Oberwolfach Workshop, June 2009 -- Strings, Fields, Topology

This entry collects material on the June 2009 workshop on [[string theory|strings]], [[quantum field theory|fields]], and [[topology]] at [[Oberwolfach]].

* Title:      Strings, Fields and Topology
* Organisers:
  * Dennis Sullivan, New York
  * Stephan Stolz, Notre Dame
  * Peter Teichner, Berkeley
* Date:       June 7th - June 13th, 2009
* ID:         0924

Writeups of most of the talks have appeared as

* _Strings, Field, Topology_,  Oberwolfach report No 28, 2009, ([pdf](https://www.mfo.de/document/0924/OWR_2009_28.pdf))

# Talks

* **Thomas Schick**, [_Smooth cohomology theories_](#schick).

* **Ullrich Bunke**, [_Smooth K-theory_](#bunke).

* **Christoph Schweigert and Ingo Runkel** , [_CFT and algebra in braided tensor categories I and II_](#schweigert).

* **Kevin Costello**, [_Factorization algebras in perturbative QFT_](#costello).

* **[[Gabriel C. Drummond-Cole]]**, [_$\infty$-Operads, $BU_\infty$ and $Hypercomm_\infty$_](#drummond-cole).

* **Alex Kahle**, [_Superconnections and index theory_](#kahle).

* **[[Chris Schommer-Pries]]**, [_Topological defects, $D$-branes, and the classifications of TFTs in low dimensions_](#schommer-pries).

* **Dan Freed**, [_Geometry and topology of orientifolds_](#freed).

* **Dan Freed**, [Evening informal session on RR fields and all that](#freed_extra).

* **Scott Wilson**, [_Some algebra related to mapping spaces and applications_](#wilson).

* **[[Urs Schreiber]]**, [_Background fields in twisted differential nonabelian cohomology_](#schreiber).

* **André Henriques**, [_Invertible conformal nets_](#henriques).

* **André Henriques**, [Evening informal session more details about conformal nets](#henriques_extra).

* **[[Konrad Waldorf]]**, [_String connections and Chern-Simons 2-gerbes_](#waldorf).

* **Kevin Walker**, [_Blob homology_](#walker)

* **Ralph Cohen**, [_String topology, field theories and Fukaya categories_](#cohen).

* **Corbett Redden**, [_String structures, 3-forms, and tmf classes_](#redden).

* **Mike Hopkins**, [_The Kervaire invariant_](#hopkins).

* **Mike Hopkins**, [Evening informal session on technical details of the solution of the Kervaire invariant problem](#hopkins_extra).

* **David Chataur**, [Evening informal session explaining recent results in opeards](#chataur_extra)

* **Kishore Marathe**, [Evening informal session on the gauge theory to string theory correspondence](#marathe_extra)

# Talks in more detail

#### Thomas Schick, _Smooth cohomology theories_ {#schick}

The basic idea of [[differential cohomology]] (also called _smooth cohomology_ ) is to combine generalized [[cohomology]] (e.g. ordinary integral cohomology, or K-theory, or tmf, or ...) and differential forms. Hopkins and Singer showed that a smooth refinement exists for any generalized cohomology theory, but didn't provide an (easy for mere mortals to understand) explicit construction. In this talk, Thomas Schick advertised geometric models for multiplicative smooth cohomology with $S^1$-integration for $K$-theory (joint with Ulrich Bunke) and MU-bordism (joint with Ulrich Bunke, Schröder? and Wethamp?). They have proved a uniqueness theorem, stating that under certain assumptions any two smooth refinements of a generalized cohomology theory are naturally isomorphic.

##### Resources

* [[oberwolfach_june2009_schick.pdf|Notes by Bruce Bartlett:file]]
* [Notes by Gabriel Drummond-Cole](http://www.math.sunysb.edu/~blafard/tex/sum09/ow_june_9.pdf)
* [[Oberwolfach Workshop, June 2009 -- Tuesday, June 9|Notes by Urs Schreiber]]
* [[oberwolfach_june2009_pavlov_schick.pdf|Notes by Dmitry Pavlov:file]]

#### Ulrich Bunke, _Smooth K-theory_. {#bunke}

In this talk, which led on from the previous one by Thomas Schick, Ulrich Bunke gave an explicit description of their smooth $K$-theory model in terms of geometric cocycles. Basically a cocycle consists of a pair $(\mathcal{E}, \rho)$ where $\mathcal{E}$ is a 'geometric family', namely a smooth proper submersion $E \rightarrow B$ with a fiberwise metric, a Clifford bundle $W$ and some connection of some sort. The object $\rho$ is an element of $\Omega(B, K) = C^\infty(B, \Lambda^*T^* B \otimes K^*)$. He also described smooth $K$-orientatiosn and the push-forward map, with an application to the $e$-invariant of Adams. There is also a Riemann-Roch theorem.

##### Resources

* [[oberwolfach_june2009_bunke.pdf|Notes by Bruce Bartlett:file]]
* [Notes by Gabriel Drummond-Cole](http://www.math.sunysb.edu/~blafard/tex/sum09/ow_june_9.pdf)
* [[Oberwolfach Workshop, June 2009 -- Tuesday, June 9|Notes by Urs Schreiber]]

#### Christoph Schweigert and Ingo Runkel,_CFT and algebra in braided tensor categories I and II_. {#schweigert}

This talk explains how correlators for a rational two-dimensional [[conformal field theory]] can be constructed in the [[FQFT|functorial TFT]] formalism. Based on a [[modular tensor category]] $C$, decoration data have been introduced in terms of special symmetric Frobenius algebras in $C$ and the correlator, as an element of $tft_C (\hat{X})$ with $\hat{X}$ a double cover of the surface $X$, has been expressed in terms ofthe invariant of a decorated 3-manifold $M_X$ with $\partial M_X = \hat{X}$. The correlators are invariant under the [[mapping class group]]s and obey the [[sewing constraint]].

[[Morita equivalence|Morita equivalent]] special symemtric Frobenius algebras lead to an equivalent description of the correlators. A Morita invariant formulation is provided by the notion of a module category $M$ over the module tensor category $C$. The [[worldsheet|world sheet]] is now decorated by categories, functors between module categories and natural transformations.

##### Resources

* Notes by Bruce Bartlett [[oberwolfach_june2009_schweigert.pdf|1:file]] and [[oberwolfach_june2009_runkel.pdf|2:file]]

* [Notes by Gabriel Drummond-Cole](http://www.math.sunysb.edu/~blafard/tex/sum09/ow_june_8.pdf)

* [[Oberwolfach Workshop, June 2009 -- Monday, June 8|Notes by Urs Schreiber]]

* [[oberwolfach_june2009_pavlov_schweigert_runkel.pdf|Notes by Dmitry Pavlov:file]]

#### Kevin Costello, _Factorization algebras in perturbative QFT_. {#costello}

A [[factorization algebra]] on a manifold $M$ is an object which associates to every ball $B \subseteq M$ a vector space (or cochain complex) $F(B)$; and to every collection $B_1 \coprod \cdots \coprod B_n \subset B_{n+1}$ of disjoint balls in a larger ball, a map $F(B_1) \otimes \cdots \otimes F(B_n) \rightarrow F(B_{n+1})$. These structures are the $C^\infty$ analogues of chiral algebras, as introduced by Beilinson and Drinfeld; they are closely related to [[algebra over an operad|algebras]] for the [[little n-disk operad]] $E_n$.

In this series of lectures, Kevin Costello set up a framework to construct [[factorization algebra|factorization algebras]] from [[perturbative quantum field theory|perturbative]] [[quantum field theory]]. The set up is analagous to the [[deformation quantization]] picture of [[quantum mechanics]]. Just as the [[observable]]s of [[quantum mechanics]] are encoded in an associative algebra, Costello argues that the observables of a QFT on $M$ are encoded in a factorization algebra on $M$, similar to but slightly different from the waqy it works in (euclidean) [[AQFT]]. This factorization algebra arises by quantizing a commutative factorization algebra associated to classical field theory.

This series of lectures finished with the statement of a theorem allowing one to quantize the commutative factorization algebra associated to a classical field theory in a range of situations, including situations of physical interest.  This is joint work with Owen Gwilliam.

##### Resources

* Notes by Bruce Bartlett [[oberwolfach_june2009_costello1.pdf|1:file]], [[oberwolfach_june2009_costello2.pdf|2:file]], [[oberwolfach_june2009_costello2.pdf|3:file]]

* [Notes by Gabriel Drummond-Cole 1](http://www.math.sunysb.edu/~blafard/tex/sum09/ow_june_8.pdf) and [2](http://www.math.sunysb.edu/~blafard/tex/sum09/ow_june_9.pdf)

* [[Oberwolfach Workshop, June 2009 -- Monday, June 8|Notes by Urs Schreiber 1]] and [[Oberwolfach Workshop, June 2009 -- Tuesday, June 9|2]]

* [[oberwolfach_june2009_pavlov_costello.pdf|Notes by Dmitry Pavlov:file]]

#### [[Gabriel C. Drummond-Cole]], _$\infty$-Operads, $BU_\infty$ and $Hypercomm_\infty$_. {#drummond-cole}

Gabriel Drummond-Cole applies the machinery of the [[model category]] of [[operad]]s to extend and explain the Barannikov-Kontsevich passage from differential BV algebras satisfying the $\partial-\overbar{\partial}$ lemma to hypercommutative algebras (Frobenius manifolds). This is joint work with Bruno Valette. The following theorem was proved:

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

Let $V$ be a differential BV-algebra over a field of characteristic zero. Let $H$ be its homology. Then:
1. If $V$ satisfies the noncommutative Hodge to de Rham degeneration condition, then there exists a hypercommutative $\infty$-structure on $H$.
2. If $V$ satisfies Park's semiclassical condition then this structure is unique up to $Hypercomm_\infty$ quasi-isomorphism.
=--

##### Resources

* [[oberwolfach_june2009_pavlov.pdf|Notes by Dmitry Pavlov:file]]

*  Gabriel Drummond-Cole's [website](http://www.math.sunysb.edu/~blafard/)

#### Alex Kahle, _Superconnections and index theory_. {#kahle}

In this talk, Alex Kahle described results from his thesis work (his supervisor was Dan Freed) about [superconnections and index theory](http://arxiv.org/abs/0810.0820). The idea is to prove a local index theorem using [[superconnection]]s via direct geometric-analytical methods (i.e. not stochastic) for use with family index problems, determinant line bundles, etc. Just like [[connection on a bundle|ordinary connection]]s on spinor buundles, a superconnection gives rise to a [[Dirac operator]] on a [[spinor bundle]]. The usual local index theorem (that the trace of the heat kernel converges as $t \rightarrow 0$ to a certain differential form) needs to be modified though, because forms of different degrees have different scaling behaviour (recall that a [[superconnection]] involves forms of different degrees). Alex worked out how to scale everything correctly so that one indeed gets a local index theorem for superconnections, leading to many potential applications.

##### Resources

* [[oberwolfach_june2009_kahle.pdf|Notes by Bruce Bartlett:file]]

* [Notes by Gabriel Drummond-Cole](http://www.math.sunysb.edu/~blafard/tex/sum09/ow_june_10.pdf)

* [[Oberwolfach Workshop, June 2009 -- Wednesday, June 10|Notes by Urs Schreiber]]

* [[oberwolfach_june2009_pavlov_kahle.pdf|Notes by Dmitry Pavlov:file]]

#### [[Chris Schommer-Pries]], _Topological defects, $D$-branes, and the classifications of TFTs in low dimensions_. {#schommer-pries}

Chris explained his classification result mentioned at [[(infinity,n)-category of cobordisms]] from his thesis of extended (he suggested calling them _local_) 2d TQFT's via the explicit generators and relations he obtained on $2Cob$. He also showed how the [[higher category theory|higher-categorical viewpoint]] unites the following two ideas: the 'open-closed' theories and the 'field theories with defects' from Ingo Runkel and Christoph Schweigert's talk. He showed how both these concepts are particular examples of a single notion, namely that of a collection of natural transformations between a restriction of [[FTQFT]] $n$-functors (called "unnatural" or "supernatural" transformations to indicate that it is a transformation not of the entire functor, but just of its restriction to the 2-category of 2-manifolds with only inverttible 2-morphisms).

This is precisely the kind of viewpoint that Urs Schreiber and Jens Fjelstad [suggested](http://golem.ph.utexas.edu/category/2007/08/dbranes_from_tin_cans_part_x.html) at the [n-category cafe](http://golem.ph.utexas.edu/category/) under the slogal [D-branes from tin-cans](http://golem.ph.utexas.edu/category/2006/10/dbranes_from_tin_cans_arrow_th.html) (a D-[[brane]] is a boundary condition ordefect, a "tin-can diagram" is the naturality diagram for a higher natural transformation).

