category|group]]).

* the ideal $J=x k[ [x] ]$ is [[isomorphism|isomorphic]]
  to $k[ [x] ]$ via division by $x$
  $\mapsto$
  similarly, the category of pointed connected spaces is equivalent
  to the category of topological groups via the loop space
  functor (it is actually an [[Quillen equivalence]]
  of [[model categories]]).

* More generally, the ideal $J^{n+1}$ is isomorphic to
  $k[ [x] ]$ via division by $x^{n+1}$.
  $\mapsto$
  similarly, the category of $n$-connected spaces is
  equivalent to the category of $(n+1)$-fold
  [[topological group]]s
  (it is actually an [[Quillen equivalence]] of
  [[model categories]]) via the $(n+1)$-fold loop space functor.

* the quotient ring $k[ [x] ]/J^{n+1}$ $\mapsto$
  the category of $n$-truncated homotopy types
  (=[[homotopy n-type]]s)

* The sequence of approximations of a
  formal power series $f(x)=a_0+a_1x+ \cdots $

  $a_0$

  $a_0+a_1 x$

  $a_0+a_1 x + a_2 x^2$

  $\cdots$

  $\mapsto$

  the [[Postnikov tower]] of a pointed homotopy type $X$:

  $[\pi_0(X)]$

  $[\pi_0(X); \pi_1(X)]$

  $[\pi_0(X); \pi_1(X); \pi_2(X)]$

  $\cdots$

  Here, $\pi_0(X)$ is the set of connected components of
  $X$, $[\pi_0(X); \pi_1(X)]$ is the [[fundamental groupoid]]
  of $X$,
  $[\pi_0(X); \pi_1(X); \pi_2(X)]$
  is the fundamental 2-groupoid of $X$, etc.

* The differences between $f(x)$ and its successives approximations

  $$
   \begin{aligned}
     R_0 = f(x)-a_0  &= a_1 x+a_2 x^2+a_3 x^3+ \cdots
     \\
     R_1 = f(x)-(a_0+a_1 x) &= a_2 x^2 + a_3 x^3 + a_4 x^4+ \cdots
     \\
     R_2 = f(x)-(a_0+a_1x+a_2x^2) &= a_3 x^3 + a_4 x^4 + a_5 x^5 +\cdots
   \end{aligned}
  $$

  $\mapsto$

  the [[Whitehead tower]] of $X$,

  $C_0=[0;\pi_1(X), \pi_2(X), \pi_3(X), \cdots]$

  $C_1=[0;0, \pi_2(X), \pi_3(X), \cdots]$

  $C_2=[0;0, 0, \pi_3(X), \cdots]$

  Here, $C_0$ is the connected component of $X$ at the base point,
  $C_1$ is the [[universal cover]] of $X$ constructed by from
  paths starting at the base point,
  $C_2$ is the universal 2-cover of $X$
  constructed from paths starting the base point, etc.

* Division by $x$ is shifting down the coefficients of a
  power series.

  If $f(x)=a_1 x+a_2 x^2 + \cdots$,
  then $f(x)/x= a_1+a_1 x^2+ \cdots $

  Similarly, the loop space functor is shifting down
  the homotopy groups
  of a pointed space:

  if $X=[a_0,a_1,a_2,...]$ then $\Omega(X)=[a_1,a_2,....]$.

  Unfortunately, the [[suspension]] functor does not shift
  up the [[homotopy group]]s of a space. It is however shifting
  the first $2n$ homotopy groups of an $n$-connected space $X$
  $(n \geq 1)$ by the
  [[Freudenthal suspension theorem]]

  For example, if $X=[0;0,a_2, a_3,...]$ then
  $\Sigma X=[0;0,0,a_2,a_3...]$,
  and if $X=[0;0,0, a_3, a_4, a_5,...]$ then
  $\Sigma X=[0;0,0, 0, a_3, a_4, a_5,...]$.

  In other words, the canonical map $X \to \Omega X$
  is a $2n$-equivalence if $X$ is $n$-[[connected]]
  $(n \geq 1)$.
  If $X[2n]$ denotes the $2n$-type of $X$
  (the $2n$-[[truncated|truncation]] of $X$),
  then we have a [[homotopy equivalence]]

  $X[2n]  \to \Omega \Sigma X[2n] \simeq \Omega \Sigma X[2n+1]$.

...

From a blog [discussion](http://golem.ph.utexas.edu/category/2008/12/smooth_structures_in_ottawa.html#c020698)

[Arone Kankaanrinta 95](#AroneKankaanrinta95) write

>The Goodwillie tower of the identity...is a tower of functors and natural transformations, which starts with stable homotopy and converges to unstable homotopy. (p. 1)

>...the Goodwillie tower is an inverse to stable homotopy in the same way as logarithm is an inverse to exponential. (p. 1)

>It is the point of this paper that the Goodwillie tower is the homotopy theoretic analog of logarithmic expansion, rather than of Taylor series. (p. 6)

What's going on, they say, is like finding a function of the form $a^{x - 1}$ which best approximates $x$. This is when $a = e$.

The functor from spaces to spaces which sends $X$ to the [[infinite loop space]] underlying its [[suspension spectrum]]

$$
\Omega^{\infty}\Sigma^{\infty} X = colim \Omega^n \Sigma^n X
$$

sends coproducts to products and is supposed to be like $e^{x - 1}$.
(The "$-1$" comes about from issues to do with basepoints.)

A homogeneous linear functor is defined to be one sending coproducts to products, so it is like an [[exponential map|exponential]].  Compared to an exponential, the [[identity functor]] is like a [[logarithm]], so it has a non-trivial Taylor series.

>...our point of view is that stable homotopy is analogous to the function $e^{x - 1}$ rather than to a linear function, and the Goodwillie tower is an infinite product, rather than an infinite sum, namely it is analogous to the product

$$
e^{x - 1} \cdot e^{-\frac{(x - 1)^2}{2}} \cdot e^{\frac{(x - 1)^3}{3}} \ldots = e^{ln(1 + (x - 1))} = x.
$$

>(p. 2)

Gregory Arone in an MO [answer](http://mathoverflow.net/questions/32071/is-the-dual-notion-of-a-presheaf-useful)

>Covariant functors from the category of pointed sets to the category of pointed topological spaces are sometimes called $\Gamma$-spaces, and they have been important in algebraic topology. One reason is that $\Gamma$-[[Gamma-space|spaces]] model infinite loop spaces (and therefore connective spectra) and are very helpful for understanding stable homotopy theory.

>$\Gamma$-spaces also serve as a model for particularly well-behaved covariant functors from the category of pointed topological spaces to itself. Of course, these functors play an important role in topology as well. I like to think of Goodwillie's Calculus of Homotopy Functors (and also of Michael Weiss's Orthogonal Calculus) as a kind of "sheaf theory for covariant functors". In these theories, covariant functors are analogous to presheaves and linear functors are analogous to sheaves (The definition of a linear functor is essentially a homotopy-invariant version of the definition of a sheaf). The process of approximating a general functor by a linear one is analogous to sheafification, and so forth. These theories provide methods for studying certain types of functors, but of course they also tell you something about the category of spaces itself.

Eric Finster's research statement

>One tantalizing aspect of the Goodwillie calculus is that it suggests the possibility of thinking geometrically about the global structure of homotopy theory. In this interpretation, the category of spectra plays the role of the tangent space to the category of spaces at the one-point space. Moreover, the identity functor from spaces to spaces is not linear...and one can interpret this as saying that spaces have some kind of non-trivial curvature.

However, Goodwillie remarks in the [report](#OWR2004) (p. 905) on a Oberwolfach meeting.:

>Rhetorical question: If the first derivative of the identity is the identity matrix, why is the second derivative not zero? Answer: Some of the terminology of homotopy calculus works better for functors from spaces to spectra than for functors from spaces to spaces. Specifically, since "linearity" means taking pushout squares to pullback squares, the identity functor is not linear and the composition of two linear functors is not linear.

