generalized global symmetry

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### Context
#### Physics
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#Contents#
* table of contents
{:toc}

## Idea

Traditionally, a *[[global symmetry|global]] [[symmetry]]* is defined as [[invariance]] (e.g. of a physical system) under a [[group action]]. This definition is well-suited for physical systems such as in [[classical mechanics]], where symmetries appear as [[symplectomorphisms]] leaving a [[Hamiltonian]] invariant.

However, many systems of interest depart, or **generalize**, from this setting in two different ways: as a [[classical field theory]], or as a [[quantum system]] (or as both, as a [[quantum field theory]]). A natural question to ask is what happens to the mathematical objects realizing the notion of symmetry when these generalizations are considered. The former case leads to the [[vertical categorification]] of groups to [[n-groups|$n$-groups]], as observed in [[higher prequantum geometry]]. The latter case leads to a [[ring]] (or [[algebra]]) structure. The concept of **generalized global symmetries** is thought to be the appropriate notion of symmetry when both of these generalizations take place at the same time.

{#OriginOfTheTerminology} The terminology *generalized global symmetry* for such a situation is due to [Gaiotto, Kapustin, Seiberg & Willett (2014)](#GaiottoKapustinSeibergWillett14). As described therein (at a very in-formal level) classical symmetries (described by a group) that survive the quantization process are thought to become "[[operators]]" that can be "inserted" (namely into the [[integrand]] of a [[path integral]], though this step and its details are left implicit) along [[submanifolds]] (of [[spacetime]]) of non-negative [[codimension]].
These operators are supposed to be "topological", in the sense that they should only depend on the homology class of said submanifold. [GKSW14](#GaiottoKapustinSeibergWillett14) propose to regard any quantum operator with the topological property as a symmetry of the quantum system even if they do not have inverses. It is in this sense that the concept of symmetry is regarded as being generalized.

As emphasized by [Freed, Hopkins, Teleman & Lurie (2009)](#FHTL09), and again in [Freed, Moore & Teleman (2022)](#FMT22), such phenomena should be controlled by [[higher algebra]]. Of course this is not quite a new observation, ideas broadly in this direction date back to [Roberts (1979)](John+Elias+Roberts#Roberts79). Concretly, "non-invertible symmetry insertions" are at least close to what elsewhere has long been known as *[[QFT with defects|defects in QFT]]* which have routinely been described via [[higher algebra|higher]] [[categorical algebra]], cf. [Fuchs, Runkel & Schweigert (2004)](#FRS04); [Fröhlich, Fuchs, Runkel & Schweigert (2007)](#FFRS07); [Lurie (2009) §4.3](QFT+with+defects#Lurie09); [Davydov, Runkel & Kong (2011)](QFT+with+defects#DavydovRunkelKong11); [Fuchs, Schweigert & Valentino (2013)](QFT+with+defects#FuchsSchweigertValentino13). The terms "defects" and "topological operator insertions" are actually often used interchangeably in the Physics literature. Indeed, many of the techniques used to deal with these generalized symmetries come from the literature on defects. A notable example is the gauging procedure of defects in two-dimensional QFT as originally described in [Fuchs, Runkel & Schweigert (2002)](#FRS02I), and directly recasted in the language of generalized symmetries in e.g. [Bhardwaj and Tachikawa (2017)](#BT17).

The upshot is that just as groups correctly describe the symmetries of classical mechanics, higher algebra is thought to describe the "generalized global symmetries" of a quantum field theory.

In this sense, the concept may be better understood as **higher quantum symmetries** (even though the name "quantum symmetries" is already used in the Physics literature for a "dual" symmetry that arises after gauging a symmetry, originating in [Vafa (1989)](#Vafa89)).

A significant part of the current Physics literature on the topic restricts to the study of *finite* generalized symmetries, thus avoiding questions regarding the *[[smooth|smoothness]]* of the corresponding higher algebra (in contrast with the situation of [[n-group|higher groups]], by which one really means a [[smooth infinity-groupoid]]). In such a case, the concrete mathematical objects of study are *bare* [[fusion categories|fusion category]] (and their vertical categorifications as in e.g. [Decoppet and Yu (2023)](#DY23)). These fusion categories are argued to contain information about the quantum operators and their fusion rules but also about any present [['t Hooft anomaly]], all (somehow) encoded in the [[topological quantum field theory]] defined through the [[Turaev-Viro model]] and their would-be higher generalizations.  Unfortunately, not only these expectations but even fundamental questions such as precisely *how* a smooth higher algebra acts on a quantum field theory (beyond the slogan of *a symmetry operator inserted on a codimension $d$ submanifold acts on quantum operators supported on dimension $d-1$ submanifolds*) are still not fully understood. An accurate description of such an action of a smooth higher algebra should in particular describe how the [[group action]] of the [[quantomorphism group]] on the [[prequantum circle n-bundle]] becomes this after [[motivic quantization]]. In this picture, the action of a higher group makes sense since the prequantum n-bundle also has a higher structure. In this sense, the action of a higher algebra should be defined on an object which has both a higher and a linear structure. This is the case for higher [[vector bundle|vector bundles]] such as [[2-vector bundle|2-vector bundles]] or, in much greater generality, an [[n-vector bundle]] over some [[E-∞ ring]].

## Freed-Moore-Teleman *quiche*

In [Freed, Moore and Teleman (2022)](#FMT22), the authors describe a proposal to make mathematical sense of the heuristics of generalized symmetries. The approach is based on the presentation of Quantum Field Theory as [[extended functorial field theory]], and relies on the idea that any QFT $T$ defines a [[topological field theory]] $Sym(T)$ defined by the generalized symmetries of $T$, called the *Symmetry TFT* and that, in turn, $T$ can be recovered from $Sym(T)$ via manipulations on its boundary.

The main concept is that of a **quiche**, which is a pair $(\sigma,\rho)$ where $\sigma$ is a fully extended $(n+1)$-dimensional TFT, and $\rho$ is a $\sigma$-"module" (where the concept of module is defined only in analogy with a [[domain wall]] between TFT's). Then the statement that an $n$-dimensional QFT $T$ has $\sigma=Sym(T)$ as its symmetry theory consists of another pair $(\tilde{T},\theta)$ where $\tilde{T}$ is a $\sigma$-module and $\theta$ is an "isomorphism"

$$
\theta: \rho\otimes_{\sigma}\tilde{T} \cong T
$$

where $\rho\otimes_{\sigma}\tilde{T}$ is schematically understood as placing $\sigma$ on a spacetime $X^n\times [0,1]$ with $\rho$ on one end and $\tilde{T}$ on the other, such that collapsing the slab gives $T$ on $X^n$.