By the way, there's much more to come from Chris, he is working on the 3d theory and much besides!

##### Resources

*  [Slides on webpage](http://sites.google.com/site/chrisschommerpriesmath/Home/Slides-MFO-6-11-09.pdf?attredirects=0)

#### Dan Freed, _Geometry and topology of orientifolds_. {#freed}

Freed described recent work with [[Jacques Distler]] and Greg Moore on a certain background structure in [[string theory|superstring theory]] called an [[orientifold], which is a [[bundle gerbe]] on a $\mathbb{Z}_2$ [[orbifold]] with a peculiar "twisted" [[equivariance]] condition. There is apparantly a beautiful string theory formula to calculate the '[[RR charge]]', namely something like
\[
 RR charge = \pm 2^{something} i_* \sqrt\left( \frac{L'(F)}{L'(\nu)} \right)
\]
involving a certain normal bundle and the 'Bott element in K-theory', which looks similar to the Hirzebruch formula for the L-genus. This uses some kind of twisted [[differential cohomology]] version of KR theory. The whole thing can be viewed as a new sort of anomaly, having to do with an exotic orientation of some kind.

##### Resources

* [[oberwolfach_june2009_freed.pdf|Notes by Bruce Bartlett:file]]

* [Notes by Gabriel Drummond-Cole](http://www.math.sunysb.edu/~blafard/tex/sum09/ow_june_8.pdf)

* [[Oberwolfach Workshop, June 2009 -- Monday, June 8|Notes by Urs Schreiber]])

* [[oberwolfach_june2009_pavlov_freed.pdf|Notes by Dmitry Pavlov:file]]

#### Dan Freed, Evening informal session on RR fields and all that.

Summary appearing soon...

##### Resources

*  [[oberwolfach_june2009_freed_extra.pdf|Notes by Bruce Bartlett:file]]

#### Scott Wilson, _Some algebra relat4ed to mapping spaces and applications_. {#wilson}

Wilson defined a _partial algebra_ as a lax [[monoidal functor]] from the category of finite sets to [[chain complex|chain complexes]], such that the map $A(j \coprod k) \rightarrow A(j) \otimes A(k)$ is a quasi-isomorphism. He showed how these pop up in [[homotopy theory]] all the time.

(Remark: this formula is the basis for [[Jacob Lurie]]'s discussion of [[commutative algebra in an (infinity,1)-category]] in general and of [[symmetric monoidal (infinity,1)-category]] in particular).

For instance, if $Y$ is any space and $A$ a partial algebra, the resulting total complex can be seen as a generalization of [[Hochschild cohomology]]. One really interesting example (for [[Bruce Bartlett|Bruce]]) was where $Y$ is the interval, and $A$ is the partial algebra of forms on a Riemannian manifold $M$. It turns out that a certain canonical equation which pops out is precisely the Navier-Stokes equation. Gulp!

##### Resources

* [[oberwolfach_june2009_wilson.pdf|Notes by Bruce Bartlett:file]]

* [Notes by Gabriel Drummond-Cole](http://www.math.sunysb.edu/~blafard/tex/sum09/ow_june_10.pdf)

* [[Oberwolfach Workshop, June 2009 -- Wednesday, June 10|Notes by Urs Scheriber]]

* [[oberwolfach_june2009_pavlov_wilson.pdf|Notes by Dmitry Pavlov:file]]

#### [[Urs Schreiber]], _Background fields in twisted differential nonabelian cohomology_. {#schreiber}

What can I ([[Bruce Bartlett|Bruce]]) say about Urs's talk? Many mere mortals might think it impressive to pass from ordinary cohomology to twisted cohomology, or to differential cohomology, or to nonabelian cohomology. Urs does this all in one step! Yes, we're talking about _twisted differential nonabelian cohomology_. As we all know and love, Urs has developed some $\infty$-machinery based on all sorts of work which enables him to 'turn the crank' and output all the cohomology theories which mathematicians and physicists currently are bumping into. A great application of his machinery is an understanding of the 'Green-Schwarz anomaly cancellation' mechanism in terms of twisted nonabelian String-gerbes with connection and the Chern-Simons 2-gerbe. Unfortunately Urs didn't have time to quite get to that point, but he did at least mention other examples like how his machinery naturally produces the 'twisted Bianchi identities' in twisted flat differential cohomology, as well as nonabelian gerbes. Urs ended his talk by explaining how nonabelian cohomology on $X$ is related to _twisted_ abelian cohomology on $X$. This ties in with Konrad's talk, where the abelian viewpoint is taken, but the nonabelian viewpoint is essential for some applications.

##### Resources

* [[oberwolfach_june2009_schreiber.pdf|Notes by Bruce Bartlett:file]]

* [Notes by Gabriel Drummond-Cole](http://www.math.sunysb.edu/~blafard/tex/sum09/ow_june_11.pdf)

* Urs's personal nLab write-up on [[schreiber:Background fields in twisted differential nonabelian cohomology|Background fields in twisted differential nonabelian cohomology]]

#### André Henriques, _Invertible conformal nets_. {#henriques}

In this talk, Henriques described joint work with Chris Douglas and Arthur Bartels on the program of establishing an equivalence between full conformal field theories and 'conformal nets'. Preliminary write-ups of this work is available on his [webpage](http://www.math.uu.nl/people/henrique/). A conformal net is something like a [[factorization algebra]] (in the sense of Costello's talk) on 1-dimensional manifolds taking values in the bicategory of Von Neumann algebras with [[bimodule]]s as 1-morphisms. The $\mu$_index $\mu(A)$ of a conformal net $A$ is defined to be the statistical dimension of the vacuum sector $H_0$, viewed as some kind of bimodule involving things which $A$ assigns to arcs on the unit circle. He stated the following result.

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

A [[conformal net]] is invertible if and only if $\mu(A)=1$, and it is fully dualizable (in the sense of Lurie) if and only if $\mu(A) \lt \infty$.
=--

##### Resources

* [Notes by Gabriel Drummond-Cole](http://www.math.sunysb.edu/~blafard/tex/sum09/ow_june_11.pdf)

*  [[Oberwolfach Workshop, June 2009 -- Thursday, June 11|Notes by Urs Schreiber]]

* [[oberwolfach_june2009_pavlov_henriques.pdf|Notes by Dmitry Pavlov:file]]

#### André Henriques, Evening informal session on more details on conformal nets.

André responded to questions from the audience about further details regarding the 'conformal nets' programme.

##### Resources
* [[oberwolfach_june2009_henriques_extra.pdf|Notes by Bruce Bartlett:file]]

* Write-up about conformal nets on André Henriques' [webpage](http://www.math.uu.nl/people/henrique/).

#### [[Konrad Waldorf]], _String connections and Chern-Simons 2-gerbes_. {#waldorf}

Konrad described the [[Chern-Simons 2-gerbe]] $CS_P$ which is a geometric object living on any manifold $M$ which is equipped with a Spin-bundle $P$. Its definition involves some kind of [[pullback]] of the basic bundle gerbe over the Spin group. He demonstrates that a [[string structure]] $S$ on $P$ is a _trivialization_ of this 2-gerbe, in the sense of a morphism from the trivial 2-gerbe living over $M$ to $CS_P$. Such a trivialization only exists precisely if the first fractional Pontryagin class of the bundle vanishes, $\frac{1}{2} p_1(P) = 0$, otherwise the Chern-Simons 2-gerbe is globally nontrivial and a string structure doesn't exist. He defined a  _string connection_ on a string structure $S$ (a connection on $P$ is needed here) on $P$ as a connection on this trivialization $S$ which is compatible with various things.

Basically, the data of a string structure together with a string connection on $P$ is a class in the _$[CS_P]$-twisted_ differential cohomology of $M$ in Urs's sense, for vanishing twist (the twist is the nontriviality of the String structure). Konrad described various results about string connections, such as the fact that they form a contractible space, confirming a conjecture by Stolz and Teichner. See his recent [arXiv article](http://arxiv.org/abs/0906.0117).

##### Resources

* [[oberwolfach_june2009_waldorf.pdf|Notes by Bruce Bartlett:file]]

* [Notes by Gabriel Drummond-Cole](http://www.math.sunysb.edu/~blafard/tex/sum09/ow_june_11.pdf)

*  [[Oberwolfach Workshop, June 2009 -- Thursday, June 11|Notes by Urs Schreiber]]

#### Kevin Walker, _Blob homology_. {#walker}

Kevin Walker described a new way to think about [[FQFT|extended TQFTs]]. Namely to an n-manifold $M$ and an [[n-category]] $C$ he defined a '[[blob complex]]' $B_*(M, C)$. The construction produces a kind of 'derived version' of an extended TQFT. The idea is that the usual quantum invariants $Z(M)$ to manifolds of dimension $\leq n$ are produced out of this chain complex, with the Atiyah-Segal axioms following as corollaries of the setup. He listed a number of examples of the TQFT's he has in mind in the beginning, such as the finite group model, one based on a *-algebra, one based on a [[pivotal category]], one based on a [[braided monoidal category|braided]] [[ribbon category]], and even one based on contact structures.

##### Resources

*  Slides available on [webpage](http://canyon23.net/math)

#### Ralph Cohen, _String topology, field theories and Fukaya categories. {#cohen}

This talk was about relating the subject of [[string topology]], which is now 10 years old, with other field theories, especially in the light of recent work by Costello, Hopkins and Lurie. [[string theory|String theory]] was described as a 'homological [[conformal field theory]]'. The slogan was that string topology simplifies when one applies Poincaré duality. A relation was sketched between string topology and [[Gromov-Witten theory|Gromov-Witten symplectic field theory]]. Ultimately it was conjectured that the symplectic field theory of $T^*M$ is equivalent to string topology of $M$.

##### Resources

* [[oberwolfach_june2009_waldorf.pdf|Notes by Bruce Bartlett:file]]

* [Notes by Gabriel Drummond-Cole](http://www.math.sunysb.edu/~blafard/tex/sum09/ow_june_12.pdf)

#### Corbett Redden, _String structures, 3-forms, and tmf classes_. {#redden}

Corbett began my defining a [[string structure]] on a manifold $M$ as a certain lift of a classifying map; string structures up to homotopy are canonically isomorphic with things called 'string classes' on $M$ - namely, elements in the third integral cohomology of the total space of the bundle which behave on the fibers as the generator of the third cohomology of the Spin group. The point about string structures is that they transgress to give a spin structure on the loop space of the manifold, which is what one wants to study things like the Witten genus. He showed a beautiful way to obtain string structures from a metric on $M$, using Hodge representatives for the forms, and then passing to the 'adiabatic limit', to obtain a certain 3-form $H$. A theorem of Mazzeo-Melrose, Dai and Forman then says that the kernel of the Hodge laplacian extends smoothly in the adiabatic limit, and comes from a filtration in the Serre spectral sequence (something like that). He gave a hypothesis that if a manifold $M^n$ admits a spin and a string structure and a Riemannian metric $g$ such that the Ricci curvature of $g$ is positive and that $H=0$, then a certain invariant $\sigma(M, S)=0$ inside $tmf^{-n}(pt)$. He showed that all these conditions are necessary!

##### Resources

* [[oberwolfach_june2009_redden.pdf|Notes by Bruce Bartlett:file]]

* [Notes by Gabriel Drummond-Cole](http://www.math.sunysb.edu/~blafard/tex/sum09/ow_june_11.pdf)

* [[Oberwolfach Workshop, June 2009 -- Thursday, June 11|Notes by Urs Schreiber]]

#### Mike Hopkins, _The Kervaire invariant_. {#hopkins}

Mike Hopkins gave a nice story of the history and origins of the [[Kervaire invariant]] in the classification of [[manifold]]s. He spoke about Pontryagin's work about the cobordism groups of stably framed manifolds in the 1930's, and the mistake he made in dimension $n=2$! Basically he thought a certain function was linear, when in fact it was quadratic. This led to the Kervaire invariant being introduced. The problem about when the Kervaire invariant vanishes has been solved in dramatic fashion by Hopkins, Hill and Ravenel very recently... if a manifold has Kervaire invariant $1$, then its dimension must be either 2, 6, 14, 30, 62 or 126, with the final of these remaining open! See the notes for the great story. Hopkins ended with a nice geometric description of the Kervaire invariant on spheres in dimension 2 and 6, relating it to complex structures and exceptional Lie groups. He wondered if these things feature in dimension 126?

##### Resources

* [[oberwolfach_june2009_hopkins.pdf|Notes by Bruce Bartlett:file]]

* [Notes by Gabriel Drummond-Cole](http://www.math.sunysb.edu/~blafard/tex/sum09/ow_june_12.pdf)

* Andrew Ranicki's [webpage](http://www.maths.ed.ac.uk/~aar/exotic.htm) on exotic spheres

* Douglas Ravenel's [webpage](http://www.math.rochester.edu/u/faculty/doug/).