>Attempted cryptic remark: Unlike the category of spectra, where pushouts are the same as pullbacks, the category of spaces may be thought of has having nonzero curvature.

>Correction: After the talk Boekstedt asked about that remark. We discussed the matter at length and found more than one connection on the category of spaces, but none that was not flat. In fact curvature is the wrong thing to look for. There are in some sense exactly two tangent connections on the category of spaces (or should we say on any model category?). Both are  flat and torsion-free. There is a map between them, so it is meaningful to subtract them. As is well-known in differential geometry, the difference between two connections is a 1-form with values in endomorphisms (whereas the curvature is a 2-form with values in endomorphisms). Thus there is a way of discussing the discrepancy between pushouts and pullbacks in the language of differential geometry, but it is a tensor field of a different type from what I had guessed.

This is from the [report](#OWR2004) (p. 905) on a Oberwolfach meeting. The table on p. 900 also makes comparisons to differential geometry.

Chris Schommer-Pries

>Any linear functor from spaces to spaces is a generalized cohomology theory. More precisely, there is a model category on the functors from spaces to spaces called the model category of W-spaces. Really I should be using pointed spaces here. This model category is one of the standard models for the category spectra and so the fibrant objects can be thought of as the (co)homology theories. The fibrant objects are precisely those functors which are linear in Goodwillie's sense. The example $S P^{\infty}$ corresponds to ordinary cohomology (well there is a $\pi_0$ issue, but let's ignore that). In general evaluating the linear functor $E$ on a space $X$ gives you a space which should be thought of as the smash product of $E$ and $X$.

>So now why should spectra/cohomology theories be thought of as linear functors? Well if you think of spectra as analogous to abelian groups, then applying a spectrum to a space (i.e. smashing with it) is a linearization of that space.

>Following this analogy, if we now have any old functor from space to spaces we can take its fibrant replacement in $W$-spaces. This is a linear functor which is the best approximation to the original functor. So it is like taking a derivative of a function. Goodwillie's insight was to extend this analogy to encompass the rudiments of calculus. There is in fact  a whole series of model categories on functors from spaces to spaces where the fibrant objects are Goodwillie's polynomial functors of degree $n$.

See also Eric Finster's blog post [Thoughts on the Goodwillie Calculus](https://web.archive.org/web/20121114044441/https://curiousreasoning.wordpress.com/2010/08/02/thoughts-on-the-goodwillie-calculus/)

## Properties

### $\infty$-Toposes of polynomial $(\infty, 1)$-functors

For each $n$, the collection of [[n-excisive (∞,1)-functors]] from bare [[homotopy types]] to bare homotopy types is an [[(infinity,1)-topos]], the _[jet topos](jet+category#JetToposes)_.

due to ( [Joyal 08, 35.5](#Joyal08), with [[Georg Biedermann]]) See also at _[[tangent (infinity,1)-category]]_, and [[Charles Rezk]], appears as ([Lurie, remark 6.1.1.11](#HigherAlg)).

## Examples

### Goodwillie derivatives of the identity functor

The Goodwillie derivatives of the [[identity functor]] on  [[pointed topological spaces]], form an [[operad]] in [[spectra]] ([Ching 05](#Ching05)).

See at _[[Goodwillie derivatives of the identity functor]]_

### Goodwillie derivatives of exponential modality
 {#GoodwillieDerivativeOfExponentialModality}

The ([[(infinity,1)-functor|$\infty$-]])functor

$$
  \Sigma^\infty \circ \Omega^\infty
  \;\colon\;
  Spectra \to Spectra
$$

[induced](monad#RelationBetweenAdjunctionsAndMonads) from the [[stabilization]]-[[adjoint (infinity,1)-functor|adjunction]] is (see [there](!-modality#RealizationInLinearHomotopyTypeTheory)) the [[exponential modality]] in [[linear homotopy type theory]] (or rather the un-pointed version is). Indeed, in Goodwillie calculus it behaves like an [[exponential function]] ("minus 1"), for instance in that

* all its higher [[Goodwillie derivatives]] are the [[unit object|unit]] [[sphere spectrum]], and

* on [[suspension spectra]] $\mathbb{X} \equiv \Sigma^\infty_+ X$ we have

  $$
    \Sigma^\infty \Omega^\infty \mathbb{X}
    \;\simeq\;
    \underset{n \geq 1}{\oplus}
    \mathbb{X}^{\wedge n} \sslash Sym(n)
    \,.
  $$

([Arone & Ching 2019, Exp. 2.6](#AroneChing19), using [Ahearn & Kuhn 2002, Cor. 1.3](#AhearnKuhn02))

### Goodwillie derivatives of mapping spaces

See at _[[stable splitting of mapping spaces]]_.

## Related concepts

* [[Goodwillie spectral sequence]]

* [[Goodwillie-differentiable (∞,1)-category]]

* [[excisive (∞,1)-functor]], [[n-excisive (∞,1)-functor]]

* [[Taylor tower]]

* [[Goodwillie differentiation]], [[Goodwillie chain rule]]

* [[Goodwillie derivative of the identity functor]]

* [[tangent (∞,1)-category]], [[tangent cohesion]]

* [[jet (∞,1)-category]]

* [[orthogonal calculus]]

* [[Weiss topology]]

* [[manifold calculus]]

## References

The original articles:

* {#Goodwillie90} [[Tom Goodwillie]], _The differential calculus of homotopy functors_, Proceedings of the International Congress of Mathematicians in Kyoto 1990, Vol. I, Math. Soc. Japan, 1991, pp. 621–6 ([article pdf](https://math.mit.edu/juvitop/old/notes/2009_Fall/goodwillie-icm.pdf), [full proceedings Vol I pdf](https://www.mathunion.org/fileadmin/ICM/Proceedings/ICM1990.2/ICM1990.2.ocr.pdf), [[GoodwillieICM1990.pdf:file]])

* [[Thomas Goodwillie]], _Calculus. I. The first derivative of pseudoisotopy theory_, K-Theory __4__ (1990), no. 1, 1-27 ([doi:10.1007/BF00534191](http://dx.doi.org/10.1007/BF00534191), MR:1076523)

* [[Thomas Goodwillie]], _Calculus. II. Analytic functors_, K-Theory __5__ (1991/92), no. 4, 295-332 ([doi:10.1007/BF00535644](http://dx.doi.org/10.1007/BF00535644), MR:1162445)

* [[Thomas Goodwillie]], _Calculus. III. Taylor series_, Geom. Topol. 7 (2003), 645--711 ([euclid:gt/1513883319](https://projecteuclid.org/euclid.gt/1513883319) [doi:10.2140/gt.2003.7.645](http://dx.doi.org/10.2140/gt.2003.7.645), [arXiv:math/0310481](http://arxiv.org/abs/math/0310481))

* {#OWR2004} The Goodwillie Calculus of Functors. Workshop Oberwolfach Report 17 (2004) [[doi:10.4171/OWR/2004/17](https://doi.org/10.14760/OWR-2004-17)]

Surveys and introductions:

* [[Brian Munson]], _Introduction to the manifold calculus of Goodwillie-Weiss_,  Morfismos, Vol . 14, No. 1, 2010, pp. 1-60 ([arXiv:1005.1698](https://arxiv.org/abs/1005.1698))

* {#MunsonVolic15} [[Brian Munson]], [[Ismar Volic]], Section 10 of: _Cubical homotopy theory_, Cambridge University Press, 2015  ([pdf](http://palmer.wellesley.edu/~ivolic/pdf/Papers/CubicalHomotopyTheory.pdf), [doi:10.1017/CBO9781139343329](https://doi.org/10.1017/CBO9781139343329))

* [[Michael Ching]], [[Nicholas Kuhn]], [[Victor Turchin]], _[Functor Calculus and Operads](http://www.birs.ca/events/2011/5-day-workshops/11w5058)_, BANFF Workshop 2011