Some remarks about this proposal are in order. First, the kind of symmetries that are usually dealt with here are of finite character (related to a [[homotopy type with finite homotopy groups]] as explained in Appendix A). This allows to avoid discussions regarding smoothness of categories. However, as highlighted in ([Kang & Kang (2023)](#KK23)), overlooking this fact has led to some inaccurate conclusions obtained from applying techniques that in reality are valid only in the discrete setting but not in the smooth setting. Work aiming towards addressing this issue is in ([Gripaios, Randal-Williams, and Tooby-Smith (2023)](#GRT23)). Second, some statements (which are not necessarily minor points) are still only schematic. This includes the isomorphism $\theta$ mentioned. This is partly due to the following assertion: *"Recall that the cobordism hypothesis enables a calculus of such functors (FTFT's) in terms of duality data inside the codomain category C. Turning to nontopological theories, a similar calculus is not in place and is a subject of wide interest.*" ([FMT22](#FMT22), p. 10). In particular, the work of [Grady and Pavlov (2021)](#GradyPavlov21) on the *geometric* [[cobordism hypothesis]] has clear relevance to these two remarks.

## Related concepts

* [[QFT with defects]]

* [[duality in physics]]

## References
 {#References}

Early discussion of "generalized global symmetries" under the name of *[[QFT with defects|defects]]*, in [[2d conformal field theory]] (via the [[FRS theorem on rational 2d CFT]]):

* {#FRS02I} [[Jürgen Fuchs]], [[Ingo Runkel]], [[Christoph Schweigert]], _TFT construction of RCFT correlators I: Partition functions_, Nucl. Phys. B **646**  (2002) 353-497 [[arXiv:hep-th/0204148](http://arxiv.org/abs/hep-th/0204148), doi:10.1016/S0550-3213(02)00744-7]

* {#FRS04} [[Jürg Fröhlich]], [[Jürgen Fuchs]], [[Ingo Runkel]], [[Christoph Schweigert]], *Kramers-Wannier duality from conformal defects*, Phys. Rev. Lett. **93** (2004) 070601 [[doi:10.1103/PhysRevLett.93.070601](https://doi.org/10.1103/PhysRevLett.93.070601), [arXiv:cond-mat/0404051](http://arxiv.org/abs/cond-mat/0404051)]

* {#FFRS07} [[Jürg Fröhlich]], [[Jürgen Fuchs]], [[Ingo Runkel]], [[Christoph Schweigert]], *Duality and defects in rational conformal field theory*, Nucl. Phys. B **763** (2007) 354-430 [[arXiv:hep-th/0607247](http://arxiv.org/abs/hep-th/0607247), [doi:10.1016/j.nuclphysb.2006.11.017](https://doi.org/10.1016/j.nuclphysb.2006.11.017)]

The terminology "categorical (symmetry)-groups" for *[[2-groups]]* goes back to the origin of the subject in

* {#Solian80} [[Alexandru Solian]], *Coherence in categorical groups*,  Communications in Algebra **9** 10 (1980) 1039-1057 [[doi:10.1080/00927878108822631](https://doi.org/10.1080/00927878108822631)]

Early appearance of the terminology "categorified symmetries" is in:

* [[Urs Schreiber]], [[Zoran Škoda]], *Categorified symmetries*, 5th Summer School of Modern Mathematical Physics, SFIN XXII Series **A1** (2009) 397-424 [[arXiv:1004.2472](https://arxiv.org/abs/1004.2472), [inspire:851901](https://inspirehep.net/literature/851901)]

The terminology of *generalized global symmetries* is due to:

* {#GaiottoKapustinSeibergWillett14} [[Davide Gaiotto]], [[Anton Kapustin]], [[Nathan Seiberg]], [[Brian Willett]], *Generalized Global Symmetries*  J. High Energ. Phys. **2015** 172 (2015) [[arXiv:1412.5148](https://arxiv.org/abs/1412.5148), doi:10.1007/JHEP02(2015)172]

Survey of the "field":

* [[Clay Cordova]], [[Thomas T. Dumitrescu]], [[Kenneth Intriligator]], [[Shu-Heng Shao]], *Snowmass White Paper: Generalized Symmetries in Quantum Field Theory and Beyond* [[arXiv:2205.09545](https://arxiv.org/abs/2205.09545)]

Further suggestion for mathematical formalization of what [GKSW14](#GaiottoKapustinSeibergWillett14) had in mind (purely at the quantum level, with caveats):

* {#FMT22} [[Daniel Freed]], [[Gregory Moore]], [[Constantin Teleman]], *Topological symmetry in quantum field theory* (2022) [[arXiv:2209.07471](https://arxiv.org/abs/2209.07471)]

rooted in

* {#FHTL09} [[Daniel Freed]], [[Mike Hopkins]], [[Constantin Teleman]], [[Jacob Lurie]], *[[Topological Quantum Field Theories from Compact Lie Groups]]*, in: *A celebration of the mathematical legacy of Raoul Bott*, AMS (2010) [[arXiv:0905.0731](http://arxiv.org/abs/0905.0731), [ISBN:978-0-8218-4777-0](https://bookstore.ams.org/view?ProductCode=CRMP/50)]

See also:

* {#Vafa89} [[Cumrun Vafa]], *Quantum Symmetries of String Vacua* (1989) [[doi:10.1142/S0217732389001842](https://doi.org/10.1142/S0217732389001842)]

* {#BT17} [[Lakshya Bhardwaj]], [[Yuji Tachikawa]]. *On finite symmetries and their gauging in two dimensions*, J. High Energ. Phys. **2018** 189 (2018) [[arXiv:1704.02330](https://arxiv.org/abs/1704.02330), doi:10.1007/JHEP03(2018)189]

* {#GradyPavlov21} [[Daniel Grady]], [[Dmitri Pavlov]], *The geometric cobordism hypothesis* [[arXiv:2111.01095](https://arxiv.org/abs/2111.01095)]

* {#PutrovWang23} [[Pavel Putrov]], [[Juven Wang]], *Categorical Symmetry of the Standard Model from Gravitational Anomaly* [[arXiv:2302.14862](https://arxiv.org/abs/2302.14862)]

In the context of [[fusion 2-categories]]:

* {#DY23} [[Thibault Decoppet]], [[Matthew Yu]], *Fiber 2-Functors and Tambara-Yamagami Fusion 2-Categories* (2023) [[arXiv:2306.08117](https://arxiv.org/abs/2306.08117)]

* Wenjie Xi, Tian Lan, Longye Wang, Chenjie Wang, Wei-Qiang Chen, *On a class of fusion 2-category symmetry: condensation completion of braided fusion category* [[arXiv:2312.15947](https://arxiv.org/abs/2312.15947)]

Work in the direction of *smooth* generalized symmetries:

* {#GRT23} Ben Gripaios, Oscar Randal-Williams, Joseph Tooby-Smith. *Smooth generalized symmetries of quantum field theories*. (2023). [arXiv:2310.16090](https://arxiv.org/abs/2310.16090#).