#### Mike Hopkins, _Evening informal session on technical details of the solution of the Kervaire invariant problem_. {#hopkins_extra}.

Notes needed... help...

##### Resources
* [[oberwolfach_june2009_freed_extra.pdf|Notes by Bruce Bartlett:file]]

#### David Chataur, _Evening informal session explaining recent results in operads_ {#chataur_extra}.

* [Notes by Gabriel Drummond-Cole](http://www.math.northwestern.edu/~gabriel/notes/sum09/ow_june_11.pdf)

#### Kishore Marathe, _Evening informal session on the gauge theory to string theory correspondence_ {#marathe_extra}.

Notes needed... help...

# Pictures

* <http://www.mfo.de/cgi-bin/tagung_espe?type=21&tnr=0924>

See also [[Oberwolfach Workshop, June 2009 -- Abstracts|talk summaries]].


Oberwolfach Workshop, June 2009 -- Thursday, June 11

Here are notes by [[Urs Schreiber]] for Thursday, June 11, from [[Oberwolfach Workshop, June 2009 -- Strings, Fields, Topology|Oberwolfach]].

## Henriques: Invertible conformal nets ##

André Henriques has notes on this work available on [his website](http://www.math.uu.nl/people/henrique/);

* Chris Douglas, André Henriques, Michael Hill, _Geometric String structures -- notes on $\mathbb{Z}_2$-graded conformal nets_ ([ps](http://www.math.uu.nl/people/henrique/TringWP.ps))

today: classification of invertible objects in a 3-category

**theorem** (André Henriques, Arthur Bartels, Chris Douglas):
_Conformal nets form the objects of a symmetric monoidal
3-category CN3_ .

* there is something about this written up to be found on Henriques' webpage.

**definition** A conformal net is a [[factorization algebra]] on the
category of 1-dimensional balls with values in
von Neumann algebras.

* 1-dim ball is just an interval, of course

* so a conformal net associates to each interval a vN algebra, and for each
inclusion of intervals there is an inclusion of vN algebras

for $I, J$ two intervals equipped with an embedding into interval $K$
the net assigns

$$
  A : (J,K \subset I) \mapsto (A(J) \otimes A(K) \to A(I))
$$

>Freed: why do you call this conformal ?

> André: (as far as I understand the reply he gives after my question) the usual covariance condition under Moebius group imposed on conformal nets is taken care of here by havin not intervals in the
real line on the line but abstract intervals with maps between them being  embeddings. The claim is that this automatically implies the right kind of covariance that is otherwise imposed by hand.

**claim**

$$
  \{unitary full CFTs\} \simeq \{conformal nets\}
$$

>question: do you really mean _full_ instead of _chiral_ CFTs on the right?

>answer: yes

symmetric monoidal structure is objectwise the tensor product

**theorem** aa conformal net $A$ is invertible  in CN3 iff its
$\mu$-index $\mu(A)$ equals 1.

It is fully dualizable iff $\mu(A) \lt \infty$

in general $\mu(A) \in \{0\} \cup [1, \infty]$

### von Neumann algebras ###

in the 2-category of vN algebras, a morphism

$$
  A \to B
$$

is a bimodule ${}_A H_B$ where $H$ is a Hilbert space

the unit
$$
   Id_A : A \to A
$$

is given by the bimodule called $L^2(A)$

this comes from the following: if $X$ is a measure space and $A$ arises as
$L^\infty(X)$ then $L^2(A)$ is $L^2(X)$.

the general $L^2(A)$ is a non-commutative analog of this construction, for
$A$ which are not of the form "measurable functions on a measure space".

One word about composition, if
$$
  A \stackrel{H}{\to} B  \stackrel{K}{\to} C
$$

given by $H \otimes_{Connes} K$ is the Connes-Fusion operation

associated to ${}_A H_B$ is

$$
  dim({}_A H_B) \in \{0\} \cup [1,\infty]
$$

$$
  dim = \left\{
    \array{
      0 & iff H=0
      \\
      1 & iff H is invertible
      \\
      \lt \infty & iff H has a dual
    }
  \right.
$$

**fact** think of an interval $I$ as the upper semicircle.
The two actions of $A(I)$ on
$H_0 = L^2(A(I))$ can be used to define an
action of $A(J)$ on $H_0$ for every $J \usbset S^1$.

Because if $J$ is in the upper semicircle, use the left action,
if in the lowwer semicircle use the right action, if it is neither,
do something else (not explained here)

so $H_0$ has an action of $A(S^1)$.

so now everything is invariant projectively
with respect to diffeomorphism group

**definition**

given two circles $S_a, S_b$, let $nw(S_a)$ be north west quarter cicle, etc

consider $H_0$ as a bimodule over

$$
  A(sw(S_a)) \otimes A(ne(S_a))
$$

and

$$
  A(nw(S_b)) \otimes A(se(S_b))
$$

claim: $\mu(A) = dim(H_0)$

now some ideas on the proof of the first theorem

define always

$$
  \bar A(I) = A(I)^{op} = A(\bar I)
$$

* if $A$ is dualizable, then $\bar A$ is its dual

* if $A$ is invertible, then $\bar A$ is its inverse

a 1-morphism in CN3, to be called a _defect_ is

consider the category of
bi-colored 1-dimensional manifold is a 1-d manifolds
(decomposition into intervals labeled by two things)
with morphisms being color-preserving maps

sufficient to look at intervals

* completely blue

* half blue, half orange

* completely orange

now a defect/1-morphism is a conformal net as above on bi-colored 1d manifolds

so suppose three nets and two defects

$$
  A \stackrel{D}{\to} B \stackrel{E}{\to} C
$$

then define the composition defect net by

$$
  D \otimes E ( blue-interval glued to yellow interval)
  =
  insert orange interval in the middle and average over all local evaluations
$$

the diffeomorphism group of the orange interval does act on the space
of all possible compositions here, which one shows to all be coherently isomorphic

now construct a defect

$$
 D:  A \otimes \bar A \to 1
$$
 by setting

$$
  D(interval)
  := A(bent-around interval)
$$

(whatever that means)

## Henriques: informal evening session ##

things we could talk aout:

* composition of defects

* examples of conformal nets

* $\mathbb{Z}_2$-graded version of all that

>Mike Hopkins: could we talk about relation full CFT to chiral CFT?

### examples for conformal nets ###

* from the standard chiral positive energy nets that are usually
called "conformal nets" in the literature we get the more
general considered here by tensoring two chiral bits ("left" and "right")
and choosing an extension

examples can be obtained as follows

* loop group nets ($S^1 \to \tilde {L G} \to L G$)

* integral lattice

  * moonshine net: take the Leech lattice and do some orbifolding

    * its full automorphism group is isomorphic to the Monster

* free fermion

* minimal model = Virasoro

general remarks:

* recall the Hillber space $H_0$ obtained from the full cnformal net as the
algebra of the upper semi circle

* but in practice one first constructs $H_0$ and then from that the net

* a functor
  $$
    \left\{
      subintervals of S^1,
      diffeos between them
    \right\}
    \to V N
  $$
  is already the same as a functor on all abstract
  intervals, since that category
  is equivalent to intervals in the circle

so given an interval $I$, construc the set of maps

$$
  I \mapsto Maps((I, \partial I), (G, \{e\}))
$$

if $I$ happens to be thought of as embedded in the circle, then
this group could be identified with the subgroup of the loop group
whose elements send the complement of $I$ to $e$.

but the point is there is way to get a conformal net here without assuming
embedding into $S^1$

so the algebra we will assign to $I$ is just the group algebra

$$
  I \mapsto \mathcal{C}(Maps((I, \partial I), (G, \{e\}))]
$$

forgetting all topology, but this still depends on the central extension
and it sits densely in the vN algebra obtained by taking the group,
letting it act on the vacuum in the Hilbert space $H_0$ through a rep of
$\tilde {L G}$

so $A(I)$ is now defined to be the completion of that algebra in $B(H_0)$

this here gives a chiral theory

>audience: one should think of intervals here as thickened points and of the vN algebras as what is assigned to the point in an extended CFT

### algebra assigned to the circle ###

here is how to get the algebra assigned to the circle:

puff up the circle to an annulus, cut this vertically in two pieces;
call the two intervals where the cut goes $I$ and $J$,

let $H_0$ be the Hilbert space assigned to the boundary circles of the
two pieces of the cut annulus. Then the algebra in question is the
Connes fusion product of $H_0$ with itself over $A(I) \otimes A(J)$

$$
  A(circle) := H_0 \otimes^{Connes}_{A(I) \otimes A(J)} H_0
$$

>audience: so the circle doesn't really appear in this formula, it's just a thing to help us think about this

>answer: yes

now recall the definition of tensor product of nets

$$
  \left(
    A \otimes A^{op}
  \right)(I)
  :=
  A(I) \otimes A(I)^{op}
$$

then consider extensions $B$ of nets of the following form

$$
  A \otimes A^{op} \hookrightarrow B
$$

such is cooked up from every Frobenius algebra object $Q$ in $Rep(A \otimes A^{op})$
(in vN theorists jargon this is a "$Q$-system")

in the 3-category VN3 this $Rep(A)$ can be written as

$$
  Rep(A)
  =
  End_{VN3}(Id_A)
  =
  \left\{
    \array{
        & \nearrow \searrow
       \\
       A &\Downarrow^{\rho}& A
       \\
       & \searrow \nearrow
    }
  \right\}
$$

to the circle $B$ assigns $Q$

##Schreiber##

detailed notes with further links and backgeound information are here:

* [[schreiber:Background fields in twisted differential nonabelian cohomology]]

The point of this is to define a notion of twisted differential nonabelian cohomology for every fibration sequence of smooth $\infty$-groupoids and then establish from that the following list of examples.

We now

* list fibration sequence of smooth $\infty$-groupoids

* and indicate properties of the corresponding differential twisted nonabelian cohomology

**Examples / Claim**

* fibration sequence: $\mathbf{B}U(n) \to \mathbf{B} PU(n) \to \mathbf{B}^2 U(1)$

  * twisting cocycle: lifting gerbe;

  * twisted cocycle: twisted bundles / gerbe modules

  * twisted Bianchi identity: $d F_\nabla = H_3$

  * occurence: Freed-Witten anomaly cancellation on [[nLab:brane|D-brane]]

* fibration sequence: $\mathbf{B}String(n) \to \mathbf{B} Spin(n) \stackrel{\frac{1}{2}p_1}{\to} \mathbf{B}^3 U(1)$

  * twisting cocycle: Chern-Simons 2-gerbe;

  * twisted cocycle: twisted nonabelian String-gerbe with conection

  * twisted Bianchi identity:
    $d H_3 \propto \langle F_\nabla \wedge F_\nabla \rangle$

  * occurence: [[nLab:Green-Schwarz mechanism|Green-Schwarz anomaly cancellation]]

  * Proof.

    * (with Danny Stevenson and Christoph Wockel: ([SSSS](http://www.math.uni-hamburg.de/home/schreiber/nactwist.pdf))) use BCSS model ([BCSS](http://arxiv.org/abs/math/0504123)) of $String(n)$ with [Brylinski-McLaughlin construction](http://golem.ph.utexas.edu/category/2008/02/construction_of_cocycles_for_c.html) of $\frac{1}{2}p_1$

    * (using ([SaScSt I](http://arxiv.org/abs/0801.3480), [SaScSt III](http://www.math.uni-hamburg.de/home/schreiber/5twist.pdf)):) compute local differential form data after differentiating smooth $\infty$-groupoids to  [[nLab:Lie infinity-algebroid|L-infinity algebroids]] using the formalism of ([SaScSt I](http://arxiv.org/abs/0801.3480))

  * for aspects of the twisted case see also

    * [AsJu](http://arxiv.org/abs/hep-th/0409200)

    * for aspects of the untwisted case see also ([Wa](http://arxiv.org/abs/0906.0117))

* fibration sequence: $\mathbf{B}Fivebrane(n) \to \mathbf{B} String(n) \stackrel{\frac{1}{6}p_2}{\to} \mathbf{B}^7 U(1)$

  * twisting cocycle: Chern-Simons 6-gerbe;

  * twisted cocycle: twisted nonabelian Fivebrane-gerbe with connection

  * occurence: dual [[nLab:Green-Schwarz mechanism|Green-Schwarz anomaly cancellation]] for NS 5-brane magnetic dual to string

  * ([SaScSt II](http://arxiv.org/abs/0805.0564) [SaScSt III](http://www.math.uni-hamburg.de/home/schreiber/5twist.pdf))

* fibration sequence: $\mathbf{B}^2 U(1) \to \mathbf{B} (U(1) \to \mathbb{Z}_2) \stackrel{}{\to} \mathbf{B} \mathbb{Z}_2$

  * twisting cocycle: $\mathbb{Z}_2$-orbifold;

  * twisted cocycle: [[nLab:orientifold|orientifold gerbe]] / Jandl gerbe with connection

  * occurence: unoriented string

  * unwrap the above abstract nonsense and use the above results to find [SchrSchwWal](http://arxiv.org/abs/hep-th/0512283) and the bosonic part of [DiFrMo](http://arxiv.org/abs/0906.0795)

***

[[Oberwolfach Workshop, June 2009 -- Wednesday, June 10|Previous day]] --- [[Oberwolfach Workshop, June 2009 -- Strings, Fields, Topology|Main workshop page]] --- [[Oberwolfach Workshop, June 2009 -- Friday, June 12|Next day]]


Oberwolfach Workshop, June 2009 -- Tuesday, June 9

Here are notes by [[Urs Schreiber]] for Tuesday, June 9, from [[Oberwolfach Workshop, June 2009 -- Strings, Fields, Topology|Oberwolfach]].