* [[Gregory Arone]], [[Michael Ching]] _[Talbot Workshop 2012: Calculus of Functors](https://math.mit.edu/events/talbot/index.php?year=2012)_ ([Talk schedule](https://math.mit.edu/events/talbot/2012/2012TalbotTalks.pdf), [Notes](https://math.mit.edu/events/talbot/index.php?year=2012&sub=talks))

* {#AroneChing19} [[Gregory Arone]], [[Michael Ching]], *Goodwillie Calculus*, in: *[[Handbook of Homotopy Theory]]* Taylor & Francis (2019) [[arXiv:1902.00803](https://arxiv.org/abs/1902.00803),[doi:10.1201/9781351251624](https://doi.org/10.1201/9781351251624)]

A [[model category]] presentation for [[n-excisive functors]]:

* [[Georg Biedermann]], [[Boris Chorny]], Oliver Röndigs, _Calculus of functors and model categories_, Adv. Math., __214__, n. 1 (2007) 92--115, [doi](http://dx.doi.org/10.1016/j.aim.2006.10.009), [math.AT/0601221](http://arxiv.org/abs/math/0601221)

* [[Gregory Arone]], [[Michael Ching]], _Operads And Chain Rules For The Calculus Of Functors_, [arXiv:0902.0399](https://arxiv.org/abs/0902.0399)

* [AroneMahowald 98](#AroneMahowald98)

* {#AroneKankaanrinta95} [[Gregory Arone]], Marja Kankaanrinta,  _The Goodwillie tower of the identity is a logarithm_, 1995 ([[Arone95.pdf:file]], [web](http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.53.8306))

See also:

* {#AhearnKuhn02} [[Stephen T. Ahearn]], [[Nicholas J. Kuhn]], *Product and other fine structure in polynomial resolutions of mapping spaces*, Algebr. Geom. Topol. **2** (2002) 591-647 [[arXiv:math/0109041](https://arxiv.org/abs/math/0109041), [doi:10.2140/agt.2002.2.591](https://doi.org/10.2140/agt.2002.2.591)]

* Andrew Mauer-Oats, _Algebraic Goodwillie calculus and a cotriple model for the remainder_, Trans. Amer. Math. Soc. __358__  (2006),  no. 5, 1869--1895 [journal](http://www.ams.org/tran/2006-358-05/S0002-9947-05-03936-X/home.html), [math.AT/0212095](http://arxiv.org/abs/math/0212095)

* [[David Barnes]], [[Rosona Eldred]], *Capturing Goodwillie's Derivative*, Journal of Pure and Applied Algebra
**220** 1 (2016) 197-222 [[arXiv:1406.0424](https://arxiv.org/abs/1406.0424), [doi:10.1016/j.jpaa.2015.06.006](https://doi.org/10.1016/j.jpaa.2015.06.006)]

Discussion in the context of [[chromatic homotopy theory]]:

* {#AroneMahowald98} [[Greg Arone]], [[Mark Mahowald]], _The Goodwillie tower of the identity functor and the unstable periodic homotopy of spheres_, Inventiones mathematicae, February 1999, Volume 135, Issue 3, pp 743-788 ([pdf](http://hopf.math.purdue.edu/Arone-Mahowald/ArMahowald.pdf), [doi:10.1007/s002220050300](https://doi.org/10.1007/s002220050300))

* [[Nicholas Kuhn]], _Goodwillie towers and chromatic homotopy: an overview_,   Proceedings of the Nishida Fest (Kinosaki 2003),  245--279, Geom. Topol. Monogr., 10, Geom. Topol. Publ., Coventry, 2007 ([arXiv:math/0410342](https://arxiv.org/abs/math/0410342), [doi:10.2140/gtm.2007.10.245](http://dx.doi.org/10.2140/gtm.2007.10.245))

Discussion via [[(∞,1)-categories]] and [[stable (∞,1)-categories]]:

* [[Jacob Lurie]], _[[(∞,2)-Categories and the Goodwillie Calculus]]_, [section 5](http://www.math.harvard.edu/~lurie/papers/GoodwillieI.pdf#page=159)

This is now section 7 of

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

See also

* Luis Pereira, _A general context for Goodwillie Calculus_ ([arXiv:1301.2832](http://arxiv.org/abs/1301.2832))

* {#Joyal08} [[André Joyal]], _Notes on Logoi_, 2008 ([pdf](http://www.math.uchicago.edu/~may/IMA/JOYAL/Joyal.pdf))

Generalization from [[(infinity,1)-categories]] to [[(infinity,n)-categories]] and relation to [[rational homotopy theory]] is discussed in

* [[Gijs Heuts]], _Goodwillie approximations to higher categories_ ([arXiv:1510.03304](http://arxiv.org/abs/1510.03304))

Relation to [[chromatic homotopy theory]] is discussed in

* [[Greg Arone]], [[Mark Mahowald]], _The Goodwillie tower of the identity functor and the unstable periodic homotopy of spheres_, Inventiones mathematicae, 1999, Volume 135, Issue 3, pp 743-788 ([pdf](http://hopf.math.purdue.edu/Arone-Mahowald/ArMahowald.pdf))

* [[Nicholas Kuhn]], _Goodwillie towers and chromatic homotopy: an overview_, Geom. Topol. Monogr. 10 (2007) 245-279 ([arXiv:math/0410342](http://arxiv.org/abs/math/0410342))

On the relation to [[configuration space (mathematics)|configuration spaces]]:

* {#Ching05} [[Michael Ching]], _Bar constructions for topological operads and the Goodwillie derivatives of the identity_, Geom. Topol. 9 (2005) 833-934 ([arXiv:math/0501429](https://arxiv.org/abs/math/0501429))

* {#Ching05b} [[Michael Ching]], _Calculus of Functors and Configuration Spaces_, Conference on Pure and Applied Topology Isle of Skye, Scotland, 21-25 June, 2005 ([pdf](https://www3.amherst.edu/~mching/Work/skye.pdf))

Discussion in terms of [[spectral Mackey functors]]

* {#Glasman15} [[Saul Glasman]], _Stratified categories, geometric fixed points and a generalized Arone-Ching theorem_ ([arXiv:1507.01976](https://arxiv.org/abs/1507.01976), [talk notes pdf](http://www-users.math.umn.edu/~sglasman/strattalk.pdf))

* {#Glasman16} [[Saul Glasman]], _Goodwillie calculus and Mackey functors_  ([arXiv:1610.03127](https://arxiv.org/abs/1610.03127))

Discussion in an [[equivariant]] setting is in

* Emanuele Dotto, _Higher Equivariant Excision_, ([arXiv:1507.01909](https://arxiv.org/abs/1507.01909))

[[!redirects Goodwillie derivative]]
[[!redirects Goodwillie derivatives]]

[[!redirects Goodwillie calculus of functors]]


Goodwillie calculus - contents

**[[Goodwillie calculus]]** -- approximation of [[homotopy theories]] by [[stable homotopy theories]]

* [[Goodwillie-differentiable (∞,1)-category]]

* [[excisive (∞,1)-functor]]

  * [[spectrum object]], [[parameterized spectrum]],

  * [[tangent (∞,1)-category]], [[tangent (∞,1)-topos]]

* [[n-excisive (∞,1)-functor]]

  * [[jet (∞,1)-category]]

  * [[polynomial (∞,1)-functor]], [[n-reduced (∞,1)-functor]], [[n-homogeneous (∞,1)-functor]]

* [[Goodwillie-Taylor tower]]

  * [[analytic (∞,1)-functor]]

  * [[Goodwillie spectral sequence]]


Goodwillie chain rule

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

# Contents
* table of contents
{: toc}

## Idea

In [[(∞,1)-category theory|(∞,1)-]][[categorification]] of the [[chain rule]] in ordinary [[differential calculus]], in [[Goodwillie calculus]] the **Goodwillie chain rule** is a formula for describing the [[Goodwillie derivatives]] of a composite [[functor]] $F \circ G$ in terms of the [[Goodwillie derivatives]] of $F$ and $G$.