Notes on a more proper treatment of *smooth* 2-groups in the present context is in:

* {#KK23} Monica Jinwoo Kang, Sungkyung Kang, *Central extensions of higher groups: Green-Schwartz mechanism and 2-connections* ([arXiv:2311.14666](https://arxiv.org/abs/2311.14666))

Further activity:

* {#CategoricalSymmetries23} *Categorical Symmetries in Quantum Field Theory* (28 Aug -1 Sept 2023) [[indico:1131193](https://indico.cern.ch/event/1131193)]

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generalized graph

#Contents#
* table of contents
{:toc}

##Idea##

A **generalized graph** in the sense of [HRY](#HRY) is a generalization of the notion of a pseudograph given at [[graph]] (see in particular [Definition in terms action on a set of half edges](https://ncatlab.org/nlab/show/graph#definition_in_terms_of_action_on_a_set_of_halfedges)). By adding structure like directions for the edges, we can define a [[wheeled graph]] which is a generalization of a [[directed pseudograph]]. What differentiates a generalized graph from a pseudograph and a wheeled graph from a [[directed pseudograph]] is the notion of an "exceptional cell," which should be thought of as a set of half edges which have no vertices.

A generalized graph can be visualized as a set of vertices with half edges attached to them, a rule for attaching half edges to glue the vertices together, and potentially some half edges (flags) that are attached to only one or zero vertices. For instance, a particularly simple generalized graph is one with no vertices and one edge.

## Definitions ##
 The following is Definition 2.2 of [HRY](#HRY):

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###### Definition

A generalized graph $G$ is a finite set $Flag(G)$ with:

   * a partition of $Flag(G)=\coprod_{\alpha\in A} F_\alpha$ into **cells** with $A$ finite,

   * a distinguished partition subset $F_\epsilon$ called the **exceptional cell**,

   * an involution $\iota$ satisfying $\iota F_\epsilon\subseteq F_\epsilon$,

   * and a free involution $\pi$ on the set of $\iota$-fixed points of $F_\epsilon$.
=--

Given a generalized graph $G$, Definition 2.3 of [HRY](#HRY) gives some useful terminology:

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###### Definition
1. The elements of $Flag(G)$ are called **flags**. A flag in a non-exceptional (resp. exceptional) cell is called an **ordinary flag** (resp. **exceptional flag**).
1. Call $G$ an **ordinary graph** if $F_\epsilon$ is empty.
1. Each non-exceptional partition subset $F_\alpha\neq F_\epsilon$ is called a **vertex**. The set of vertices is denoted by $Vt(G)$. An empty vertex is an **isolated vertex**. A flag in a vertex is said to be **adjacent to** or **attached to** that vertex.
1. An $\iota$-fixed point is a **leg** of $G$. The set of legs is denoted by $Leg(G)$. An **ordinary leg** (resp. **exceptional leg**) is an ordinary (resp. exceptional) flag that is also a leg. For an $\iota$-fixed point $x\in F_\epsilon$, the pair $\{x,\pi x\}$ is called an **exceptional edge**.
1. A 2-cycle of $\iota$ consisting of ordinary flags is called an **ordinary edge**. A 2-cycle of $\iota$ contained in a vertex is a **loop**. A vertex that does not contain any loop is called **loop free**. A 2-cycle of $\iota$ contained in the exceptional cell is called an **exceptional loop**.
1. An **internal edge** is any 2-cycle of $\iota$. An **edge** is an internal edge, an exceptional edge, or an ordinary leg. The set of edges of $G$ (resp. internal edges) is denoted $Edge(G)$ (resp. Edge_i(G))$.
1. An ordinary edge $e=\{e_0,e_1\}$ is said to be **adjacent to** or **attached to** a vertex $v$ if either $e_0$, $e_1$ or both are adjacent to $v$.
=--

##Extra Structure##

There are a number of important structures that a generalized graph can possess which will be useful in using them to describe [[properads]]. The following is again from [HRY](#HRY):

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###### Definition

Suppose $G$ is a generalized graph. Fix a (possibly infinite) set of _colors_ $\mathcal{C}$.

1. A **coloring** of $G$ is a function $Flag(G)\overset{\kappa}\to \mathcal{C}$ that is constant on orbits of $\iota$ and $\pi$.

1. A **direction** for $G$ is a function $Flag(G)\overset{\delta}\to \{-1,1\}$ such that
    * if $\iota x\neq x$, then $\delta(\iota x)=-\delta(x)$,
    * and if $x \in F_\epsilon$, then $\delta(\pi x)=-\delta (x)$.

1. For $G$ with direction, an **input** (resp. **output**) of a vertex $v$ is a flag $x\in v$ such that $\delta(x)=1$ (resp. $\delta(x)=-1$). An **input** (resp. **output**) of $G$ is a leg $x$ such that $\delta(x)=1$ (resp. $\delta(x)=-1$). For $u\in Vt(G)\cup \{G\}$, the set of inputs (resp. outputs) of $u$ is written $in(u)$ (resp. $out(u)$).

1. A **listing** for $G$ with direction is a choice for each $u\in Vt(G)\cup \{G\}$ of a bijection of pairs of sets
$$
(in(u),out(u))\overset{l_u}\to(\{1,\ldots,|in(u)|\},\{1,\ldots,|out(u)|\}).
$$
=--

Thus, using these properties, we can model [[PROPs]] and [[properads]] with generalized graphs. See there and [[wheeled graph]] for more.

## Related concepts

* [[graph]]

* [[quiver]]

## References

* {#HRY}  [[Philip Hackney]], [[Marcy Robertson]] and [[Donald Yau]]. _Infinity Properads and Infinity Wheeled Properads_,  Lecture Notes in Mathematics, 2147. Springer, Cham, 2015. [(arxiv version)](http://arxiv.org/pdf/1410.6716v2.pdf)

* {#kock} [[Joachim Kock]]. _Graphs, hypergraphs, and properads_, [arXiv:1407.3744](https://arxiv.org/pdf/1407.3744v3.pdf).


generalized Gray tensor product

# Idea #

For any $n$, the category $Str n Cat$ of [[strict n-categories]] is [[cartesian closed category|cartesian closed]].  If $C$ and $D$ are strict $n$-categories, the objects of the [[exponential object]] $D^C$ are strict $n$-functors $C\to D$, its morphisms are *strict* $n$-natural transformations, and likewise its higher cells are *strict* higher [[transfors]].

On the other hand, one expects the weak $(n+1)$-category $n Cat$ of [[weak n-categories]] to be cartesian closed in the $(n+1)$-categorical sense, so that one has equivalences of hom-$n$-categories
\[n Cat(C\times D, E) \simeq n Cat(C, Hom(D,E)).\]
In this case, $Hom(D,E)$ should consist of weak functors, pseudo transformations (which are natural up to coherent [[equivalence]]), and pseudo higher transfors.  Note that although $C\times D$ (probably) means the same thing in the strict and weak worlds, $D^C$ and $Hom(D,E)$ are not the same, and in general not even [[equivalence of categories|equivalent]].