## Schick: differential cohomology ##

smooth cohomology

* idea:

  * combine cohomology + differential forms

main diagram

$$
  \array{
    \hat H(M) &\stackrel{I}{\to}& H^\bullet(M)
    \\
    \downarrow^{R} && \downarrow
    \\
    \Omega^\bullet_{d=0}(M)
    &\stackrel{}{\to}&
    H^\bullet_{dR}(M)
    \simeq H^\bullet(M,\mathbb{R})
  }
$$

so differential cohomology $\hat H^\bullet(M)$ combines the ordinary cohomology $H^\bullet(M)$
with a differential form representative of its image in real cohomology.

* $I$ projects a differential cohomology to its underlying ordinary cohomology class;

* $R$ send the differential cohomology class to its __curvature_ differential form data

we want an exact sequence

$$
 \array{
   H^{\bullet-1}(M)
   &\stackrel{ch}{\to}&
   \Omega^{\bullet-1}(M)/{im(d)}
   &\stackrel{d}{\to}&
   \hat H(M)
   &\stackrel{I}{\to}&
   H^\bullet(M) \to 0
   \\
   &&&
   {}_{d}\searrow
   &
   \downarrow^R
   \\
   &&&&
   \Omega^\bullet_{d=0}(M)
 }
$$

**definition**

Given cohomology theory $E^\bullet$, a smooth refinement
$\hat E^\bullet$ is a functor $\hat E : Diff \to Grps$
with transformations $I, R$ such that

$$
  \array{
    \hat E(M) &\stackrel{I}{\to}& E^\bullet(M)
    \\
    \downarrow^{R} && \downarrow
    \\
    \Omega^\bullet_{d=0}(M, V)
    &\stackrel{}{\to}&
    E^\bullet_{dR}(M)
    \simeq E^\bullet(M,\mathbb{R})
  }
$$

where

$V = E^\bullet(pt)\otimes \mathbb{R}$

is the graded non-torsion cohomology of $E$ on the point. So now all the
gradings above denote total grading.

and such that there is a transformation

$$
  a : \Omega^{\bullet -1}(M)/{im(d)}
  \to
  \hat E^*(M)
$$

that gives the above kind of exact sequence

**definition**

if $E^*$ is multiplicative, we say $\hat E^*$ is multiplicative with
product $\vee$
if $\hat E$ takes values in graded rings and the transformations are
compatible with multiplicative structure, where

$$
  a(\omega) \vee x = a(\omega \wedge R(x))
$$

**definition**

$\hat E$ has $S^1$-integration if there is a natural (in $M$) transformation

$$
  \int : \hat E^*(M \times S^1) \to \hat E^{\bullet -1}(M)
$$

compatible with $\int$ of forms  and for $E$ it is given by the suspension isomorphism

and

$$
  \int \circ p^* = 0
$$

for $p : M \times S^1 \to M$ and

$$
  \int \circ (  id \times (z \mapsto \bar z) )^* = - \int
$$

**remark** ordinary cohomology theories are supposed to be homotopy
invariant, but differential forms are not, so in general the differential
cohomology is not

**Lemma** Given $\hat E$ a smooth cohomology theory.
The **homotopy formula**:

given $h : M \times [0,1] \stackrel{smooth}{\to} N$
a smooth homotopy we have

$$
  h^*_1(X) - h^*_0(X)
  =
  a( \int_{M \times [0,1]/M}  h^*(R(x)))
$$

**corollary** $ker(R)$ (i.e. flat cohomology) is a homotopy invariant
functor

**def** $\hat _H{flat} := ker(R)$

**proof of lemma**

suffices to show

$$
  \iota_1^*(x) - \iota_0^*(x) = a(\int_{M\times [0,1]/M} R(x))
$$

for all $x \in \hat E(M \times [0,1])$

observe if $x = p^* y$ the left hand side vanishes, $\int R(p^* y) = 0$

for general $x$ $\exists y \in \jhat E(M)$; $x - p^*(y) = a (\omega)$
$\omega \in \Omega(M \times [0,1])$

Stokes' theorem gives $i^*_1 \omega - i^*_0 \omega = \int_{[0,1]} d \omega$
$= \int R(a(\omega)) = \int R(x-p^* \omega) = \int R(x)$

on the other hand

$$
  i^*_1(x) - i^*_0(x) = i^*_1(a(\omega))
    - i^*_0(a(\omega))
    = a(\int R(x))
$$

a calculation: $\hat H^1_{flat}(pt) = \hat H^1(pt) = \mathbb{R}/\mathbb{Z} = \hat K^1(pt)$

**Theorem (Hopkins-Singer)**

For each generalized cohomology theory $E^*$
a differential version  $\hat E^*$ as in the above definition does exist

Moreover $\hat E_{flat}^* = E \mathbb{R}/\mathbb{Z}^{\bullet -1}$

**remark** it's not evident hot to obtain more structure like multiplication

**theorem** using geometric models, multiplicative smooth extensions
with $S^1$-integration are constructed for

* K-theory (Bunke-Schick)

* MU-bordisms (unitary bordisms)
  (Bunke-Schröder-Schick-Wiethaupts; and from there Landweber exact cohomology theories)

**uniqueness theorem (Bunke-Schick)**
(Simons-Sullivan proved this for ordinary integral cohomology)

assume $E^*$ is _rationally even_, meaning that

$$
  E^k(pt)\otimes \mathbb{Q} = 0 \;\; for odd k
$$

plus one further technical assumption

then any two smooth extensions $\hat E^*$, $\tilde E^*$ are naturally
isomorphic

such that if required to be compatible with integration the ismorphism is unique

if $\hat E, \tilde E$ are multiplicative, then this isomorphism is,
as well.

**example** if we don't require compatibility with $S^1$-integration, then there
are "exotic" abelian group structures on $\hat K^1$

## Bunke: smooth K-theory ##

(I gave up taking notes in that one, maybe somebody else
has notes?)

## Costello, part II ##

alternative to yesterday's axioms:

replace $B(M)$ by $Embeddings(\bar D^n, M)$

and replace $B_n(M)$ by

$$
  Embed(D^n, M) \times Enb(\bar D^n \coprod \cdots \coprod \bar D^n, \bar D^n)
$$

* what is the classical analog of a factorization algebra?

* and how do we get classical QFT?

**basic idea**

factorization algebras form a symmetric monoidal category

so we can look at [[algebra over an operad]] in the category of factorization
algebras

if $F, F'$ are factorization algebras, then
$$
  (F\otimes F')(B) = F(B) \otimes F'(B)
$$

**def** a **classical factorization algebra** is a commutative algebra
in the category of factorization algebras

recall, an $E_\infty$ object in $E_n$-algebras is an $E_\infty$-algebra

idea of how to associate a classical factorization algebra to a classical field
theory is as follows

suppose we have classical field theory, e.g. space of fields is
section of a vector bundle $E \to M$

$$
  S : \Gamma(M,E) \to \mathbb{R}
$$

is the classical action

$S$ is local: obtained by $int$ of a Lagrangian

if $B  \subset M$ is a ball, let

$$
  EL(B) =
  \left\{
    \phi \in \Gamma(interior(B), E)
    such that \phi satisfies Euler-Lagrange equations
  \right\}
$$

>Freed: notice that you are doing here classical QFT in Euclidean signature
>Costello: yes

**rough idea**

the classical factorization algebra $X_S$ associated to $S$
assigns to $B$, the algebra

$$
  O(EL(B))
$$

of functions on the set of solutions to $EL$.

we want maps

$$
  X_S(B_1) \otimes \cdots \otimes X_S(B_n)
  \to
  X_S(B_{n+1})
$$

for $B_i$ in $B_{n+1}$

we have a map

$$
  EL(B_{n+1}) \to EL(B_1)\times \cdots \times EL(B_n)
$$

this yields a map

$$
  O(EL(B_1)) \otimes \cdots \otimes O(EL(B_{n+1}))
$$

as desried

**simple example**

fields are $C^\infty$ functions on $M$

$$
  S(\phi) := \int_M \phi \Delta \phi
$$

Euler-Lagrange equation is $\Delta \phi = 0$

$$
  EL(B) = \{Harmonic functions on the interior of B\}
$$

$$
  O(EL(B)) = \prod_{n \geq 0} Hom(EL(B)^{\otimes n}, \mathbb{R})^{S_n}
$$

where Hom means continuous linear maps, and where $\otimes$ is the completed
tensor product

later, for more complex examples, what we really want to do is
to take the _derived_ space of EL solutions

**question**

Why does this classical factorization algebra want to become just a
factorization algebra?

recall that fact-algebras form a symmetric monoidal category

the $E_0$-[[operad]] is defined by

* $E_0(n) = \emptyset$ for $n \geq 1$

* $E_0(0) = pt$

so for instance an $E_0$-algebra in $Vect$ is a vector space with an element

forgot to mention that factorization algebras need to have a unit,
a section $F$ on $B(M)$ which is a unit for the product

So: an $E_0$-algebra in factorization algebras is just a factorization algebra

$$
 \array{
     graded commutative algebra with Poisson bracket of deg +1
     &\stackrel{quantize}{\to}&
     E_0-algebras (in co[[chain complex]]es)
     \\
     Poisson &\stackrel{quantize}{\to}& E_1-algebras
     \\
     graded comm algebra with Poisson bracket of deg -1
    &\stackrel{quantize}{\to}&
    E_2-algebras
     \\
     graded comm algebra with Poisson bracket of deg -2
    &\stackrel{quantize}{\to}&
    E_3-algebras
  }
$$

Beilinson and Drinfeld define an operad over (i.e. in the category of
cochain complex moudles over) the ring of formal power
series over $\hbar$
$$
  \mathbb{R}[\![\hbar]\!]
$$

as follows:

* generated by $\cdot$, a commutative product

* $\{-,-\}$ a Poisson bracket of deg +1

* with differential $d(-) = \hbar \{-,-\}$

call this the **BD operad**

$$
  BD/\hbar BD = operad of commutative algebras with \{-,-\} of deg +1
$$

$$
  H_\bullet(BD(n) \otimes_{\mathbb{R}[\![\hbar]\!]} \mathbb{R}(\hbar))  = 0
$$

so

$$
  BD \otimes_{\mathbb{R}[\![\hbar]\!]} \mathbb{R}[\![\hbar]\!]
  \simeq E_0
$$

**rant on [[BV-theory]] termionology**

framed $E_2$, which is often called the _BV-operad_
has **nothing** to do with the [[BV-theory]]

instead: the BD-operad from above is the one related to [[BV-theory]]