## Related concepts

* [[Goodwillie derivatives of the identity functor]]

## References

* John R. Klein, [[John Rognes]], _A chain rule in the calculus of homotopy functors_, Geom. Topol. 6 (2002) 853-887 ([arXiv:math/0301079](http://arxiv.org/abs/math/0301079))

* [[Jacob Lurie]], section 6.3 of _[[Higher Algebra]]_

* [[Gregory Arone]], [[Michael Ching]], _Operads And Chain Rules For The Calculus Of Functors_, [arXiv:0902.0399](https://arxiv.org/abs/0902.0399)


Goodwillie derivative of the identity functor

[[!redirects Goodwillie derivatives of the identity functor]]

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

#Contents#
* table of contents
{:toc}

## Idea

While the [[Goodwillie calculus of functors]] is an accurate [[(infinity,1)-category theory|(∞,1)-]][[categorification]] of ordinary [[differential calculus]], in that it obeys various analogous rules, such as notably the [[chain rule]], it fails one such rule, in an interesting way:

While in ordinary [[differential calculus]] the [[derivative]] of the [[identity function]] $\mathbb{R}^1 \overset{id}{\to} \mathbb{R}^1$ on the [[real line]] (considered as a map between [[tangent spaces]]) is itself the identity, this is not the case for the [[Goodwillie derivative]] of the [[identity functor]] $Top^{\ast/} \overset{Id}{\to} Top^{\ast/}$ on the [[category]] of [[pointed topological spaces]].

But the [[Goodwillie chain rule]] implies then that the nontrivial Goodwillie derivatives $D_\bullet Id$ of the identity functor satisfy a good algebraic compositional law. Indeed, they form an [[operad]] in [[spectra]] ([Ching 05a](#Ching05a)).

## Properties
 {#Properties}

### Action on configuration spaces of points
 {#ActionOnConfigurationSpacesOfPoints}

For $X$ a [[parallelizable manifold]] of [[dimension]] $dim(X) \in \mathbb{N}$ (without [[manifold with boundary|boundary]]) and for $n \in \mathbb{N}$ any [[natural number]], the $n \cdot dim(X)$-shifted [[suspension spectrum]] of the [[configuration space of points|configuration space]] $Conf_n(X)$ of $n$ points in $X$ (whose elements are such configurations, and a basepoint is freely adjoined) is canonically an [[algebra over an operad|algebra]] over the operad of Goodwillie derivatives

\[
  \label{ConfigurationSpaceAsAlgebraOverGodwillieDerivativesOfIdentity}
  \Sigma^{-n dim(X)}\Sigma^{\infty} Conf_n(X)
    \;\in\;
   Alg\big(
     D_\bullet Id
   \big)
  \,.
\]

([Ching 05b, Prop. 3.1](#Ching05b))

### Relation to the little $n$-disk operad

For each [[natural number]] $d \in \mathbb{N}$ is a canonical homomorphism of [[operads]] from the Goodwillie derivatives of the identity functor to the the $d$-fold [[looping]] of the [[little n-disk operad|little d-disk operad]], incarnated as the [[Fulton-MacPherson operad]]

\[
  \label{HomomorphismOfOperadsToLittleNDiskOperad}
  \phi \;\colon\; D_\bullet Id \longrightarrow \Sigma^{-d} E_d
\]

where the shifted operad on the right has component space in degree $k$ given by the iterated [[loop space]]

\[
  \label{HomomorphismToLittleNDiskOperad}
  \left( \Sigma^{-d}E_d \right)(k)
    \;\coloneqq\;
  \Omega^{d(k-1)} \left( E_n(k) \right)
  \,.
\]

([Ching 05b, below Lemma 4.1](#Ching05b))

Now the [[configuration space]] $Con_n(X)$ from [above](#ActionOnConfigurationSpacesOfPoints) is _also_ canonically an [[algebra over an operad|algebra]] over the [[little n-disk operad]] ([Markl 99](configuration+space+of+points#Markl99)):

\[
  \label{ConfigurationSpaceAsAlgebraOverLittleNDiskOperad}
  \Sigma^{-n dim(X)}\Sigma^{\infty} Conf_n(X)
    \;\in\;
  Alg\left(
    \Sigma^{-dim(X)}E_{dim(X)}
  \right)
  \,.
\]

Conjecturally, the [[operad]]-[[homomorphism]] (eq:HomomorphismToLittleNDiskOperad), via its induced [[functor]] on [[algebras over an operad]]

$$
  \phi^\ast
  \;\colon\;
  Alg\left(
    \Sigma^{-dim(X)}E_{dim(X)}
  \right)
    \longrightarrow
   Alg\big(
     D_\bullet Id
   \big)
$$

identifies these two [[algebra over an operad]]-[[structures]] on shifted [[configuration spaces of points]] from (eq:ConfigurationSpaceAsAlgebraOverGoodwillieDerivativesOfIdentity) and from  (eq:ConfigurationSpaceAsAlgebraOverLittleNDiskOperad)

$$
  \Sigma^{-n dim(X)}\Sigma^{\infty} Conf_n(X)
    \;\in\;
  Alg\left(
    \Sigma^{-dim(X)}E_{dim(X)}
  \right)
    \overset{ \phi^\ast }{\longrightarrow}
   Alg\big(
     D_\bullet Id
   \big)
$$

([Ching 05b, Conjecture 4.2](#Ching05b))

## References

* {#AroneMahowald98} [[Gregory Arone]], [[Mark Mahowald]], _The Goodwillie tower of the identity functor and the unstable periodic homotopy of spheres_, Inventiones mathematicae February 1999, Volume 135, Issue 3, pp 743-788  ([pdf](http://hopf.math.purdue.edu/Arone-Mahowald/ArMahowald.pdf))

* {#AroneKankaanrinta95} [[Gregory Arone]], Marja Kankaanrinta,  _The Goodwillie tower of the identity is a logarithm_, 1995 ([[Arone95.pdf:file]], [web](http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.53.8306))

* {#Ching05a} [[Michael Ching]], _Bar constructions for topological operads and the Goodwillie derivatives of the identity_, Geom. Topol. 9 (2005) 833-934 ([arXiv:math/0501429](https://arxiv.org/abs/math/0501429))

* {#Ching05b} [[Michael Ching]], _Calculus of Functors and Configuration Spaces_, Conference on Pure and Applied Topology Isle of Skye, Scotland, 21-25 June, 2005 ([pdf](https://www3.amherst.edu/~mching/Work/skye.pdf))


Goodwillie spectral sequence

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

#Contents#
* table of contents
{:toc}

## Idea

The [[spectral sequence]] induced from the [[filtration]] given by the [[Goodwillie-Taylor tower]].

## Application

### To homotopy groups of spheres

Applied to the identity functor at the point, the Goodwillie spectral sequence computes [[homotopy groups of spheres]] and as such interacts with the [[EHP spectral sequence]], this is discussed in [Behrens 10](#Behrens10)

## Related concepts

* [[Goodwillie calculus]]

## References

* {#Behrens10} [[Mark Behrens]], _The Goodwillie tower and the EHP sequence_ ([arXiv:1009.1125](http://arxiv.org/abs/1009.1125))


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

#Contents#
* table of contents
{:toc}

## Idea

In the context of [[Goodwillie calculus]], an [[(∞,1)-category]] is called _Goodwillie-differentiable_ if [[(∞,1)-functors]] to it admit "derivatives" in the form of [[n-excisive approximations]].

(This concept is not related to that of [[smooth ∞-groupoids]].)

## Definition

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

An [[(∞,1)-category]] $\mathcal{C}$ is **Goodwillie-differentiable** if

1. it has [[finite (∞,1)-limits]];

1. it has [[sequential limit|sequential]] [[(∞,1)-colimits]];

1. the [[(∞,1)-colimit]] [[(∞,1)-functor]] $\underset{\longrightarrow}{\lim} Func(\mathbb{N}, \mathcal{C}) \longrightarrow \mathcal{C}$ is a [[left exact (∞,1)-functor]], hence commutes with [[finite (∞,1)-limits]].