Gray tensor products are part of the general philosophy of trying to understand weak higher structures without getting bogged down in the theory of weaker higher-er structures.  One instance of this philosophy is the use of [[model categories]] (or more general types of [[homotopy theory]]) to use strict functors to model weak ones. Although not every weak functor is equivalent to a strict one (already when $n=2$), one expects this to become true if the domain is suitably [[cofibrant object|cofibrant]].  This is a theorem for small values of $n$ and can be regarded as a definition of "weak functor" for large values of $n$; see the [[folk model structure]] for 2-categories and [this paper](http://www.dpmms.cam.ac.uk/~rhgg2/Womaps/Womaps.html).

Gray tensor products attempt to extend this to deal with weak $k$-transfors for $k\gt 0$, starting with pseudo (or lax or oplax) natural transformations, by equipping higher categories of some sort with a non-cartesian [[monoidal category|monoidal]] and/or [[closed category|closed]] structure.  This began with John Gray, who showed that the category $Str2Cat$ is a non-symmetric [[closed monoidal category]] in which one internal-hom $Lax(C,D)$ consists of [[strict 2-functors]], [[lax natural transformations]], and [[modifications]] (the other contains oplax transformations instead).  A straightforward modification produces a closed symmetric monoidal structure based on *pseudo* natural transformations; the monoidal product is now called the [[Gray tensor product]].

Later various authors constructed analogous tensor products for [[strict ∞-categories]] and strict ω-groupoids, here called the [[Crans-Gray tensor product]].  Here the internal-homs consists of strict $\omega$-functors and lax/oplax $k$-transfors for $k\gt 0$.  (It appears to be unknown whether there is a pseudo version.)

# The Problem #

The (pseudo version of the) Gray tensor product of 2-categories is used to define [[Gray-categories]], which are a version of [[semistrict n-category|semistrict]] 3-categories.  This is very convenient because Gray-categories are just [[enriched category|categories enriched]] over $(Str2Cat,\otimes_{Gray})$, so that all of enriched category theory applies to them (as it does to 2-categories, being categories enriched over $Cat$).  Thus, it is natural to hope that one might be able to iterate this process, defining a "Gray tensor product" of semistrict $n$-categories and then defining semistrict $(n+1)$-categories to be categories enriched over semistrict $n$-categories.

However, there is a problem:

* _No category $SS n Cat$ of [[algebraic definition of higher category|algebraic]] $n$-categories with weak [[interchange law|interchange]] and strict functors can admit a [[closed category|biclosed structure]] that contains non-strict transformations._

Now there are lots of adjectives in this statement; for now just notice that the progression "categories, 2-categories, Gray-categories, Gray-cat-categories?" *is* producing categories of algebraic $n$-categories with weak interchange and strict functors.  So if this no-go statement is true, then Gray-cat does not admit a biclosed structure of the desired sort, and Gray-cat-categories probably can't be defined correctly.

To see why this no-go statement holds, imagine some hypothetical algebraic notion of semistrict $n$-category with weak interchange, and consider what sort of transformation (that is, 1-transfor) we want to model.  Let $C$ and $D$ be semistrict $n$-categories and $F,G:C\to D$ (strict) functors; then a transformation $\alpha:F\to G$ should probably consist of

* components $\alpha_x: F(x) \to G(x)$,
* equivalences $\alpha_f: G(f) \circ \alpha_x \simeq \alpha_y F(f)$ for $f:x\to y$,
* and higher data,
* satisfying axioms.

Now the question is, how is $\alpha_{g f}$ related to the composite of $\alpha_g$ and $\alpha_f$?  For pseudonatural transformations between 2-categories, they are equal (both being 2-cells, there is no room for anything weaker).  For a "fully weak" notion of transformation in higher dimensions, one might expect them to be only equivalent---but for the sorts of transfors used in the internal-hom for the Crans-Gray tensor product of $\omega$-categories, it turns out that they are also *equal*.  In fact, if the transformations are to give us a biclosed category of $n$-categories and strict functors, it turns out that they *must* be equal.

Consider the following: a transformation $F\Rightarrow G: C\to D$ should be the same as a functor $2 \to [C,D]$ where $2$ is the [[interval category|interval]] $n$-category.  But if we are to have a closed category $SS n Cat$, giving such a functor should be equivalent to giving a functor $C\to [2,D]$.  (This doesn't depend on symmetry, but if things are not symmetric then $[-,-]$ means something different in the two cases.)  Now inspection of how this would work reveals that the relationship of $\alpha_{g f}$ to $\alpha_g$ and $\alpha_f$ in the first picture corresponds to the *functoriality* of $C\to [2,D]$ in the second picture.  That is, if $\alpha_{g f} = \alpha_g . \alpha_f$ (and so on) then the functor $C\to [2,D]$ is a strict functor, while if it is only equivalent, then $C\to [2,D]$ will be only a weak functor.  Hence, if we allow only strict functors, then all our transformations must be "semistrict" in the sense that $\alpha_{g f} = \alpha_g . \alpha_f$.

Now, however, we meet the second horn of the dilemma: in an $n$-category with weak interchange, semistrict transformations need not be composable!  Suppose that $\alpha: F \to G$ and $\beta: G\to H$ are semistrict transformations.  Then we can define the components of a transformation $\beta.\alpha:F\to H$ in the obvious way, but is it semistrict?  If you try to prove that it is, you'll find that you need a strict interchange law.  (There is a pentagon involving all the ways to compose a 2x2 grid of square 2-cells, a picture of which can be found in Crans' paper cited below.)  So we are stuck.

What can we do?  Well, for one thing, the theorem is only about *closed* structures, not about *monoidal* structures.  And Sjoerd Crans did manage to construct a non-closed "Gray-like" monoidal structure on Gray-cat.  But it's not clear whether the resulting notion of "Gray-cat-category" is in fact semistrict (i.e. whether arbitrary 4-categories could be strictified to them).

Another way to escape would be to allow weak functors.  This is not much good all by itself, since algebraic $n$-categories and weak functors do not generally form very good categories.  They don't have many limits and colimits, for one thing, and the theory of enriched categories isn't much good when the enriching category doesn't have limits and colimits.  However, if we instead use a [[geometric definition of higher category]], then we can get well-behaved 1-categories containing weak functors.  And it is known for many nonalgebraic notions of higher category that there are good Gray-like tensor products; in fact, in many cases it is actually the cartesian product (an important exception being the lax version of the [[Verity-Gray tensor product]] of weak complicial sets).  This is one reason why non-algebraic notions of higher category are very attractive.

Finally, a third escape that suggests itself would be to look for a notion of algebraic semistrict $n$-categories which nevertheless has strict interchange.  As far as I know, the only work on strictly interchanging semistrict categories is one nonalgebraic notion which depends on weak units.  The prospects for algebraic notions don't seem that appealing, however, since the whole idea of the Gray-approach was to build in the semistrictness via the iterated enrichment, but while it is easy to imagine weakening interchange this way, it is hard to see how this could produce strict interchange but weak units.