**def** the $P_0$ (or $Poisson_0$) operad is the operad
of commutative Poisson algebras with $\{-,-\} of deg 1$

so $P_0 = BD/\hbar$

**general fact**

let $M$ be a manifold, and $f : M \to \mathbb{R}$ a function, then
$O(derived critical locus of f)$ is a $P_0$-operad

derived critical locus has as functions the [[differential graded algebra]]

$$
  \cdots
   \Gamma(M,\Lambda^2 T M)
  \stackrel{\vee d f}{\to}
   \Gamma(M,\Lambda^ T M)
  \stackrel{\vee d f}{\to}
   O(M)
  =
  \Gamma(M, \Lambda^\bullet T M)
$$

here $\Lambda^k T M $ is in degree $-k$ with differential $\vee d f$

$$
  \Gamma(M, \Lambda^\bullet T M)
$$

has Schouten bracket, which is of degree +1

This "wants to become" $E_0$

**observation** if $M$ has a [[measure]], then
$O(crit^n(f))$ has a canonical quantization to an
algebra over $BD$.

then quantization is

$$
  \Gamma(M, \Lambda^\bullet T M), \vee d f + \hbar \Delta
$$

here $\Delta$ arises whenever $M$ has a measure

$$
  \Delta X = Div X
$$

$$
  X \in \Gamma(M, T M)
$$

## Costello, part III ##

so recall that the derived critical locis if a function is a $P_0$-algebra,
so it wants to quantize to $E_0$

if we have a classical field theory, the derived space of
solutions to EL yields a $P_0$ algebra in factorizatoin algebra

so it wants to become a factorization algebra

**Example** $\phi \in C^\infty(M)$, $S(\phi) = \int \phi \Delta \phi$

derived space of solutions to EL is the complex

$$
 \array{
  C^\infty(M) &\stackrel{\Delta}{\to}&
    C^\infty(M)
  \\
    0 && 1
 }
$$

if $B \subset M$ is a ball, then

$$
  O(EL^n(B))
  =
  \Pi_{n \geq 0}(
     Hom( C^\infty(int B) \stackrel{\Delta}{\to} C^\infty(int B))^{\otimes n},
     \mathbb{R}
  )^{S_n}
$$

this is a commutative dga and defines a commutative factorizaton algebra

if we add an interaction term to the action functional

$$
  S(\phi)
  =
  \int \phi \Delta \phi + \phi^3
$$

then we get the same algebra of functions but the differential changes

in Yang-Mills theory with gauge Lie algebra $g$:
first, we consider the derived quotient of
$\Omega^1(M)\otimes g$ by $\Omega^0(M)\otimes g$, then, take derived critical
locus of YM action

What we get, when linearized looks like

$$
  (
  \array{
     \Omega^0(M) &\to& \Omega^1(M) &\stackrel{d \star d}{\to}& \Omega^3(M)
       &\stackrel{}{\to}& \Omega^4(M)
     \\
     -1 && 0 && 1 && 2
  }) \otimes g
$$

the algebra of functions iss
$$
  \Pi_{n \geq 0} Hom(E^{\otimes n}; \mathbb{R})^{S_n}
$$

with diffeential including YM action

**theorem**
if we take the derived space of solutions to the EL equations,
looking infinitesimally near a fixed solution, then we find
a $P_0$-algebra internal to factorization algebras on $M$

this amounts to quantizing the action $S$ into a solution
of the quantum master equation

this requires machinery of counter-terms, Wilsonian effective
actions, to even define the quantum master equation

>see Kevin Costello's book linked to on hos website for details

**theorem** (joint with O. Gwilliam)

("wave" version)

conssider the scalar field theory, with an action of the form

$$
  S(\phi) = \int \phi (\Delta \phi + m^2 \phi)
  + arbitrary local higher terms
$$

use the above theorem, around the 0-solution

Let $X_S$ be the classical factorization algebra associated to it
(it is a $P_0$-algebra)

Let $Q^{(n)}(X_S)$ be the set of quantizations

= $\{lifts of $X_S$ to an algebra over BD/{\hbar^{n+1}}\}$

there is a sequence $T^{(n)} \to T^{(n-1)} \to \cdots \to T^{(1)} \to pt$

where $T^{(n)}$ maps to $Q^{(n)}(X_S)$, so that the obvious diagram commutes

where $T^{(n)} \to T^{(n-1)}$ is a torsor for the abelian group of local
functions of the field $\phi$

so $T^{(\infty)} = lim_n T^n$ then

$$
  T^{(\infty)} \simeq \{\sum_{k \geq 1} \hbar^k S^{(k)}\}
$$

$
 S^{(k)}
$ is a local function, but this is non-canonical

**more sophisticated version**

consider any reasonable classical theory, with its
classical factorization algebra $X_S$

let $Q^{(n)}(X_S) = simplicial set of possible quantizations defined mod \hbar^{n+1}$

$
  Der_{loc}(X_S)
$
is the cochain complex of derivations of $X_S$, preserving
$P_0$-structure (is, in fact, local functions on an extended space)

theorem

there exists a sequence of simplicial sets

$$
  T^{(n)} \to T^{(n-1)}
  \to
  \cdots \to
  T^{(1)}
  \to pt
$$

with maps $T^{(n)} \to Q^{(n)}(X_S)$

such that $T^{(n)}$ fits into a [[homotopy limit|homotopy fiber diagram]]

$$
  \array{
    T^{(n)}
    &\to&
    0
    \\
    \downarrow
    &&
    \downarrow
    \\
    T^{n-1} &\stackrel{obstruction}{\to}& Der_{loc}(X_S)[2]
  }
$$

so we get obstructions; for instance for $\phi^4$-theory the obstruction is
the famous $\beta$-function

**theorem**
Let $g$ be a simple Lie algebra. Then there is a
quantization of Yang-Mills theory on $\mathbb{R}^4$
which is "renormalizable"
(behaves well with respect to scaling)

the set of quantizations is 1-dimensional term by term

The set of such quantizations is $\hbar \mathbb{R}[\![\hbar]\!]$

**correlation functions**

Where do correlation functions appear?

If $F$ is a factorization algebra on $M_S$, corresponding to some
QFT, then $F(B) = \{measurements we can make on the ball B\}$

if $B_1, B_2 \subset B$  are disjoint, the maps
$$
  F(B_1)\otimes F(B_2) \to F(B)
$$

is defined by doing both observations

correlation functions should be cochain maps

$$
  \langle
    F(B_1) \otimes \cdots \otimes F(B_n) \to \mathbb{R}
  \rangle
$$

if $B_1, \cdots, B_n$ are disjoint, if
$
  O_i \in F(B_i)
$

then

$$
  \langle O_1, \cdots, O_n\rangle
$$
is a measurement of how to observe $O_i$ correlations

if $B_1, B_2 \subset \tilde B$ the diagram

$$
  \array{
    F(B1) \otimes \cdots \otimes F(B_n)
    &\stackrel{\langle \cdots \rangle}{\to}&
    \mathbb{R}
    \\
    \downarrow && \uparrow^{\langle \cdots \rangle}
    \\
    F(\tilde B) \otimes F(B_3) \otimes \cdots \otimes F(B_n)
  }
$$

should commute (operator product expansion)

we can consider correlation functions with coefficients in any
cochain complex, we require they must satisfy this equation

**def** (Beilinson-Drinfeld)

$$
  CH_\bullet(M,F)
  =
  homotopy universal recipient of correlation functions
$$

$$
  = colim_{B_1, \cdots B_n \subset M disjoint}
F(B_1) \otimes \cdots F(B_n)
$$

(Kevin Walker (blob homology), Jacob Lurie (topological chiral homology))

**lemma**
for a massive scalar field,

$$
  CH_\bullet(M,F) \simeq \mathbb{R}[\![\hbar]\!]
$$

in general

$
  CH_\bullet(M,F)
$ looks like measures on the space of critical points of the classical action

if we perturb around isolated critical points,
$
  CH_\bullet(M,F)
  =
  \mathbb{R}[\![\hbar]\!]
$

in this situation correlation functions exist and are unique

general program: correlation functions define a measure
on the space of classical solutions

(Feynman graphs appear here as homotopies between operads, or something,
see his book)

***

[[Oberwolfach Workshop, June 2009 -- Monday, June 8|Previous day]] --- [[Oberwolfach Workshop, June 2009 -- Strings, Fields, Topology|Main workshop page]] --- [[Oberwolfach Workshop, June 2009 -- Wednesday, June 10|Next day]]


Oberwolfach Workshop, June 2009 -- Wednesday, June 10

Here are notes by [[Urs Schreiber]] for Wednesday, June 10, from [[Oberwolfach Workshop, June 2009 -- Strings, Fields, Topology|Oberwolfach]].

## Alexander Kahle: superconnections and index theory ##

* 1) superconnections

* 2) index theory

* 3) sketch some proofs

### 1) superconnections ###

**definition** A **superconnection** $\nabla_s$ on a
$\mathbb{Z}_2$-graded vector bundle $V \to M$ is an odd derivation
on $\Omega^\bullet(M,V)$

superconnections form an affine space modeled on $\Omega^\bullet(M, End(V))^{odd}$

$$
  End(V)
$$

$$
  \nabla_s = \omega_0 + \nabla + \omega_2 + \omega_3
$$

class in $K$-theory given by a map $V \stackrel{f}{\to} W$

unitary superconnection on $\mathbb{Z}_2$-graded unitary bundles $V$
with map as a above look like

$$
  \nabla_s
  =
  \left(
    \array{
      & f^*
      \\
      f &
    }
  \right)
  + \nabla
$$

**Chern character** by the usual formulas

$$
  ch(\nabla_s) := sTr e^{\nabla^2}
$$

### 2) index theory ###

**definition** Let $M$ be smooth Riemannian and $Spin$,
The Dirac operator associated to $(V \to M, \nabla_s)$
is defined by

* - $$ D(\nabla_s) : \Gamma(S \otimes V)
  \stackrel{\nabla_s \otimes 1 \oplus 1 \otimes \nabla_s}{\to} \Omega^\bullet(M, S \otimes V)
  \stackrel{c(.)}{\to} \Gamma(S \otimes V)
  $$

This is

* an elliptic operator;

* formally self adjoint

* of the form

  $$ D(\nabla_s) = \left(
    \array{
       & D'(\nabla_s)
       \\
       D'(\nabla_s)
    }
   \right)
  $$

* **theorem** (corollary of Atiyah-singer index theory)
  $$
    index(D(\nabla_s))
    =
    index(D(\nabla))
    =
    \int_M \hat A(\Omega^m) ch(\nabla_s)
  $$

so superconnections don't give new topological data: they are geometric objects with the same underlying topology as ordinary connections but refined "geometry"

recall that Atiyah-Singer says that

$$
  Tr \exp(-t D(\nabla_s)^2 ) = index(D(\nabla_s))
$$

the heat semi-group is smoothing, therefore it is represented by
a kernel

$$
  \exp(-t D(\nabla_s)^2) \psi(x)
  =
  \int_M p_t(x,y) \psi(y) d y
$$

$$
  Tr \exp(-t  D(\nabla_s)^2) =
  \int_M Tr p_t(x,x) d vol
$$

the following expected formula which holds for ordinary
connections (due to Ezra Getzler)
_no longer_ holds directly for superconnections

$$
  \lim_{t \to 0}
  Tr p_t(x,x) d vol
  \neq
  (2 \pi i)^{-n/2}
  [
    \hat A(\Omega^m) ch(\nabla_s)
  ]_n
$$

here $n = dim X$ is the dimension of the manifold

problem is that components in a superconnections scale in a different

to make it true, we need to rescale

$$
  \nabla_s^t := |t|^{-1/2} \omega_0 + \nabla + |t|^{1/2} \omega_2 + \cdots
$$

A Riemannian map is a triple $(\pi, g, P)$

$
  \pi : M  \to B
$

a family with fibers close Spin manifolds,
$
  g^{M/B}
$
a metric onm the fibers,
$$
  p : T(M) \to T(M/B)
$$

$$
  \array{
    V, \nabla_s
    \\
    \downarrow
    \\
    M
    \\
    \downarrow^\pi
    \\
    B
  }
$$

  $\pi_* (V)$ : a fibre at $y \in B$ is

  $\Gamma_y(S^{M/B} \otimes V)$

due to Bismut we get from a connection on the top a superconnecction
on the bottom (which is one of the main original motivations to be interested
in superconnection in the first place),
which we tweak here a bit to get a superconnection on $B$
from a superconnection on $V$

$$
  \pi_! \nabla_s = \pi_! \nabla + \pi_! \omega
$$

with $\nabla_s = \nabla + \omega$

$$
  [\pi_! \omega_!]_{\omega}(\xi_1, \cdots, \xi_i)
  =
  c^{M/B}(2 (\tilde \xi_1), 2(\tilde \xi_2) \cdots 2(\tilde \xi_k))
$$

$$
  \pi^r = (\pi, r g^{M/B}, P)
$$

$$
  \lim_{t \to 0}
  ch(\pi_!^t \nabla_s)
  =
  (2 \pi i)^{dim M/B}
  \pi_*
  [
    \hat A(\Omega^{M/B} ch(\nabla_s))
  ]
$$

the scalings are related by

$$
  \pi_!^t(\nabla_s) = [\pi_! \nabla_s^{1/t}]^t
$$

### determinant line bundles ###

(...skipping a bunch of remarks...)

### 3) sketch of some proofs###

(no time, as expected)

### $\infty$-operads ###

Baronikov-Kontsevich passage

## Gabriel Drummond-Cole; $\infty$-operads, $BV_\infty$ and $HyperComm_\infty$ ##

>(was hard to take typed notes of this otherwise pretty cool talk, does anyone have handwriitten notes?)