=--

([Lurie, def. 6.1.1.6](#Lurie))

## Examples

+-- {: .num_example }

###### Example

Every [[(∞,1)-topos]] is a Goodwillie-differentiable $(\infty,1)$-category.

=--

([Lurie, example 7.1.1.8](#Lurie))

## Properties

### $n$-Excisive reflection / Taylor tower

By [[Goodwillie calculus]], [[(∞,1)-functors]] to Goodwillie-differentiable $(\infty,1)$-categories have [[n-excisive approximations]]/[[Taylor towers]].

## Related concepts

* [[tangent (∞,1)-category]]

* [[jet (∞,1)-category]]

* [[Goodwillie calculus]]

## References

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

[[!redirects differentiable (infinity,1)-category]]

[[!redirects differentiable (infinity,1)-categories]]
[[!redirects differentiable (infinity,1)-category]]

[[!redirects differentiable (∞,1)-category]]
[[!redirects differentiable (∞,1)-categories]]

[[!redirects Goodwillie-differentiable (∞,1)-categories]]

[[!redirects Goodwillie-differentiable (infinity,1)-categories]]
[[!redirects Goodwillie-differentiable (infinity,1)-category]]

[[!redirects Goodwillie-differentiable (∞,1)-category]]

[[!redirects Goodwillie differentiation]]
[[!redirects Goodwillie differentiations]]


Goodwillie-Taylor tower

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

#Contents#
* table of contents
{:toc}

## Idea

In the context of [[Goodwillie calculus]] the _Taylor tower_ of an [[(∞,1)-functor]] $F$ to a [[Goodwillie-differentiable (∞,1)-category]] is its stagewise approximation by [[n-excisive (∞,1)-functors]] $P_n F$ (its [n-excisive projections](n-excisive+%28?%2C1%29-functor#nExcisiveApproximation))

$$
  \cdots \to P_{n+1} F \to P_n F \to \cdots \to P_1 F \to P_0 F
  \,.
$$

which is [[analogy|analogous]] to the approximation of a [[smooth function]] by its [[Taylor series]].

If this tower converges to $F$, then $F$ is analogous to an [[analytic function]] and is called an _[[analytic (∞,1)-functor]]_.

The [[spectral sequence]] induced by the [[filtration]] given by the Goodwillie-Taylor tower is the _[[Goodwillie spectral sequence]]_.

## Properties

### Convergence

convergence for $\rho$-[[analytic (∞,1)-functors]] on $\rho$-[[n-connected object of an (infinity,1)-topos|connective objects]] (...) ([Goodwillie 03, theorem 1.13](#Goodwillie03), see also [Munson-Volic 15, theorem 10.1.51 and section 10.2.4](#MunsonVolic15))

## Examples

### Stable splitting of mapping spaces
 {#StableSplittingOfMappingSpaces}

Under some conditions, the Goodwillie-Taylor tower of [[mapping spaces]] into sufficiently high [[suspensions]] is a [[direct sum]] of [[suspension spectra]] of [[configuration spaces]]. This _[[stable splitting of mapping spaces]]_ is originally due to ([Snaith 74](#Snaith74), [Bödigheimer 87](#Boedigheimer87)) and was understood as a Goodwillie-Taylor tower in [Arone 99](#Arone99), see also [Ching 05](#Ching05).

## Related concepts

* [[n-excisive (∞,1)-functor]]

* [[Postnikov tower]], [[chromatic tower]]

## References

### General

The concept originates in

* {#Goodwillie03} [[Thomas Goodwillie]], _Calculus III: Taylor Series_, Geom. Topol. 7(2003) 645-711 ([arXiv:math/0310481](http://arxiv.org/abs/math/0310481))

Review includes

* {#MunsonVolic15} [[Brian Munson]], [[Ismar Volic]], _Cubical homotopy theory_, Cambridge University Press, 2015 [pdf](http://palmer.wellesley.edu/~ivolic/pdf/Papers/CubicalHomotopyTheory.pdf)

See also

* [[Gregory Arone]], [[Michael Ching]], _A classification of Taylor towers of functors of spaces and spectra_ ([arXiv:1209.5661](http://arxiv.org/abs/1209.5661))

### Stable splitting of mapping spaces

The [[stable splitting of mapping spaces]] is originally due to

* {#Snaith74} [[Victor Snaith]],  _A stable decomposition of $\Omega^n S^n X$_, Journal of the London Mathematical Society 7 (1974), 577 - 583 ([pdf](https://www.maths.ed.ac.uk/~v1ranick/papers/snaiths.pdf))

Review and generalization is due to

* {#Boedigheimer87} [[Carl-Friedrich Bödigheimer]], _Stable splittings of mapping spaces_, Algebraic topology. Springer 1987. 174-187 ([pdf](http://www.math.uni-bonn.de/~cfb/PUBLICATIONS/stable-splittings-of-mapping-spaces.pdf))

Further generalization and interpretation in terms of the Goodwillie-Taylor tower of mapping spaces is due to

* {#Arone99} [[Greg Arone]], _A generalization of Snaith-type filtration_, Transactions of the American Mathematical Society 351.3 (1999): 1123-1150. ([pdf](https://www.ams.org/journals/tran/1999-351-03/S0002-9947-99-02405-8/S0002-9947-99-02405-8.pdf))

* {#Ching05} [[Michael Ching]], _Calculus of Functors and Configuration Spaces_, Conference on Pure and Applied Topology Isle of Skye, Scotland, 21-25 June, 2005 ([pdf](https://www3.amherst.edu/~mching/Work/skye.pdf))

[[!redirects Goodwillie-Taylor towers]]

[[!redirects Taylor tower]]
[[!redirects Taylor towers]]

[[!redirects Goodwillie-Taylor series]]


Goong Chen

* [personal page](https://www.math.tamu.edu/~goong.chen/)

## Selected writings

On [[quantum computation]]:

* [[Ranee K. Brylinski]], [[Goong Chen]] (eds.), *Mathematics of Quantum Computation*, Chapman and Hall/CRC 2002 ([doi:10.1201/9781420035377](https://doi.org/10.1201/9781420035377))

* [[Goong Chen]], [[Louis Kauffman]], [[Samuel J. Lomonaco]] (eds.), *Mathematics of Quantum Computation and Quantum Technology*, Chapman and Hall/CRC (2007) (ISBN:9780367388614, [doi:10.1201/9781584889007](https://doi.org/10.1201/9781584889007))

category: people


Gopakumar-Vafa duality

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

#Contents#
* table of contents
{:toc}

## Idea

A [[duality in string theory]].

Gopakumar–Vafa duality is a [[duality in string theory]] between [[perturbative quantization of 3d Chern-Simons theory|perturbative]] [[Chern-Simons theory|3d Chern-Simons]] [[Wilson loop observables]] ([[Vassiliev link invariants]]) and relative [[Gromov–Witten invariants]] of a 6–dimensional [[symplectic manifold]] relative to a [[Lagrangian submanifold]].

## Related concepts

* [[Gopakumar-Vafa invariant]]

## References

The original articles are

* [[Rajesh Gopakumar]], [[Cumrun Vafa]], _Topological Gravity as Large N Topological Gauge Theory_, Adv. Theor. Math. Phys. 2:413-442, 1998 ([arXiv:hep-th/9802016](https://arxiv.org/abs/hep-th/9802016))

* [[Rajesh Gopakumar]], [[Cumrun Vafa]], _On the Gauge Theory/Geometry Correspondence_, Adv. Theor. Math. Phys. 3 (1999) 1415-1443 ([arXiv:hep-th/9811131](https://arxiv.org/abs/hep-th/9811131))

Review includes

* Dave Auckly, Sergiy Koshkin, _Introduction to the Gopakumar-Vafa Large $N$ Duality_, Geom. Topol. Monogr. 8 (2006) 195-456 ([arXiv:0701568](https://arxiv.org/abs/math/0701568))

  (relation to [[perturbative quantization of 3d Chern-Simons theory]], [[weight systems]] on [[chord diagrams]], etc.)