# References

* [[Sjoerd Crans]], _A tensor product for $Gray$-categories_ ([TAC](http://www.tac.mta.ca/tac/volumes/1999/n2/5-02abs.html))

* [[Joachim Kock]], _Weak identity arrows in higher categories_ (semistrict categories with weak units): [math/0507116](http://arxiv.org/abs/math/0507116)

* [[Michael Batanin]], Denis-Charles Cisinski, Mark Weber, _Algebras of higher operads as enriched categories II_
([arXiv](http://arxiv.org/abs/0909.4715))


Generalized higher loop spaces

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

* table of contents
{:toc}

Generalized higher loop spaces are analogs of [[loop spaces]] adapted to the setting of [[overconvergent global analytic geometry]].

## Overview

Recall that one may define higher Hochschild homology of a derived scheme as functions on its completed formal higher loop space
$$\hat{L}^n X:=\widehat{Hom}_{dSt}(S^n,X).$$
There is a natural action of the symmetry group of $S^n$ on this space. One may try to define analogs of this construction for other symmetry groups, at least on a local field, to try to use these spaces to study representation theory.

The basic idea here is to start with the non-strict analytic space $GL_n$, and look at the mapping space
$$L_{GL_n}X:=Hom_{dSt}(GL_n,X).$$
It may only be definable for usual spaces, not Artin stacks, but one has to check that.

In any case, there is a natural (in the setting of [[generalized global analytic geometry]]) subgroup $U(n)\subset GL_n$, and one may also consider
$$L_{U(n)}X:=\widehat{Hom}_{dSt}(U(n),X).$$
Normally, since $U(n)$ is proper, this space is finite dimensional in the differential stacky sense. One may then define linear and unitary higher Hochschild cohomology of dimension $n$ to be given by functions on the moduli spaces
$$M_{GL_n}X:=[L_{GL_n}X/GL_n]$$
and
$$M_{U(n)}X:=[L_{U(n)}X/U(n)].$$
Up to $D^1$-homotopy, both spaces should be identical. Now the unitary moduli space has a strict global definition, which means it is a very rigid object (essentially, a kind of Arakelov stack, given by a scheme-moduli stack together with an archimedean structure).

The determinant map $det:GL_n\to GL_1$ induces a natural morphism
$$det^*:L_{GL_n}X\to L_{GL_1}X,$$
which may give also a morphism
$$det^*:M_{GL_n}X\to M_{GL_1}X.$$

Remark now that there exists an exact sequence of (non-strict) analytic groups
$$1\to U(1)\to GL_1\to N_1\to 1$$
where $N$ is the group of norms of elements in $GL_1$. In dimension $n$, we
only have a groupoid $N_n$. However, there is always the possibility to define a new
moduli space given by
$$D_{GL_n}X:=[L_{GL_n}X/U(n)].$$

This moduli space is equipped with an action of the groupoid $N_n$. In dimension $1$ over $C$, if $X$ is, say, projective, we get, as functions on $D_{GL_1}(X)$, a (probably infinite dimensional) space over $C$ together with an action of $N_1\cong \mathbb{R}_+^*$. In the $p$-adic situation, we have $N_1=p^\mathbb{Z}$. On a global field, one may work over adeles to get a sensible construction.

One should also not forget that there is (in characteristic $0$, e.g., on adeles of $\mathbb{Q}$) a logarithm map
$$log:\overset{\circ}{D}(1,1)_{GL_1}\to Lie(GL_1)$$
that allows us to relate the above constructions to the notion of [[generalized lambda-structure]].

## Relation with usual Hochschild and cyclic homology

Remark that, even over $\mathbb{C}$, the $D^1$-homotopy invariant version of $M_{U(1)}X$ is not equivalent to the (simplicial) loop space $L^1 X$. Indeed, to get a sensible relation between our moduli spaces and loop spaces (i.e., usual Hochschild and negative cyclic homology), one has to be very careful: inverting $D^1$ is not the right thing to do because it just gives usual loops up to homotopy, not the derived ones, which are necessary for differential constructions. A nice way to circumvent this problem is given by the following construction: let $R$ and $S$ be two base ind-Banach ring, and consider the category of derived stacks on $R$ with values in the (say) stable $D^1$-homotopy category of usual stacks over $S$. If we start with $S=\mathbb{C}$ and $R=(\Z,|\cdot|_0)$, we get a setting that is appropriate for explaining the relation between derived loop spaces and generalized higher loop spaces. Indeed, one may see $GL_n$ as a constant derived stack given by the associated group object of the $D^1$-homotopy category, denoted $GL_n^c$. Then the corresponding moduli space $M^c_{GL_n}X$ is related to the derived loop space in the case $n=1$: they are homotopy equivalent because the $D^1$-homotopy category over $C$ is equivalent (following Ayoub) to the usual homotopy category of simplicial sets.

If one wants to avoid getting back the usual loop space, one may work with the category
$$Shv(DAff^\dagger_S\times SAff^\dagger_R,{}^\infty Grpd)$$
of sheaves of sheaves on smooth affinoid spaces over $R$ on derived affinoid algebras over $S$. This may be seen as an extension of scalars of the $\infty$-category of derived analytic stacks over $S$ along the inclusion of infinity-groupoids into usual analytic higher stacks over $R$ (that form a presentable infinity-category, so that the tensor product operation works).

Recall that we may define two types of linear group stack-valued derived stacks by setting, for $(A,B)\in DAff^\dagger_S\times SAff^\dagger_R$, the constant linear stack to be defined associated to the prestack
$$GL_n^c(A,B)=GL_n(B),$$
and the usual derived analytic stack
$$GL_n^d(A,B)=GL_n(A).$$
Having a derived and a non-derived direction seems to be quite important. Indeed, the derived direction is needed for doing differential calculus, and the non-derived direction is needed for doing $D^1$-homotopy theory. Combining the two may give an optimal setting for relating generalized loop spaces with usual derived loop space.

There is a relation between the homotopy and classical generalized higher loop spaces given by the following: if we assume given a morphism $R\to S$, we may define the diagonal (extension of scalars) functor
$$\Delta:SAff^\dagger_R\longrightarrow DAff^\dagger_R\times SAff^\dagger_S,$$
to get a natural morphism of stack-valued derived stacks induced by
$$
\Delta^*GL_n^d(A)\cong GL_n(A)\longrightarrow GL_n(A_S)\cong\Delta^*GL_n^c(A).
$$

We may now define a version of the generalized higher loop spaces that is directly related to Hochschild cohomology: we may define
$$
L^{dc}_{GL_n}(X):=
\{(f,g)\in Hom(GL_n^d,X)\times Hom(GL_n^c,X),\; \Delta^*f\cong \Delta^* g\}.
$$
This means that we are choosing (for $n=1$), for every classical derived loop $g:GL_n^c\to X$ (up to $D^1$-homotopy, say), a corresponding generalized derived loop $f:GL_n\to X$, that we may call its representative. Forgetting the representative gives back the usual derived loop spaces if $R=(\Z,|\cdot|_0)$ and $S=(\C,|\cdot|_\infty)$, and $X$ is a derived scheme over $\Z$, and forgetting the classical derived loop gives the generalized higher derived loop space defined at the beginning of this page. We thus have two projections
$$L^d_{GL_n}X\leftarrow L^{dc}_{GL_n}(X)\to L^c_{GL_n}(X)$$
that may be used to define a correspondence between modules on the generalized higher derived loop space and on the classical derived loop space.