## Scott Wilson: Categorical algebra, mapping spaces and applications ##

(for closely related blog entry see

* [Higher Hochschild Cohomology and Differential Forms on Mapping Spaces](http://golem.ph.utexas.edu/category/2008/04/higher_hochschild_cohomology_a.html)

)

outline

* language for some elementary algebraic topology

* application to generalizatons of Hochschild complexes

* Examples

  * invariants on mapping spaces

  * contributions related to def of Laplacian

**def/lema**

A commutative associative
[[differential graded algebra]] is (equivalently given by) a
strict monoidal functor

$$
  (FinSet, \coprod) \to (ChainComplexes, \otimes)
$$

generalize this

**def** a **partial DGA** is a  monoidal functor
with coherence map given by weak equivalence in the
model structure

$$
 A :  (FinSet, \coprod) \to (ChainComplexes, \otimes)
$$

i.e. there exists a natural weak equivalence

$$
  A(j \sqcup k) \stackrel{T}{\to} A(j) \otimes A(k)
$$

that respects the obvious coherence properties

generalized

* 1) co-algebras

* 2) any operad

* 3) note that $FinSet_*$ (pointed finite sets) is a module over $FinSet$, so generalize
  to modules, comodules, etc.

Then weak partial algebras can be functorially replaced by $E_\infty$-algebras

**example**

$X$ be a space $j \stackrel{f}{\to} k$

$$
  X^j = Map(j,X) \leftarrow Map(k,X) = X^k
$$

pass to the chains version of this

$$
  Ch_*(X^j) \leftarrow Ch_*(X^k)
$$

$$
  Ch^*(X^j) \to Ch^*(X^k)
$$

by Kuenneth formula we have a chain equivalence

$$
  C_*(X^j) \otimes C_*(X^k) \to C_*(X^{j+k})
$$

and similarly for cochains.

so this gives two things:

* a partial coalgebra on $C_*(X)$

* a partial algebra on $C^*(X)$

Let $Y$ be any finite simplicial space. A partial algebra

$$
  \Delta \stackrel{\gama}{\to} FinSet \stackrel{A}{\to} ChainCompl
$$

simplicial object in $ChainCompl$, so total complex

$$
  CH^\gamma(A)
$$

meaning generalization of Hochschild complex

* this is joint work with Tradler and Zanelli (spelling? probably wrong)

goes back to Pitashvili and more recently Gregory Ginot

For $A = \Omega(X)$, then $CH^\gamma(A)$ computes cohomology of
$X^\gamma $, if $X$ is sufficiently connected

**example**

let $A$ be a strict algebra, and $\gamma = Y = S^1$ then

$$
  CH^{S^1}(A) = \prod_{n \geq 0} A \otimes A^{\otimes n}
$$

is the Hochschild complex

there is also a shuffle product  in the game, so this implies there is an
exponential map

calculate:

$$
  \exp(| \otimes x)
  =
  | + | \otimes x + | \otimes  x \otimes x + | \otimes  x \otimes x \otimes x + \cdots +
$$

$$
  D \exp(1 \otimes x) = (1 \otimes d x + x \cdot x) \cdot e^{1 \otimes x}
$$

then: if $d x + x \cdot x = 0$ then $D e^{1 \otimes x} = 0$

this reminds us of curvature and connection

this can be taken further

let $A = \Omega^\bullet(M)$ be differential forms on $M$

$$
  \array{
    && CH^{S^1}(A) (\simeq \Omega(M^{S^1}))
    \\
    &\nearrow & \downarrow
    \\
    K(M)&\stackrel{ch}{\to}&\Omega(M)
  }
$$

commutes (due to some people)

**example 2**

$Y = I$ (the interval)

then $CH^I(A)$ is the 2-sided bar construction

more generally $CH(A, M, N) = \prod_{n \geq 0} M \otimes A^{\otimes n} \otimes N$

with $M$ and $N$ $A$-modules sitting on the end of the interval

consider the case $A = \Omega^\bullet(Riemannian manifold)$
and $M = A$ and $N = (\Omega^\bullet(...), d^* , (x\in A) \cdot (y\in N) = \star^{-1}(x \wedge \star y)))$

(the operatoin on $N$ here is the intersection product of forms)

Let $D$ be differential on $CH^I$

let $D$ be differential on $CH^I$ for normal structure, and
and $D^*$ for $A, M, N$ as just described.

Set

$$
  \Delta = [D, D^*]
$$

then acting with this $\Delta$ on something produces interesting non-linear
differential equations related to Witten't Morse-theory deformation
of susy quantum mechanics and to Navier-Stokes' equations in fluid dynamics...

***

[[Oberwolfach Workshop, June 2009 -- Tuesday, June 9|Previous day]] --- [[Oberwolfach Workshop, June 2009 -- Strings, Fields, Topology|Main workshop page]] --- [[Oberwolfach Workshop, June 2009 -- Thursday, June 11|Next day]]


object

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

# Contents
* table of contents
{: toc}

## Definition

A [[category]] consists of a collection of **objects** together with [[morphisms]] between these objects.  Thus, naively, we may think of objects as the '[[elements]]' of a category.

More generally, in [[higher category theory]] the objects of an $(n,r)$-[[(n,r)-category|category]] are the $0$-dimensional cells of that structure, the $0$-[[0-morphism|morphisms]].

## Examples

* If a [[set]] is regarded as a [[discrete category]] (with no nontrivial morphisms) then the objects of that category are precisely the elements of the set.

* In the [[fundamental groupoid]] $\Pi_1(X)$ of a [[topological space]] $X$, the objects are the points of $X$.

* In the category [[Set]], the objects are sets; in [[Vect]] the objects are [[vector space]]s; in [[Top]] the objects are [[topological space]]s, etc.

* If a [[simplicial set]] that is a [[Kan complex]] is regarded as an $\infty$-[[infinity-groupoid|groupoid]], then its vertices are the objects of that $\infty$-groupoid.

* Similarly if a simplicial set that is a [[quasi-category]] is regarded as an $(\infty,1)$-[[(∞,1)-category|category]], then its vertices are the objects of that $(\infty,1)$-category.

* If a [[globular set]] is equipped with the structure of a [[strict ∞-category]], then its $0$-cells are the objects of that [[∞-category]].

[[!redirects object]]
[[!redirects objects]]
[[!redirects Object]]
[[!redirects Objects]]


object of finite length > history

see _[[length of an object]]_


object of finite type

> This entry is about "objects of finite type" in [[algebra]], [[homological algebra]] and [[rational homotopy theory]]. For [[finite homotopy types]] and [[π-finite homotopy types]] in [[homotopy theory]] see there. For related notions in [[category theory]] see at *[[compact object]]*.
For finite [[types]] in [[type theory]] and in [[homotopy type theory]] see at *[[inductive family]]*. For more disambigation see at *[[finite type]]*.

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

## Contents
* table of contents
{:toc}

## Idea

The terminology of "objects of finite type" or of objects that "have finite type"  is common in various contexts and usually means something like *[[finitely generated object]]*; but beware that conventional usage across contexts is not fully systematic.

## Definitions

### In algebraic geometry

See at _[[morphism of finite type]]_ for the notion in [[algebraic geometry]].

### In Abelian categories

An object $X$ in an [[AB5-category]] $C$ is __of finite type__ if one of the following equivalent conditions hold:

(i) any complete directed set $\{X_i\}_{i\in I}$ of [[subobject]]s of $X$ is stationary

(ii) for any complete directed set $\{Y_i\}_{i\in I}$ of subobjects of an object $Y$ the natural morphism $colim_i C(X,Y_i) \to C(X,Y)$ is an isomorphism.

An object $X$ is __finitely presented__ if it is of finite type and if for any
epimorphism $p:Y\to X$ where $Y$ is of finite type, it follows that $ker\,p$ is also of finite type. An object $X$ in an AB5 category is __coherent__ if it is of finite type and for any morphism $f: Y\to X$ of finite type $ker\,f$ is of finite type.

For an exact sequence $0\to X'\to X\to X''\to 0$ in an AB5 category the following hold:

(a) if $X'$ and $X''$ are finitely presented, then $X$ is finitely presented;

(b) if $X$ is finitely presented and $X'$ of finite type, then $X''$ is finitely presented;

(c) if $X$ is coherent and $X'$ of finite type then $X''$ is also coherent.

For a module $M$ over a ring $R$ this is equivalent to $M$ being finitely generated
$R$-module. It is finitely presented if it is finitely presented in the usual
sense of existence of short exact sequence of the form $R^I\to R^J\to M\to 0$
where $I$ and $J$ are finite.

### In homotopical algebra and rational homotopy theory
 {#InRationalHomotopyTheory}

A _graded object_ is often said to be of **finite type** if it is _degreewise_ of [[finite number|finite]] [[dimension]]/[[rank]], in some sense. This terminology is used specifically in [[rational homotopy theory]].

Notably a [[rational topological space]] is said to be of _finite type_ if all its rational [[homotopy groups]] are [[finite number|finite]] [[dimension|dimensional]] [[vector spaces]] over the [[rational numbers]].

Accordingly, a [[chain complex]] of vector spaces, possibly those generating a [[semifree dga]] is said to be of finite type if it is degreewise [[finite number|finite]] [[dimension|dimensional]].

Beware however that the terminology clashes somewhat with the use in [[homotopy theory]], there the concept of _[[finite homotopy type]]_ is crucially different from _[[homotopy type with finite homotopy groups]]_.

### In stable homotopy theory

A [[spectrum]] of finite type (in the sense of [[stable homotopy theory]]) is one whose cohomology is finitely generated in each degree, but could exist in infinitely many degrees.

> check

See also/instead at _[[finite spectrum]]_.

## References

* [[Nicolae Popescu]], _Abelian categories with applications to rings and modules_, London Math. Soc. Monographs 3, Academic Press 1973. xii+467 pp. [MR0340375](http://www.ams.org/mathscinet-getitem?mr=0340375)

* MathOverflow [m-oplus-n-is-of-finite-type-if-m-n-are-of-finite-type](http://mathoverflow.net/questions/46136/m-oplus-n-is-of-finite-type-if-m-n-are-of-finite-type)

[[!redirects objects of finite type]]

[[!redirects of finite type]]


object of isomorphisms

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

\tableofcontents

## Idea

In the same way that [[internal homs]] in a [[cartesian monoidal category]] behave as an internal version of [[hom-sets]] between two objects, the *object of isomorphisms* should behave as an internal version of the sub-hom-set of isomorphisms between two objects.

These generalize [[bijection sets]] in the [[category of sets]] [[Set]].

## Definition

Given a [[cartesian monoidal category]] $C$ and two objects $A \in C$ and $B \in C$, there is a functor
$$X \mapsto \mathrm{Hom}_{\mathrm{Core}(C/X)}(A \times X, B \times X)$$
which takes an object $X \in C$ to a morphism between objects $A \times X$ and $B \times X$ in the [[core]] of the [[slice category]] $C/X$. A [[representing object]] of the above functor, if it exists, is the **object of isomorphisms** between $A$ and $B$ and is denoted as $[A, B]_\mathrm{iso}$.

## Properties

If the category is a [[cartesian closed category]], then there is a [[monomorphism]] $i:\mathrm{Mono}_C([A, B]_\mathrm{iso}, [A, B])$ between the object of isomorphisms between $A$ and $B$ and the [[internal hom]] from $A$ to $B$.

## See also

* [[bijection set]]
* [[equivalence type]]
* [[exponential object]], [[internal hom]], [[object of monomorphisms]]

[[!include types and logic - table]]

[[!redirects object of isomorphisms]]
[[!redirects objects of isomorphisms]]


object of monomorphisms

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

\tableofcontents

## Idea

In the same way that [[internal homs]] in a [[cartesian monoidal category]] behave as an internal version of [[hom-sets]] between two objects, the *object of monomorphism* should behave as an internal version of the sub-hom-set of monomorphisms between two objects.

These generalize [[injection sets]] in the [[category of sets]] [[Set]].

## Definition

Given a [[cartesian monoidal category]] $C$ and two objects $A \in C$ and $B \in C$, there is a functor
$$X \mapsto \mathrm{Mono}_{C/X}(A \times X, B \times X)$$
which takes an object $X \in C$ to a [[monomorphism]] between objects $A \times X$ and $B \times X$ in the [[slice category]] $C/X$. A [[representing object]] of the above functor, if it exists, is the **object of monomorphisms** between $A$ and $B$ and is denoted as $[A, B]_\mathrm{mono}$.

## Properties

If the category is a [[cartesian closed category]], then there is a [[monomorphism]] $i:\mathrm{Mono}_C([A, B]_\mathrm{mono}, [A, B])$ between the object of monomorphism between $A$ and $B$ and the [[internal hom]] from $A$ to $B$.