Gopakumar-Vafa invariant

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

## Related entries

* [[topological string]]

* [[black holes in string theory]]

## Related concepts

* [[Gopakumar-Vafa duality]]

## References

* Wikipedia, _[Gopakumar-Vafa invariant](https://en.wikipedia.org/wiki/Gopakumar%E2%80%93Vafa_invariant)_

[[!redirects Gopakumar-Vafa invariants]]

[[!redirects Gopakumar-Vafa invariants]]


Gor Sarkissian

* [inspire page](https://inspirehep.net/authors/990294)

* [GoogleScholar page](https://scholar.google.com/citations?user=T08KTiAAAAAJ)

## Selected writings

On [[NS5-brane]]-[[brane intersections|intersections]] with [[D-branes]]:

* {#EGKRS00} [[Shmuel Elitzur]], [[Amit Giveon]], [[David Kutasov]], [[Eliezer Rabinovici]], [[Gor Sarkissian]], *D-Branes in the Background of NS Fivebranes*, JHEP 0008 (2000) 046 [[arXiv:hep-th/0005052](https://arxiv.org/abs/hep-th/0005052), [doi:10.1088/1126-6708/2000/08/046](https://doi.org/10.1088/1126-6708/2000/08/046)]

category: people


Goran Senjanović

[[!redirects Goran Senjanovic]]

* [Wikipedia entry](https://en.wikipedia.org/wiki/Goran_Senjanovi%C4%87)

## Selected talks

On [[neutrinos]] and the [[LHC experiment]]:

* [[Goran Senjanovic]], _LHC and the neutrino paradigm_, talk at CERN EP seminar 2011 ([cern:148714](https://indico.cern.ch/event/148714), [recording](https://youtu.be/VVoMUHokn2U))

category: people


Gord Sinnamon

* [institute page](https://www.math.uwo.ca/faculty/sinnamon/)

* [Mathematics Genealogy page](https://www.mathgenealogy.org/id.php?id=68239)

## Selected writings

On the [[D-topology]] on [[diffeological spaces]]:

* {#CSW13} [[J. Daniel Christensen]], [[Gord Sinnamon]], [[Enxin Wu]], _The $D$-topology for diffeological spaces_, Pacific Journal of Mathematics 272 (1), 87-110, 2014 ([arXiv:1302.2935](http://arxiv.org/abs/1302.2935), [doi:10.2140/pjm.2014.272.87](https://msp.org/pjm/2014/272-1/p04.xhtml))

category: people


Gordon D. James

[[!redirects G. D. James]]

* [personal page](http://wwwf.imperial.ac.uk/~gdj/)

## Selected writings

On the [[representation theory of the symmetric groups]]:

* [[Gordon D. James]], *The Representation Theory of the Symmetric Groups*,  Lecture Notes in Mathematics, volume 682, Springer 1978 ([doi:10.1007/BFb0067708](https://link.springer.com/book/10.1007/BFb0067708), [pdf](http://www-users.math.umn.edu/~webb/oldteaching/Year2010-11/the-representation-theory-of-the-symmetric-groups-SLN.pdf))

category: people


Gordon Kane

* [Webpage](http://particle-theory.physics.lsa.umich.edu/kane/)

* [Wikipedia entry](http://en.wikipedia.org/wiki/Gordon_L._Kane)

## Selected writings

* Nature article, _[Particle physics is at a turning point](http://www.nature.com/news/particle-physics-is-at-a-turning-point-1.9675)_, Dec. 2011

* [Sakurai prize 2017](https://www.aps.org/programs/honors/prizes/prizerecipient.cfm?last_nm=Kane&first_nm=Gordon&year=2017) "For instrumental contributions to the theory of the properties, reactions, and signatures of the Higgs boson."

On the history of the concept of [[supersymmetry]]:

* {#KaneShifman00} [[Gordon Kane]], [[Mikhail Shifman]], _The Supersymmetric World: The Beginnings of The Theory_, World Scientific 2000 ([doi:10.1142/4611](https://doi.org/10.1142/4611))

  * {#KaneShifman01} [[Gordon Kane]], [[Mikhail Shifman]], _Introduction to "The Supersymmetric world"_ ([arXiv:hep-ph/0102298](https://arxiv.org/abs/hep-ph/0102298)), chapter in: [Kane-Shifman 00](#KaneShifman00)

Kane has been arguing that [[string phenomenology]]-[[model (physics)|models]] of the kind of [[M-theory on G2-manifolds]], notably the [[G2-MSSM]], "generically" predict [[Higgs particle]] [[masses]] around 120-130 GeV. He stated this in

* {#Kane11} [[Gordon Kane]], _String theory and generic predictions for our world – superpartner masses, LHC signatures, dark matter, EWSB, cosmological history of universe, etc_, talk at String phenomenology 2011, August 2011 ([pdf](http://conferencing.uwex.edu/conferences/stringpheno2011/documents/kane.pdf))

a few months before the actual detection of the Higgs particle (and its [[mass]] of $\sim 125$[[GeV]]) at the [[LHC]], and with more details in

* [[Gordon Kane]], [[Piyush Kumar]], Ran Lu, Bob Zheng, _Higgs Mass Prediction for Realistic String/M Theory Vacua_, Phys. Rev. D 85, 075026 ([arXiv:1112.1059](http://arxiv.org/abs/1112.1059))

a few days before the result was officially announced (a useful informed comment is [here](http://www.math.columbia.edu/~woit/wordpress/?p=4262&cpage=1#comment-101525)).

See also

* [[Gordon Kane]], _From Planck-scale M-theory to predictions for superpartners at LHC, soon_, June 2015 ([web](http://agenda.albanova.se/getFile.py/access?contribId=278&resId=250&materialId=slides&confId=4682))

* [[Gordon Kane]], _String/M-theories About Our World Are Testable in the traditional Physics Way_, talk at _[Why Trust a Theory? Reconsidering Scientific Methodology in Light of Modern Physics](http://www.whytrustatheory2015.philosophie.uni-muenchen.de/index.html)_, Munich 2015  ([arXiv:1601.07511](https://arxiv.org/abs/1601.07511), [video](https://videoonline.edu.lmu.de/en/node/7485))

* [[Gordon Kane]], _Compactifying M-theory on a $G_2$ manifold to describe/explain our world – Predictions for LHC (gluinos, winos, squarks),  and dark matter_ ([pdf](https://indico.cern.ch/event/506172/contributions/2026001/attachments/1256948/1855957/CMS_talk.pdf))

* {#Kane17} [[Gordon Kane]], _String theory and the real world_, Morgan & Claypool, 2017, 2021 (doi:0.1088/978-1-6817-4489-6, [ISBN:978-0-7503-3583-6](https://iopscience.iop.org/book/978-0-7503-3583-6))

* {#Kane18} [[Gordon Kane]], _Exciting Implications of LHC Higgs Boson Data_ ([arXiv:1802.05199](https://arxiv.org/abs/1802.05199))

On [[Yukawa couplings]] in the [[G2-MSSM]]:

* Eric Gonzalez, [[Gordon Kane]], Khoa Dang Nguyen, [[Malcolm Perry]], _Quark and lepton mass matrices from localization in M-theory on $G_2$ orbifold_ ([arXiv:2002.11820](https://arxiv.org/abs/2002.11820))

and on [[neutrino]] masses:

* Eric Gonzalez, [[Gordon Kane]], Khoa Nguyen, *Neutrino mass matrices from localization in M-theory on G2 orbifold* ([arXiv:2107.12893](https://arxiv.org/abs/2107.12893))

## Related $n$Lab entries

* [[string phenomenology]]

* [[M-theory on G2-manifolds]], [[G2-MSSM]]

* [[Higgs field]], [[vacuum stability]]