## Related entry

[[overconvergent derived loop spaces]]


generalized homology

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

#Contents#
* table of contents
{:toc}

## Idea

A _generalized homology theory_ is a certain [[functor]] from suitable [[topological spaces]] to [[graded abelian groups]] which satisfies most, but not all, of the abstract properties of [[ordinary homology]] functors (e.g. [[singular homology]]).

By the [[Brown representability theorem]], under certain conditions every [[spectrum]] $K$ is the coefficient object of a [[generalized (Eilenberg-Steenrod) cohomology|generalized]] [[cohomology theory]] and [[Spanier-Whitehead duality|S-dually]] of a generalized homology theory. For $K = H R$ an [[Eilenberg-MacLane spectrum]] this reduces to [[homology|ordinary homology]].

See at _[[generalized (Eilenberg-Steenrod) cohomology]]_ for more.

## Definition

### Reduced homology

Throughout, write [[Top]]${}_{CW}$ for the category of [[topological spaces]] [[homeomorphism|homeomorphic]] to [[CW-complexes]]. Write $Top^{\ast/}_{CW}$ for the corresponding category of [[pointed topological spaces]].

Recall that [[colimits]] in $Top^{\ast/}$ are computed as colimits in $Top$ after adjoining the base point and its inclusion maps to the given diagram

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

The [[coproduct]] in [[pointed topological spaces]] is the _[[wedge sum]]_, denoted $\vee_{i \in I} X_i$.

=--

Write

$$
  \Sigma \coloneqq S^1 \wedge (-)
  \;\colon\;
  Top^{\ast/}_{CW} \longrightarrow Top^{\ast/}_{CW}
$$

for the [[reduced suspension]] functor.

Write $Ab^{\mathbb{Z}}$ for the category of [[integer]]-[[graded abelian groups]].

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

A **reduced homology theory** is a [[functor]]

$$
  \tilde E_\bullet
   \;\colon\;
  (Top^{\ast/}_{CW}) \longrightarrow Ab^{\mathbb{Z}}
$$

from the category of [[pointed topological spaces]] ([[CW-complexes]]) to $\mathbb{Z}$-[[graded abelian groups]] ("[[homology groups]]"), in components

$$
  \tilde E _\bullet
    \;\colon\;
  (X \stackrel{f}{\longrightarrow} Y)
    \mapsto
  (\tilde E_\bullet(X)
    \stackrel{f_\ast}{\longrightarrow}
  \tilde E_\bullet(Y))
  \,,
$$

and equipped with a [[natural isomorphism]] of degree +1, to be called the **[[suspension isomorphism]]**, of the form

$$
  \sigma
    \;\colon\;
  \tilde E_\bullet(-)
    \overset{\simeq}{\longrightarrow}
  \tilde E_{\bullet +1}(\Sigma -)
$$

such that:

1. **([[homotopy invariance]])** If $f_1,f_2 \colon X \longrightarrow Y$ are two morphisms of pointed topological spaces such that there is a (base point preserving) [[homotopy]] $f_1 \simeq f_2$ between them, then the induced [[homomorphisms]] of abelian groups are [[equality|equal]]

   $$
     f_1_\ast = f_2_\ast
     \,.
   $$

1. {#ReducedExactnessAxiom} **(exactness)** For $i \colon A \hookrightarrow X$ an inclusion of pointed topological spaces, with $j \colon X \longrightarrow Cone(i)$ the induced [[mapping cone]], then this gives an [[exact sequence]] of graded abelian groups

   $$
     \tilde E_\bullet(A)
      \overset{i_\ast}{\longrightarrow}
     \tilde E_\bullet(X)
       \overset{j_\ast}{\longrightarrow}
     \tilde E_\bullet(Cone(i))
     \,.
   $$

We say $\tilde E_\bullet$ is **additive** if in addition

* **([[wedge axiom]])** For $\{X_i\}_{i \in I} $ any set of pointed CW-complexes, then the canonical morphism

  $$
    \oplus_{i \in I} \tilde E_\bullet(X_i)
      \longrightarrow
    \tilde E^\bullet(\vee_{i \in I} X_i)
  $$

  from the [[direct sum]] of the value on the summands to the value on the [[wedge sum]], example \ref{WedgeSumAsCoproduct}, is an [[isomorphism]].

We say $\tilde E_\bullet$ is **ordinary** if its value on the [[0-sphere]] $S^0$ is concentrated in degree 0:

* **(Dimension)**  $\tilde E_{\bullet\neq 0}(\mathbb{S}^0) \simeq 0$.

A [[homomorphism]] of reduced cohomology theories

$$
  \eta \;\colon\; \tilde E_\bullet \longrightarrow \tilde F_\bullet
$$

is a [[natural transformation]] between the underlying functors which is compatible with the suspension isomorphisms in that all the following [[commuting square|squares commute]]

$$
  \array{
    \tilde E_\bullet(X) &\overset{\eta_X}{\longrightarrow}&  \tilde F_\bullet(X)
    \\
    {}^{\mathllap{\sigma_E}}\downarrow && \downarrow^{\mathrlap{\sigma_F}}
    \\
    \tilde E_{\bullet + 1}(\Sigma X)
    &\overset{\eta_{\Sigma X}}{\longrightarrow}&
    \tilde F_{\bullet + 1}(\Sigma X)
  }
  \,.
$$

=--

### Unreduced homology

In the following a _pair_ $(X,A)$ refers to a [[subspace]] inclusion of [[topological spaces]] ([[CW-complexes]])  $A \hookrightarrow X$.  Whenever only one space is mentioned, the subspace is assumed to be the [[empty set]] $(X, \emptyset)$. Write $Top_{CW}^{\hookrightarrow}$ for the category of such pairs (the [[full subcategory]] of the [[arrow category]] of $Top_{CW}$ on the inclusions). We identify $Top_{CW} \hookrightarrow Top_{CW}^{\hookrightarrow}$ by $X \mapsto (X,\emptyset)$.

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

A **homology theory** (unreduced, [[relative cohomology|relative]]) is a [[functor]]

$$
  E_\bullet : (Top_{CW}^{\hookrightarrow}) \longrightarrow Ab^{\mathbb{Z}}
$$

to the category of $\mathbb{Z}$-[[graded abelian groups]], as well as a [[natural transformation]] of degree +1, to be called the **[[connecting homomorphism]]**, of the form

$$
  \delta_{(X,A)}
    \;\colon\;
  E_{\bullet + 1}(X, A)
    \longrightarrow
  E_\bullet(A, \emptyset)
  \,.
$$

such that:

1. **(homotopy invariance)** For $f \colon (X_1,A_1) \to (X_2,A_2)$ a [[homotopy equivalence]] of pairs, then

   $$
     E_\bullet(f)
      \;\colon\;
     E_\bullet(X_1,A_1) \stackrel{\simeq}{\longrightarrow} E_\bullet(X_2,A_2)
   $$

   is an [[isomorphism]];

1. {#ExactnessUnreduced} **(exactness)** For $A \hookrightarrow X$ the induced sequence

   $$
     \cdots
       \to
     E_{n+1}(X, A)
       \stackrel{\delta}{\longrightarrow}
     E_n(A)
       \longrightarrow
     E_n(X)
       \longrightarrow
     E_n(X, A)
       \to
     \cdots
   $$

   is a [[long exact sequence]] of [[abelian groups]].