## See also

* [[injection set]]
* [[embedding type]]
* [[exponential object]], [[internal hom]], [[object of isomorphisms]]

[[!redirects object of monomorphisms]]
[[!redirects objects of monomorphisms]]


object-oriented programming

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Constructivism, Realizability, Computability
+-- {: .hide}
[[!include constructivism - contents]]
=--
=--
=--

\tableofcontents

\section{Idea}

Object oriented [[programming language|programming languages]] are, roughly speaking, ones in which the programmer works primarily with 'objects' which have the following characteristic aspects:

1. One constructs an object by providing certain input. E.g. one might construct an object of type 'mathematician' with input: 'name', 'university', 'research field'. This input is typically 'stored' inside the object.
1. One defines for an object certain 'methods', which, given an object and optionally some further input, a programmer may call on that object, to obtain certain output. These methods typically use the 'stored' input. E.g. for our object of type 'mathematician', we may define a method 'worksInSameCountryAs' which takes as additional input some other university X, and which returns a [[boolean]] according to whether the stored university is in the same country as university X. (And to do this, university itself should be an object which has a method providing as output the country to which the university belongs, etc).
1. The input (which may be mutable or immutable) stored inside the object is not typically exposed directly when a programmer uses the object, although one often provides a 'getter' method to return this stored variable at a given point in time (if it is mutable, it may have changed since the object was constructed). This is known as 'encapsulation'. One idea behind this is that the internals of an object can change without a programmer using an object needing to change their code accordingly.

Often, an object oriented programming language admits a notion of blueprint for an object, i.e. a description of the input needed to construct one and possibly also of the methods which the programmer may call given one, as well as the definition of these methods: a language construct known for example as a 'class' or 'prototype'. An object is then simply an 'instance' of this class or 'extension' of this prototype, i.e. it is what one obtains by actually providing some input to the blueprint (class, prototype, ...). Often 'classes' or similar play the role of '[[types]]' in a typed object oriented language.

One may also often 'extend' objects and their blueprints to make them more specific, adding methods which rely on this greater specificity. This is known as 'inheritance'. In the case that an object blueprint is a prototype, inheritance can even be viewed as involved in _construction_ of an object. It is, however, a tricky implementation problem to have an object extend (inherit from) more than one blueprint, and many languages (e.g. Java) forbid this.

Often object oriented languages have further levels of abstraction, e.g. the notion of an 'interface' or, in a slightly different way, of an 'abstract class' in Java. These typically can be thought as blueprints for classes, or of blueprints of objects which do not provide the details of how methods are defined, describing only the type of the input to the method and the type of the output. One says that a class 'implements' an interface. This is entirely distinct from 'inheritance'. When the language has them, interfaces can also be used as types in the language, although classes are often still types too: it is usually good programming practise to 'code to an interface', e.g. use as interface as the output type of a method, rather than use a class which implements the interface, as this provides greater flexibility.

For a discussion of how object oriented programming languages may be able to be thought of via [[type theory]], see the *[Examples](#Examples)* below.

Proponents of object oriented programming often argue that code in this style is more 'readable' and more 'flexible' than code in other styles, though often also more verbose. These points are, naturally, fiercely contended.

As touched upon above, there exist various different approaches to object-oriented programming, including the class-based approach of [Java](https://en.wikipedia.org/wiki/Java_(programming_language)), the prototype-based approach of [Self](https://en.wikipedia.org/wiki/Self_(programming_language)) or JavaScript, and the 'everything is an object'-approach of [Smalltalk](https://en.wikipedia.org/wiki/Smalltalk).

\section{Relation to category theory}

Henry Story has [noted](https://www.quora.com/Why-is-functional-programming-seen-as-the-opposite-of-OOP-rather-than-an-addition-to-it/answer/Henry-Story) that, since OOP is coalgebraic and [[functional programming]] (FP) is algebraic, OOP and FP can in some sense be regarded as dual to each other.

Although, in the context of [[computer science]], [[category theory]] is perhaps most often associated with [[functional programming]], there has been work on analysing object-oriented programming from a category-theoretic viewpoint, e.g. in the work of [[Bart Jacobs]] (e.g [Jacobs 2003](#Jacobs2003)).

\section{Examples}
 {#Examples}

Let us give here an implementation of *[[natural numbers]]* in a couple of object oriented programming languages. First, in Java, we can define a class as follows.

    public final class NaturalNumber {
        private final NaturalNumber successorOf;

        private NaturalNumber(final NaturalNumber successorOf) {
            this.successorOf = successorOf;
        }

        public static NaturalNumber zero() {
            return new NaturalNumber(null);
        }

        public NaturalNumber successor() {
            return new NaturalNumber(this);
        }

        public boolean isZero() {
            return successorOf == null;
        }

        public NaturalNumber getSuccessorOf() {
            return successorOf;
        }

        public NaturalNumber add(final NaturalNumber n) {
            NaturalNumber remainingPartToAdd = n;
            NaturalNumber withAddedPart = this;
            while (!remainingPartToAdd.isZero()) {
                withAddedPart = withAddedPart.successor();
                remainingPartToAdd = remainingPartToAdd.getSuccessorOf();
            }
            return withAddedPart;
        }
    }

There are a number of aspects to this which illustrate aspects of object oriented programming in languages such as Java. Firstly, if we try to see past the code itself and understand what is being expressed in terms of [[type theory]], we can observe the following.

1. A class corresponds roughly to a [[type]]. In other words, the class 'NaturalNumber' here corresponds roughly to a [[natural numbers type]].
1. An object of this class corresponds roughly to a [[term]] of this type. In other words, a 'NaturalNumber' object corresponds roughly to a term of a natural numbers type, i.e. to a natural number.
1. There is no sense in which this class 'NaturalNumber' is defined in terms of a canonical type construction, such as an [[inductive type]]. We are simply introducing a type in the empty context, and then defining [[natural deduction|rules]] for working with that type. This a common theme in object oriented programming: the core language typically provides very few canonical type constructions, but the programmer (or standard library!) has freedom and flexibility for introducing types and their rules. Java for example does not even have a 'pair' or 'product' type in its standard library, though it is easily implemented.
1. In this particular example, there are roughly speaking two [[introduction rule|introduction rules]]: one given by the method 'zero', and one given by the method 'successor'. The method zero occurs in the empty context, i.e. corresponds to a rule of the form
$$
\frac{}{zero : \mathbb{N}}
$$
whilst the method successor corresponds to a rule of the following form.
$$
\frac{n : \mathbb{N}}{successor(n) : \mathbb{N}}
$$
1. The method getSuccessorOf is roughly akin to a [[computation rule]]. Indeed, it can be thought of as being of the following form.
$$
\frac{n : \mathbb{N} \vdash successor(n) : \mathbb{N}}{getSuccessorOf(successor(n)) = n : \mathbb{N}}
$$
1. The method add is roughly akin to an [[elimination rule]] combined with two computation rules. Indeed, the elimination rule can be thought of as being of the form
$$
\frac{n : \mathbb{N} \vdash m : \mathbb{N}}{n + m : \mathbb{N}},
$$
and the computation rules can be thought of as being of the form
$$
\frac{n : \mathbb{N}, zero : \mathbb{N} \vdash n + zero : \mathbb{N}}{n + zero = n : \mathbb{N}}
$$
and of the following form.
$$
\frac{n : \mathbb{N}, m : \mathbb{N}, successor(m) : \mathbb{N} \vdash n + successor(m) : \mathbb{N}}{n + successor(m) = successor(n + m): \mathbb{N}}
$$

Secondly, if we look at the code itself, we can observe a few typical features.

1. The distinction between private and public methods and variables is often important in object oriented programming ('encapsulation'). Private methods and variables are those which can only be used internally in an object, i.e. they are 'implementation details' which cannot be seen by any code using an object. In this example, both of the two introduction rules (the methods zero and successor) actually rely on a third, purely internal, introduction rule, namely the class constructor. The code for a constructor is often 'boilerplate', i.e. almost always follows exactly or almost exactly the pattern used here: it instantiates the private variables of the class. The same goes for 'getters' like 'getSuccessorOf'. The idea is that by only exposing certain methods for public use, one controls more precisely how an object can be used, reducing the possibility for bugs, making code more predictable and easier/quicker to read.
1. The fact that 'add' is a method of the class NaturalNumber rather than a static method of form `public static NaturalNumber add(final NaturalNumber n, final NaturalNumber m)`
in some other class is archetypical object oriented programming. For another example, observe in the [Java API documentation](https://docs.oracle.com/javase/8/docs/api/java/lang/String.html#concat-java.lang.String-) that concatenation of two strings is implemented in the Java standard library as a method `concat` in a String class which takes as input a single string, not as a static method which takes as input two strings. This is a fundamental difference in design from the way things are done in a [[functional programming|functional programming language]]. Use of static methods (ones not associated to an object of a class) is often a 'code smell' in Java and other object oriented programming languages, though there are exceptions, such as in the factory method 'zero' in the example above. We can see in the example above why this pattern is used in object oriented programming languages: it allows us to use the 'internal' private variables of an object, in this case successorOf. Though in this case it doesn't make much difference since we could have used the getter getSuccessorOf, the principle in general is that this allows for greater code flexibility: we can change the internals of our class without breaking code which uses objects of the class.
1. Observe that 'add' relies on mutability (the variables remainingPartToAdd and withAddedPart) and a while loop rather than recursion. This is again very typical for an object oriented programming language vs a functional programming language.

We can carry out the same kind of implementation in Javascript. Javascript is not statically typed, i.e. has no explicit types or compile time type checking, and also has no explicit concept of class, relying instead on a notion of prototyping of objects. Nevertheless, the fundamental principles are the same, and in particular we claim that if we see past the code and try to understand things type theoretically, the story is exactly the same as for Java.

    function NaturalNumber(successorOf) {
      if (successorOf == undefined) {
        this.successorOf = null;
      } else if (!(successorOf instanceof NaturalNumber)) {
        throw "A natural number must be constructed either as zero or as the successor of a natural number";
      }
      this.successorOf = successorOf;
    }

    NaturalNumber.prototype.isZero = function () {
      return this.successorOf == null;
    }

    NaturalNumber.prototype.successor = function () {
      return new NaturalNumber(this);
    }

    NaturalNumber.prototype.add = function(n) {
      if (!(n instanceof NaturalNumber)) {
        throw "Can only add a natural number to a natural number";
      }
      var remainingPartToAdd = n;
      var withAddedPart = this;
      while (!remainingPartToAdd.isZero()) {
        withAddedPart = withAddedPart.successor();
        remainingPartToAdd = remainingPartToAdd.successorOf;
      }
      return withAddedPart;
    }

\section{Comparison with abstract data types}

Object oriented programming languages (OOP) should be contrasted with the use of [[abstract data types]] (ADTs), since they are deceptively similar. Both involve a distinction between a private implementation and a public interface, but with ADTs, the implementation is "private to the type", while with OOP it's "private to the object", and an object is not a type. Assuming the language in question has [[types]], different objects of the same type generally have different implementations of the same interface.

As a general rule, it's easy to add new data variants with OOP, because that just involves adding more objects, while it's harder to add new operations, as that involves adding to the implementations of all the existing objects. With ADT's, the reverse is true, as new operations are easy to add, while adding data variants involve changing the definition of the ADT and the implementations of all existing operations. See also [William R. Cook's essay](#Cook2009) on the differences between objects and ADTs.

This tension leads to the so-called [expression problem](http://homepages.inf.ed.ac.uk/wadler/papers/expression/expression.txt) in programming: How to design a language that makes both kinds of extension simultaneously easy.

The complementarity between user-defined types and objects was observed already by [Reynolds in 1978](#Reynolds1978).