* [[naturalness]]

category: people


Gordon Plotkin

* [Publications](http://homepages.inf.ed.ac.uk/gdp/publications/)

* [Scottish Wikipedia page](http://en.wikipedia.org/wiki/Gordon_Plotkin)

## Selected writings

On [[logical frameworks]]:

* [[Robert Harper]], [[Furio Honsell]], [[Gordon Plotkin]], _A framework for defining logics_, J. ACM __40__:1 (1993) 143--184 [[doi:10.1145/138027.138060](https://doi.org/10.1145/138027.138060)]

On [[higher-order abstract syntax]]:

* [[Marcelo Fiore]], [[Gordon Plotkin]], [[Daniele Turi]]. _Abstract syntax and variable binding_, Proceedings. 14th Symposium on Logic in Computer Science (Cat. No. PR00158). IEEE (1999) [[doi:10.1109/LICS.1999.782615](https://doi.org/10.1109/LICS.1999.782615), [pdf](https://homepages.inf.ed.ac.uk/gdp/publications/Abstract_Syn.pdf), [webpage](https://www.dcs.ed.ac.uk/home/dt/abstractsyn.html)]

On [[monads in computer science]], such as the (local) [[state monad]]:

* {#PlotkinPower02} [[Gordon Plotkin]], [[John Power]], Section 3 of: *Notions of computation determine monads* In: M. Nielsen, U. Engberg, (eds.) FOSSACS 2002. LNCS, Lecture Notes in Computer Science **2303** 342-356 Springer, (2002) [[doi:10.1007/3-540-45931-6_24p](https://doi.org/10.1007/3-540-45931-6_24)]

On [[category theory|category theoretic]] [[probability theory]] (cf. *[[convex powerset of distributions monad]]*):

* [[Regina Tix]], [[Klaus Keimel]], [[Gordon D. Plotkin]]: *Semantic domains for combining probability and non-determinism*, Electronic Notes in Theoretical Computer Science, **222** (2009) 3–99 [[doi:10.1016/j.entcs.2009.01.002](https://doi.org/10.1016/j.entcs.2009.01.002)]

On effect handling via [[module over a monad|modules]] over [[monads in computer science]]:

* [[Gordon D. Plotkin ]], [[Matija Pretnar]], *Handling Algebraic Effects*, Logical Methods in Computer Science, **9** 4 (2013) lmcs:705 [[arXiv:1312.1399](https://arxiv.org/abs/1312.1399), doi:10.2168/LMCS-9(4:23)2013]

category: people

[[!redirects Gordon D. Plotkin]]


Gordon Semenoff

* [webpage](https://www.phas.ubc.ca/users/gordon-semenoff)

## Selected writings

Interpretation of the [[DBI-action]] on [[flavor branes]] as witnessing the [[holographic Schwinger effect]]:

* {#SemenoffZarembo11} [[Gordon Semenoff]], [[Konstantin Zarembo]], _Holographic Schwinger Effect_, Phys. Rev. Lett. 107, 171601 (2011) ([arXiv:1109.2920](https://arxiv.org/abs/1109.2920),  [doi:10.1103/PhysRevLett.107.171601](https://doi.org/10.1103/PhysRevLett.107.171601))

category: people


Gorenstein ring

#Contents#
* table of contents
{:toc}

## Idea
A Gorenstein ring, $R$, is a commutative Noetherian local ring of finite injective dimension over itself. It follows then that its injective dimension is equal to $r$ = Krull dimension$(R)$, and that $Ext^{\ast}_R(k, R)$ is one dimensional as a $k$-vector space, where $k$ is the residue field of $R$.

##Related concepts

* [[Gorenstein ring spectrum]]

## References

* Hyman Bass, 1963, _On the ubiquity of Gorenstein rings_, Mathematische Zeitschrift, 82: 8–28.

* [[William Dwyer]], [[John Greenlees]], [[Srikanth Iyengar]], _Duality in algebra and topology_, Advances in Maths
200 (2006) 357-402, ([ arXiv:math/0510247](https://arxiv.org/abs/math/0510247))

[[!redirects Gorenstein rings]]


Gorenstein ring spectrum

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

#Contents#
* table of contents
{:toc}

## Idea

The definition of a **Gorenstein** [[ring spectrum]] is motivated by the [[Gorenstein ring|Gorenstein condition]] for [[rings]]. A Gorenstein ring, $R$, is a commutative [[Noetherian ring|Noetherian]] [[local ring]] such that the [[Ext]]-group $Ext^{\ast}_R(k, R)$ is one [[dimension|dimensional]] as a $k$-[[vector space]], where $k$ is the [[residue field]] of $R$. This last condition may be restated as the property that the [[homology]] of the ([[derived functor|right derived]]) Hom complex $Hom_R(k, R)$ is equivalent to a [[suspension]] of $k$.

Thus, a [[ring spectrum]] $\mathbf{R} \to \mathbf{k}$ is said to be Gorenstein if there is an [[equivalence in an (infinity,1)-category|equivalence]] of $\mathbf{R}$-[[module spectra]] $Hom_{\mathbf{R}}(\mathbf{k}, \mathbf{R}) \simeq \Sigma^a\mathbf{k}$ for some [[integer]] $a$.

A ring, $R \to k$, is [[Gorenstein ring|Gorenstein]] if and only if the ring spectrum $H R \to H k$ is Gorenstein.

## Examples

Examples from [[representation theory]], from [[chromatic stable homotopy theory]] and from [[rational homotopy theory]] are given in ([Greenlees16, Sec. 23](#Greenlees16)).

## References

* [[William Dwyer]], [[John Greenlees]], [[Srikanth Iyengar]], _Duality in algebra and topology_, Advances in Maths
200 (2006) 357-402, ([ arXiv:math/0510247](https://arxiv.org/abs/math/0510247))

* {#Greenlees16} [[John Greenlees]], _Homotopy Invariant Commutative Algebra over fields_, ([arXiv:1601.02473](https://arxiv.org/abs/1601.02473))

[[!redirects Gorenstein ring spectra]]


Gorgias dialogue

#Contents#
* table of contents
{:toc}

## Idea

*Georgias* is a dialogue on rhetoric by [[Plato]]. The main characters are Socrates and Gorgias, a sophist who teaches rhetoric to young people.

## The topic

Whereas for Socrates a discussion is about progressing together towards the truth regarding a concrete question, for Gorgias it is more about convincing each other, with the clarification of the actual question just being part of the process.

Thus, the dialogue can be seen as a _mise en abyme_ of the subject it is discussing. Early in the dialogue, Plato specifies the conditions of a good discussion between two partners. Perhaps, it would be good to agree on this before any discussion, including mathematical discussions.

> SOCRATES: You, Gorgias, like myself, have had great experience of disputations, and you must have observed, I think, that they do not always terminate in mutual edification, or in the definition by either party of the subjects which they are discussing; but disagreements
are apt to arise -- somebody says that another has not spoken truly or clearly; and then they get into a passion and begin to quarrel, both parties conceiving that their opponents are arguing from personal feeling only and jealousy of themselves, not from any interest in the question at issue. And sometimes they will go on abusing one another until the company at last are quite vexed at themselves for ever listening to such fellows. Why do I say this? Why, because I cannot help feeling that you are now saying what is not quite consistent or accordant with what you were saying at first about rhetoric. And I am afraid to point this out to you, lest you should think that I have some animosity against you, and that I speak, not for the sake of discovering the truth, but from jealousy of you. Now if you are one of my sort, I should like to cross-examine you, but if not I will let you alone. And what is my sort? you will ask. **I am one of those who are very willing to be refuted if I say anything which is not true, and very willing to refute any one else who says what is not true, and quite as ready to be refuted as to refute; for I hold that this is the greater gain of the
two, just as the gain is greater of being cured of a very great evil than of curing another. For I imagine that there is no evil which a man can endure so great as an erroneous opinion about the matters of which we are speaking; and if you claim to be one of my sort, let us have the discussion out, but if you would rather have done, no matter;--let us make an end of it.**