1. {#excision} **([[excision]])** For $U \hookrightarrow A \hookrightarrow X$ such that $\overline{U} \subset Int(A)$, then the natural inclusion of the pair $i \colon (X-U, A-U) \hookrightarrow (X, A)$ induces an isomorphism

   $$
     E_\bullet(i)
      \;\colon\;
     E_n(X-U, A-U)
      \overset{\simeq}{\longrightarrow}
     E_n(X, A)
   $$

We say $E_\bullet$ is **additive** if it takes [[coproducts]] to [[direct sums]]:

* **(additivity)** If $(X, A) = \coprod_i (X_i, A_i)$ is a [[coproduct]], then the canonical comparison morphism

  $$
    \oplus_i E_n(X_i, A_i) \overset{\simeq}{\longrightarrow} E_n(X, A)
  $$

  is an [[isomorphism]]from the [[direct sum]] of the value on the summands, to the value on the total pair.

We say $E_\bullet$ is **ordinary** if its value on the point is concentrated in degree 0

* **(Dimension)**: $E_{\bullet \neq 0}(\ast,\emptyset) = 0$.

A [[homomorphism]] of unreduced homology theories

$$
  \eta \;\colon\; E_\bullet \longrightarrow F_\bullet
$$

is a [[natural transformation]] of the underlying functors that is compatible with the connecting homomorphisms, hence such that all these [[commuting square|squares commute]]:

$$
  \array{
     E_{\bullet +1}(X,A)
       &\overset{\eta_{(X,A)}}{\longrightarrow}&
     F_{\bullet +1}(X,A)
     \\
     {}^{\mathllap{\delta_E}}\downarrow && \downarrow^{\mathrlap{\delta_F}}
     \\
     E_\bullet(A,\emptyset) &\overset{\eta_{(A,\emptyset)}}{\longrightarrow}& F_\bullet(A,\emptyset)
  }
  \,.
$$

=--

+-- {: .num_defn #AlternativeFormulationOfExcisionAxiom}
###### Lemma

The excision axiom in def. \ref{GeneralizedHomologyTheory} is equivalent to the following statement:

For all $A,B \hookrightarrow X$ with $X = A \cup B$, then the inclusion

$$
  i \colon (A, A \cap B) \longrightarrow (X,B)
$$

induces an isomorphism,

$$
  i_\ast
    \;\colon\;
  E_\bullet(A, A \cap B)
    \overset{\simeq}{\longrightarrow}
  E_\bullet(X, B)
  \,.
$$

=--

(e.g [Switzer 75, 7.2, 7.5](#Switzer75))

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

First consider the statement under the condition that $X = Int(A) \cup Int(B)$.

In one direction, suppose that $E^\bullet$ satisfies the original excision axiom. Given $A,B$ with $X = \Int(A) \cup Int(B)$, set $U \coloneqq X-A$ and observe that

$$
  \begin{aligned}
     \overline{U}
        & = \overline{X-A}
     \\ & = X- Int(A)
     \\ & \subset Int(B)
  \end{aligned}
$$

and that

$$
  (X-U, B-U) = (A, A \cap B)
  \,.
$$

Hence the excision axiom implies $  E^\bullet(X, B) \overset{\simeq}{\longrightarrow} E^\bullet(A, A \cap B)$.

Conversely, suppose $E^\bullet$ satisfies the alternative condition. Given $U \hookrightarrow A \hookrightarrow X$ with $\overline{U} \subset Int(A)$, observe that we have a cover

$$
  \begin{aligned}
          Int(X-U) \cup Int(A)
      & = (X - \overline{U}) \cap \Int(A)
   \\ & \supset (X - Int(A)) \cap Int(A)
   \\ & = X
  \end{aligned}
$$

and that

$$
  (X-U, (X-U) \cap A) = (X-U, A - U)
  \,.
$$

Hence

$$
  E^\bullet(X-U,A-U)
  \simeq
  E^\bullet(X-U, (X-U)\cap A)
  \simeq
  E^\bullet(X,A)
  \,.
$$

This shows the statement for the special case that $X = Int(A)\cup Int(U)$.  The general statement reduces to this by finding a suitable homotopy equivalence to a slightly larger covering pair (e.g [Switzer 75, 7.5](#Switzer75)).

=--

+-- {: .num_prop #ExactSequenceOfATriple}
###### Proposition
**(exact sequence of a triple)**

For $E_\bullet$ an unreduced generalized cohomology theory, def. \ref{GeneralizedHomologyTheory}, then every inclusion of two consecutive subspaces

$$
  Z \hookrightarrow Y \hookrightarrow X
$$

induces a [[long exact sequence]] of homology groups of the form

$$
  \cdots
   \to
  E_q(Y,Z)
    \stackrel{}{\longrightarrow}
  E_q(X,Z)
    \stackrel{}{\longrightarrow}
  E_q(X,Y)
    \stackrel{\bar \delta}{\longrightarrow}
  E_{q-1}(Y,Z)
    \to
  \cdots
$$

where

$$
  \bar \delta
    \;\colon \;
  E_{q}(X,Y)
   \stackrel{\delta}{\longrightarrow}
  E_{q-1}(Y)
   \longrightarrow
  E_{q-1}(Y,Z)
  \,.
$$

=--

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

Apply the [[braid lemma]] to the interlocking long exact sequences of the three pairs $(X,Y)$, $(X,Z)$, $(Y,Z)$:

<img src="http://www.ncatlab.org/nlab/files/BraidDiagramForHomologyOnTripled.jpg" width="500">

(graphics from [this Maths.SE comment](http://math.stackexchange.com/a/1180681/58526))

See [here](braid+lemma#ExactSequenceForTripleInGeneralizedHomology) for details.

=--

## Properties

### Expression by ordinary homology via Atiyah-Hirzebruch spectral sequence

The [[Atiyah-Hirzebruch spectral sequence]] serves to express generalized homology $E_\bullet$ in terms of ordinary homology with coefficients in $E_\bullet(\ast)$.

### Whitehead theorem
 {#WhiteheadTheorem}

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

Let $\phi \colon E \longrightarrow F$ be a morphism of reduced generalized (co-)homology functors, def. \ref{ReducedGeneralizedHomology} (a [[natural transformation]]) such that its component

$$
  \phi(S^0) \colon E(S^0) \longrightarrow F(S^0)
$$

on the [[0-sphere]] is an [[isomorphism]]. Then $\phi(X)\colon E(X)\to F(X)$ is an [[isomorphism]] for $X$ any [[CW-complex]] with a [[finite number]] of cells. If both $E$ and $F$ satisfy the [[wedge axiom]], then $\phi(X)$ is an isomorphism for $X$ any [[CW-complex]], not necessarily finite.