\section{Related concepts}

* [[lens(in computer science)]]

* [[functional programming]]

* [[imperative programming]]

* [[programming language]]

\section{References}

* Wikipedia, _[Object-oriented programming](https://en.wikipedia.org/wiki/Object-oriented_programming)_

* {#Cook2009} [[William R. Cook]], _[On  Understanding  Data  Abstraction,  Revisited](https://www.cs.utexas.edu/~wcook/Drafts/2009/essay.pdf)_, OOPSLA, 2009

* Alan Kay, _[The Computer Revolution hasn't happened yet](https://www.youtube.com/watch?v=oKg1hTOQXoY)_, OOPSLA Keynote, 1997

* {#Jacobs1995a} [[Bart Jacobs]], _Inheritance and Cofree Constructions_, 1995 ([pdf](https://ir.cwi.nl/pub/4938/4938D.pdf))

* {#Jacobs1995b} [[Bart Jacobs]], _Objects and Classes, Co-algebraically_, Object Orientation with Parallelism and Persistence, 1995 ([pdf](https://static.aminer.org/pdf/PDF/000/137/588/objects_and_classes_co_algebraically.pdf))

* {#Jacobs2003} [[Bart Jacobs]], Erik Poll, _Coalgebras and monads in the semantics of Java_, Theoretical Computer Science Volume 291 Issue 3, 2003 ([doi:10.1016/S0304-3975(02)00366-3](https://www.sciencedirect.com/science/article/pii/S0304397502003663))

* {#Reynolds1978} [[John C. Reynolds]], _[User-Defined Types and Procedural Data Structures as Complementary Approaches to Data Abstraction](https://www.cs.tufts.edu/~nr/cs257/archive/john-reynolds/procedural-data-structures.pdf)_, In: Gries D. (eds) Programming Methodology. Texts and Monographs in Computer Science. Springer, New York, NY

* {#Story2018} Henry Story, _[Why is functional programming seen as the opposite of OOP rather than an addition to it?](https://www.quora.com/Why-is-functional-programming-seen-as-the-opposite-of-OOP-rather-than-an-addition-to-it/answer/Henry-Story)_, 2018

* [[Anton Setzer]], _Object-oriented programming in dependent type theory_, in Henrik Nilsson (Ed.): Trends in Functional Programming. Volume 7, Series Trends in Functional Programming, Intellect, Intellect, Bristol and Chicago, 2007, pp. 91 -- 108. ISBN 978-184150-188-8, ([pdf](http://www.cs.swan.ac.uk/~csetzer/articles/objectOrientedProgrammingInDepTypeTheoryTfp2006PostProceedings.pdf))

Early discussion of [[lenses in computer science]] (before that term was coined) as  formalizing aspects of object-oriented programming:

* {#HofmannPierce96} [[Martin Hofmann]], [[Benjamin Pierce]], pp. 11-12 in: *Positive Subtyping*, Information and Computation **126** 1 (1996) 11-33 [[doi:10.1006/inco.1996.0031](https://doi.org/10.1006/inco.1996.0031)]

[[!redirects object-oriented program]]
[[!redirects object-oriented programs]]


objective and subjective logic

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Philosophy
+-- {: .hide}
[[!include philosophy - contents]]
=--
#### Mathematics
+-- {: .hide}
[[!include mathematicscontents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

In his _[[Science of Logic]]_ [[Georg Hegel]] makes a distinction between _subjective logic_, which involves _[[type|concepts]]_, _[[judgements]]_ and _[[deduction]]_ (as usual for [[logic]]), and _objective logic_ which is concerned with something like the _[[logos]]_, a kind of [[ontology]] or [[metaphysics]], not unlike the ontological deductions of [[Spinoza's system]].

This _objective logic_ as presented by Hegel is hence rather unlike common [[predicate logic]], a fact that led to the rejection of this school of [[objective idealism]] by [[Bertrand Russell]] and the school of [[analytic philosophy]].

In ([Lawvere 94](#Lawvere94)) it is suggested that [[universal constructions]] in [[categorical logic]] and [[topos theory]], such as [[adjunctions]], should be thought of as the formal incarnation of Hegel's _objective logic_. Related suggestions are implicit in many of [[Lawvere]]'s writings, see at _[Lawvere -- Mathematics relating to philosophy](William+Lawvere#RelationToPhilosophy)_.

> {#ItIsMyBelief} [Lawvere 92](#Lawvere92): It is my belief that in the next decade and in the next century the technical advances forged by category theorists will be of value to dialectical philosophy, lending precise form with disputable mathematical models to ancient philosophical distinctions such as general vs. particular, objective vs. subjective, being vs. becoming, space vs. quantity, equality vs. difference, quantitative vs. qualitative etc. In turn the explicit attention by mathematicians to such philosophical questions is necessary to achieve the goal of making mathematics (and hence other sciences) more widely learnable and useable. Of course this will require that philosophers learn mathematics and that mathematicians learn philosophy.

In ([Lawvere-Rosebrugh 03, section C.1](#LawvereRosebrugh03)) the following characterization is given:

> **Logic:**

> The science of logic in the ancient philosophical sense means the study of the general laws of the development of thinking. Thinking (1) reflects reality (i.e., has a content) but also (2) is itself part of reality and so has some motions that are oblivious to content. Therefore the science of logic finds two aspects of thought's motion: (1) the struggle to form a conceptual image of reality that is ever more refined, whose laws we may call objective logic, and (2) the motion of thought in itself (for example the inference of statements from statements), whose laws we may call subjective logic. Although grammar and some aspects of algebra might be considered as subjective logic, we will limit ourselves to the part we will sometimes refer to as logic in the narrow sense – that which is related to the inference of statements from statements by means dependent on their form rather than on their content. (Logic in the narrow sense is explained in more detail in Appendix A.) Logic in the narrow sense is useful (at least in mathematics) if it is made explicit, and the work of Boole and Grassmann in the 1840s, Schröder in the late 1800s, Skolem in the early 1900s, Heyting in the 1930s (and of many others) has led to a high development, most aspects of which were revealed to be special cases of adjoint functors by 1970 [La69b]. The use of adjoint functors assists in reincorporating the subjective into its rightful place as a part of the objective so that it can organically reflect the objective and in general facilitate the mutual transformation of these two aspects of logic.

> **Logic, Objective:**

> The long chains of correct reasonings and calculations of which subjective logic is justly proud are only possible within a precisely defined universe of discourse, as has long been recognized. Since there are many such universes of discourse, thinking necessarily involves many transformations between universes of discourse as
well as transformations of one universe of discourse into another. The results of
applying logic in the narrow sense to the laws of these objective transformations are necessarily inadequate; for example, such attempts have led to the use of phrases such as "let X be a set in which there exists a group structure," which are essentially meaningless. Rather than using "there exists" in such contexts, one needs instead a logic of "given." Before category theory, at least one systematic discussion of the laws of these objective transformations was given by Bourbaki, who discussed how one structure could be deduced from another. The concepts of categories, functors, homomorphisms, adjoint functors, and so on, provide a rich beginning to the project of making objective logic explicit, but there is probably much more to be discovered.

Related commentary is in ([Rodin 14](#Rodin14)).

## References

* [[Georg Hegel]], _[[Science of Logic]]_

* {#Lambek82} [[Joachim Lambek]], _The Influence of Heraclitus on Modern Mathematics_, In _Scientific Philosophy Today: Essays in Honor of Mario Bunge_, edited by Joseph Agassi and Robert S Cohen, 111–21. Boston: D. Reidel Publishing Co. (1982) ([doi:10.1007/978-94-009-8462-2_6](https://link.springer.com/chapter/10.1007/978-94-009-8462-2_6))

* {#Lawvere92} [[William Lawvere]], _Categories of space and quantity_, in J. Echeverria et al (eds.), _The Space of mathematics_, de Gruyter, Berlin, New York, pages 14-30, 1992.

* {#Lawvere94} [[William Lawvere]], _[[Tools for the advancement of objective logic]]: closed categories and toposes_, in J. Macnamara and [[Gonzalo Reyes]] (Eds.), _The Logical Foundations of Cognition_, Oxford University Press 1993 (Proceedings of the Febr. 1991 Vancouver Conference "Logic and Cognition"), pages 43-56, 1994.

* {#LawvereRosebrugh03} [[William Lawvere]], [[Robert Rosebrugh]], _[[Sets for Mathematics]]_ , Cambridge UP  2003. ([web](http://books.google.de/books?id=h3_7aZz9ZMoC&pg=PP1&dq=sets+for+mathematics))

* {#Rodin14} [[Andrei Rodin]], section 5.8 of:  _Categorical Logic and Hegelian Dialectics_ of _Axiomatic Method and Category Theory_, Springer 2014 ([arXiv:1210.1478](http://arxiv.org/abs/1210.1478), [doi:10.1007/978-3-319-00404-4](https://www.springer.com/gp/book/9783319004037))

[[!redirects subjective and objective logic]]

[[!redirects objective logic]]
[[!redirects subjective logic]]


objective idealism

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

#Contents#
* table of contents
{:toc}

## Idea

(...)

A point of view in [[philosophy]].

(...)

A prominent example is [[Georg Hegel]]'s [[natural philosophy]] as expressed in his _[[Science of Logic]]_. In opposition to this is the school of [[analytic philosophy]] which aims to make use of formal reasoning in [[first-order logic]]. However [[William Lawvere]] argued that key Hegelian concepts such as _[[unity of opposites]]_, _[[Aufhebung]]_, _[[category of being]]_, do have useful formalization in [[modal type theory]] (or its [[categorical semantics]]).

## Relation to physics

> There are quite a few steps between an empiricist and a Hegelian (or objective idealist). The former's view of scientific [[theory (physics)|theory]] generally is that it's an efficient encoding of the [[observations]] we make of the world, accompanied typically by a wariness about taking the [[theory (physics)|theory]] too seriously, e.g., as to what it says about the unobservable. Observations, on the other hand, are fairly straightforwardly given to us by the world. Science's basic task is to codify them, so that we can predict and sometimes control nature.

> Germany had for many years gone down a non-empiricist path. Kant had wanted to know, given the existence of [[mathematics]] and [[mechanics|Newtonian theory]], what must be the case about our cognitive faculties. He worked out that we impose certain structures in our construction of experience, time and space, causality, etc. No simple empiricism then. We can't see the world as it is in itself. He read off twelve "categories" from the types of judgement we make.

> [[Hegel]] criticises him for only *listing* the 12 categories. The thing was to deduce them from first principles. What could come first? A Logic of the Idea. We starting with Being and Nothingness, note their identity and simultaneously their difference, deduce Becoming, and off we go, through a large number of twists and turns to explain why the world must be as it is.

> The world is secondary. The Idea has an internal dynamic which is driven by the dialectical process. It requires a world to play itself out in. We are a vehicle for the Idea.

> For Hegel, there's no Kantian separation between us and the thing-in-itself, unexperienced through our faculties. We and our experience are just one part of the working out of the Idea.

> ([[David Corfield]], [forum comment, June 2013](http://nforum.mathforge.org/discussion/4973/synthetic-quantum-field-theory/?Focus=40500#Comment_40500))

## Related concepts

* [[doctrine of ideas]]

* [[objective logic]]

* [[idealism]]

  * objective idealism

  * [[subjective idealism]]

  * [[absolute idealism]]

## References

* Wikipedia, _[Objective idealism](http://en.wikipedia.org/wiki/Objective_idealism)_


objective number theory

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

* table of contents
{: toc}

## Idea

**Objective number theory** is the study of [[addition]] and [[multiplication]] (and eventually [[exponentiation]]) of [[objects]] in suitable [[categories]]. ([[Stephen Schanuel]] [2000](#Schanuel00), p.295)

## Related entries

* [[number theory]]

* [[extensive category]]

* [[distributive category]]

* [[Burnside rig]]

* [[polynomial functor]]

* [[FinSet]]

* [[decategorification]]

## References

* [[John Baez]], [[James Dolan]], _From finite sets to Feynman diagrams_ , pp.29-50 in Engquist, Schmid (eds.), _Mathematics Unlimited - 2001 and Beyond_  , Springer Heidelberg 2001.

* [[Andreas Blass]], *Seven trees in one*, Journal of Pure and Applied Algebra,  **103** 1 (1995) 1–21 [[arXiv:math/9405205](https://arxiv.org/abs/math/9405205), doi:10.1016/0022-4049(95)00098-H]

* [[Marcelo Fiore]], [[Tom Leinster]], _An Objective Representation of the Gaussian Integers_ ,  J. Symbolic Computation **37** no.6 (2004) pp.707-716. ([arXiv](http://arxiv.org/abs/math/0211454v2))

* [[Marcelo Fiore]], [[Tom Leinster]], _Objects of categories as complex numbers_, Advances in Mathematics **190** (2005) pp.264-277. ([arXiv](http://arxiv.org/abs/math/0212377v1))

* Robbie Gates, _On the generic solution to $P(X)\simeq X$ in distributive categories_ , JPAA **125** (1998) pp.191-212.

* Robbie Gates, _On generic seperable objects_ , TAC **4** no.10 pp.208-248 (1998). ([abstract](http://www.tac.mta.ca/tac/volumes/1998/n10/4-10abs.html))

* [[F. William Lawvere]], _Qualitative Distinctions between some Toposes of Generalized Graphs_ , Cont. Math. **92** (1989) pp.261-299.

* [[Matías Menni]], _Every Rig with a One-Variable Fixed Point Presentation is the Burnside Rig of a Prextensive Category_ , Appl. Cat. Struc. **25** (2017) pp.663-707.

* [[Stephen Schanuel]], _Negative sets have Euler characteristic and dimension_ , Category theory. Proc. Int. Conf. Como/Italy 1990, LNM **1488** pp.379–385 (1991).

* {#Schanuel00}[[Stephen Schanuel]], _Objective number theory and the retract chain condition_ , JPAA **154** pp.295–298 (2000).

* [[Stephen Schanuel]], _Transcendence in objective number theory_ , in Categorical studies in Italy, Perugia 1997. Suppl. Rend. Circ. Mat. Palermo, II. Ser. 43–48, 64 (2000).

[[!redirects Objective number theory]]
[[!redirects Objective Number Theory]]