Goro Ishiki

* [inspire page](https://inspirehep.net/authors/1034322)

## Selected writings

Using the [[KK-compactification]] of [[D=4 N=4 super Yang-Mills theory]] to the [[BMN matrix model]] for [[lattice gauge theory]]-computations in [[D=4 N=4 SYM]] and for numerical checks of the [[AdS-CFT correspondence]]:

* Masazumi Honda, [[Goro Ishiki]], Sang-Woo Kim, Jun Nishimura, Asato Tsuchiya, _Direct test of the AdS/CFT correspondence by Monte Carlo studies of N=4 super Yang-Mills theory_, JHEP 1311 (2013) 200 [[arXiv:1308.3525](https://arxiv.org/abs/1308.3525)]

On [[M2-M5-brane bound states]] in the [[BMN matrix model]]:

* {#AIST17b} [[Yuhma Asano]], [[Goro Ishiki]], [[Shinji Shimasaki]], [[Seiji Terashima]], _Spherical transverse M5-branes from the plane wave matrix model_, JHEP 02 (2018) 076 [[arXiv:1711.07681](https://arxiv.org/abs/1711.07681), doi:10.1007/JHEP02(2018)076]

* {#AIST17a} [[Yuhma Asano]], [[Goro Ishiki]], [[Shinji Shimasaki]], [[Seiji Terashima]], _On the transverse M5-branes in matrix theory_, Phys. Rev. D **96** 126003 (2017) [[arXiv:1701.07140](https://arxiv.org/abs/1701.07140), [doi:10.1103/PhysRevD.96.126003](https://doi.org/10.1103/PhysRevD.96.126003)]

category: people


Goro Nishida

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

## related $n$Lab entries

* [[Nishida nilpotence theorem]]

category: people


Goss zeta function

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

#Contents#
* table of contents
{:toc}

## Idea

The Goss zeta function is an [[analogy|analog]] (under the [[function field analogy]]) of the [[Riemann zeta function]] for _[[number fields]]_.

Like the [[Weil zeta function]].

## Properties

### Function field analogy

[[!include function field analogy -- table]]

## References

* Wikipedia, _[Goss zeta function](http://en.wikipedia.org/wiki/Goss_zeta_function)_

* Jeffrey T. Sheats, _The Riemann Hypothesis for the Goss zeta function for $\mathbb{F}_q[T]$_ ([arXiv:math/9801158](http://arxiv.org/abs/math/9801158))

[[!redirects Goss zeta functions]]


Gottfried Curio

* [webpage](http://www.theorie.physik.uni-muenchen.de/MATH/members/former_mem/former_asc/curio_gottfried/index.html)

## related $n$Lab entries

* [[membrane instanton]], [[M-theory on G2-manifolds]]

category: people


Gottfried Leibniz

__Gottfried Leibniz__ (1646-1716)

* [Wikipedia entry](http://en.wikipedia.org/wiki/Gottfried_Wilhelm_Leibniz)

is responsible for a great many constructions and concepts of current interest:

*  [[differential calculus]] and [[integral calculus]], especially the basic notation that we use today;

*  the [[product rule]] of calculus may be called the _Leibniz rule_ or _Leibniz law_;

*  [[differentiation under the integral sign]] follows the _Leibniz integral rule_;

*  [[identity of indiscernibles]] may be called the _Leibniz law_;
*  a version of [[infinitesimals]] sometimes regarded as a precursor to [[synthetic differential geometry]];

* {#OnMonads} a philosophy of 'monads' (*[monadology](https://en.wikipedia.org/wiki/Monadology)*)

  (compare also in [[Hegel]], _[[Science of Logic]]_ the section _[The monad of Leibniz](Science+of+Logic#TheMonadOfLeibniz))

  from which it certainly seems (but remains implicit) that [Robinson 1966 (p. 57)](nonstandard+analysis#Robinson66) motivated the terminology of *[[monads in nonstandard analysis]]* ([[infinitesimal neighbourhoods]])

  and which may or may not have motivated [Bénabou 1967 (§5.4)](monad#Bénabou67) to coin the term *[[monad]]* in [[category theory]] (see [there](monad#Etymology) for more).

## Metaphysics

> An instructive starting point distinction made in the contemporary literature ... is between three 'levels of reality' in Leibniz's mature metaphysics. At the most basic level, what is real for Leibniz are simple substances which alone have true unity. These are the famously obscure monads. Next, we have the 'phenomenal level' that is made up of _phenomena bene funda_ – well founded phenomena – that, due to pre-established harmony, are accurate reflections of the real and actual monadic states. Finally, we have the ideal level which, by contrast, is made up of _entia rationis_ – abstract or fictional things – that include 'phenomena' founded upon possible but non-actual monadic states. Crucially, although both the phenomenal and the ideal levels can include things which are infinite, all concepts that depending upon the continuum are only applicable to the ideal realm. Thus, if we were to define time as the real line, $\mathbb{R}$, then this concept of time could only be represented for Leibniz as an entia rationis and thus ideal. Furthermore, phenomenal things for Leibniz can only acquire their status as phenomena bene funda by their grounding upon the actual. They must always be understood as representations or perceptions of the monads of the actual world. ([Thébault 19, p. 2](#Thebault))

## References

* [[Georg Hegel]], _[[Lectures on the History of Philosophy]] -- [Leibniz](Lectures+on+the+History+of+Philosophy#Leibniz)_

On [[infinitesimals]]:

* {#KatzSherry13} [[Mikhail Katz]], David Sherry, _Leibniz's Infinitesimals: Their Fictionality, Their Modern Implementations, And Their Foes From Berkeley To Russell And Beyond_, D. Erkenn (2013) 78: 571 ([arXiv:1205.0174](https://arxiv.org/abs/1205.0174), [doi:10.1007/s10670-012-9370-y](https://link.springer.com/article/10.1007/s10670-012-9370-y))

* {#Thebault} Karim Thébault, _The Problem of Time_, ([PhilSci Archive](http://philsci-archive.pitt.edu/15795/))

category: people

[[!redirects Gottfried Wilhelm Leibniz]]
[[!redirects Gottfried Leibniz]]
[[!redirects Leibniz]]

[[!redirects G. W. Leibniz]]


Gottlob Frege

[[!redirects Frege]]

* wikipedia, _[Gottlob Frege](http://en.wikipedia.org/wiki/Gottlob_Frege)_

* Metaphysics Research Lab, _[Friedrich Ludiwg Gottlog Frege](http://mally.stanford.edu/frege.html)_

## Selected writings

* {#Frege1879} [[Gottlob Frege]], *Begriffsschrift* (1879) [[web scan](https://gdz.sub.uni-goettingen.de/id/PPN538957069?tify=%7B%22pages%22%3A%5B4%5D%2C%22view%22%3A%22info%22%7D), [Wikipedia entry](https://en.wikipedia.org/wiki/Begriffsschrift)]

## Related entries

* [[formal logic]]

* [[judgement]]

category: people


Goursat category

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

#Contents#
* table of contents
{:toc}

## Idea

(...)

## Related concepts

* [[Malcev category]]

## References

General:

* [[Marino Gran]], [[Diana Rodelo]], *A New Characterisation of Goursat Categories*, Applied Categorical Structures **20** (2012) 229–238 [[doi:10.1007/s10485-010-9236-x](https://doi.org/10.1007/s10485-010-9236-x)]

On the [[Beck-Chevalley condition]] in Goursat categories:

* [[Marino Gran]], [[Diana Rodelo]], *Beck-Chevalley condition and Goursat categories*, Journal of Pure and Applied Algebra **221** 10 (2017) 2445-2457 [[arXiv:1512.04066](https://arxiv.org/abs/1512.04066), [doi:10.1016/j.jpaa.2016.12.031](https://doi.org/10.1016/j.jpaa.2016.12.031)]

[[!redirects Goursat categories]]