=--

For $E$ and $F$ [[ordinary cohomology]]/[[ordinary homology]] functors a proof of this is in ([Eilenberg-Steenrod 52, section III.10](#EilenbergSteenrod52)). From this the general statement follows (e.g. [Kochman 96, theorem 3.4.3, corollary 4.2.8](#Kochman96)) via the [[natural transformation|naturality]] of the [[Atiyah-Hirzebruch spectral sequence]] (the classical result gives that $\phi$ induces an isomorphism between the second pages of the AHSSs for $E$ and $F$). A complete proof of the general result is also given as ([Switzer 75, theorem 7.55, theorem 7.67](#Switzer75))

## Examples

* [[stable homotopy homology theory]] is the homology theory represented by the [[sphere spectrum]]

* [[ordinary homology]] is the homology theory represented by an [[Eilenberg-MacLane spectrum]]

* [[bordism homology theory]] is the homology theory represented by a [[Thom spectrum]];

## Related concepts

* [[generalized cohomology]]

* [[Kronecker pairing]]

* [[Atiyah-Hirzebruch spectral sequence]]

* [[Steenrod algebra]]

* [[phantom map]]

* [[bivariant cohomology]]

* [[Bousfield equivalence]]

* [[Hurewicz homomorphism]], [[Boardman homomorphism]]

## References

(For more see the references at  _[[generalized (Eilenberg-Steenrod) cohomology]]_.)

Original articles include

* [[George Whitehead]], _Generalized homology theories_  (1961) ([pdf](http://www.maths.ed.ac.uk/~aar/papers/gww9.pdf))

Textbook accounts include

* {#Kochmann96} [[Stanley Kochmann]], section 3.4 of _[[Bordism, Stable Homotopy and Adams Spectral Sequences]]_, AMS 1996

* {#Switzer75} [[Robert Switzer]], chapter 7 (and 8-12) of _Algebraic Topology - Homotopy and Homology_, Die  Grundlehren der Mathematischen Wissenschaften in Einzeldarstellungen, Vol. 212, Springer-Verlag, New York, N. Y., 1975.

* [[Stefan Schwede]], chapter II, section 6 of  _[[Symmetric spectra]]_, 2012  ([pdf](http://www.math.uni-bonn.de/~schwede/SymSpec-v3.pdf))

See also

* [[Friedrich Bauer]], _Classifying spectra for generalized homology theories_ Annali di Maternatica pura ed applicata
(IV), Vol. CLXIV (1993), pp. 365-399

* [[Friedrich Bauer]], _Remarks on universal coefficient theorems for generalized homology theories_ Quaestiones Mathematicae
Volume 9, Issue 1 & 4, 1986, Pages 29 - 54

A general construction of homologies by "geometric cycles" similar to the [[Baum-Douglas geometric cycles]] for [[K-homology]] is discussed in

* S. Buoncristiano, C. P. Rourke and B. J. Sanderson, _A geometric approach to homology theory_, Cambridge Univ. Press, Cambridge, Mass. (1976)

Further generalization of this to [[bivariant cohomology theories]] is in

* Martin Jakob, _Bivariant theories for smooth manifolds_, Applied Categorical Structures 10 no. 3 (2002)

[[!redirects generalized homology theory]]
[[!redirects generalized homology theories]]

[[!redirects Whitehead-generalized homology theory]]
[[!redirects Whitehead-generalized homology theories]]

[[!redirects homology theory]]
[[!redirects homology theories]]

[[!redirects generalized homologies]]

[[!redirects generalised homology]]
[[!redirects generalised homologies]]


generalized kernel

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

# Contents
* table of contents
{: toc}

## Idea

In an [[Ab-enriched category]], it is natural to produce an [[image]] factorization of a morphism by first forming its [[kernel]] and then the [[cokernel]] of the kernel.  Similarly, in a [[regular category]] we can produce an image factorization by first forming the [[kernel pair]] and then the [[coequalizer]] of the kernel pair.  Several similar situations arise in the study of [[2-categories]] as well.  The theory of **generalized kernels** in [[enriched categories]] subsumes all of these examples.

## Generalized kernels and their quotients

Let $V$ be a [[cosmos]] and let $\mathbf{2}$ denote the [[interval category]] regarded as a $V$-category, i.e. it has two objects $0$ and $1$, with $hom(0,0) = hom(1,1) = hom(0,1) = I$ (the unit object of $V$) and $hom(1,0) = \emptyset$ (the initial object of $V$).  Let $H$ be a $V$-category which contains $\mathbf{2}$ as a full subcategory, and let $J$ be the full subcategory of $H$ containing all the objects except $1$.  Then the inclusions of $J$ and $\mathbf{2}$ into $H$ induce a [[profunctor]] $K\colon \mathbf{2} ⇸ J$.  This is the input data for a notion of kernels.

Now suppose that $C$ is a $V$-category with sufficiently many limits and colimits, and that $f\colon x\to y$ is a morphism in $C$.  Then $f$ can be identified with a $V$-functor $\ulcorner f \urcorner \colon \mathbf{2}\to C$.  The **kernel** of $f$ is defined to be the $K$-[[weighted limit]] of $\ulcorner f \urcorner$, which is a $V$-functor $ker(f)\colon J\to C$.  By construction of $K$, it can equivalently be described as the (enriched, pointwise) right [[Kan extension]] of $\ulcorner f \urcorner$ from $\mathbf{2}$ into $H$, followed by restriction to $J$.  Since the inclusion of $\mathbf{2}$ into $H$ is fully faithful, it follows that $ker(f)(0) = \ulcorner f \urcorner (0) = x$.

Similarly, let $M$ be a $V$-functor $J\to C$; we call such a functor **kernel data**.  The **quotient** of $M$ defined to be its $K$-weighted colimit. This is a functor $\mathbf{2}\to C$, i.e. a single morphism in $C$.  Again, by construction of $K$, the quotient of $M$ can equivalently be described by the left Kan extension of $M$ to $H$, followed by restriction to $\mathbf{2}$, and since $J\to H$ is fully faithful, the source of the quotient of $M$ is the object $M(0)$.

Since colimits and limits weighted by a fixed profunctor are adjoint to each other (or equivalently, since left and right Kan extension are left and right adjoint to restriction), we obtain an adjunction
$$ quot: [J,C] \;\rightleftarrows\; [\mathbf{2},C] :ker $$
Moreover, by the remarks above, for any object $x\in C$ this adjunction restricts to an adjunction
$$ quot: [J,C]_x \;\rightleftarrows\; x/C :ker $$
where $[J,C]_x$ denotes the subcategory of $[J,C]$ on those functors $M$ such that $M(0)=x$.  Of course the analogous subcategory $[\mathbf{2},C]_x$ is just the [[coslice category]] $x/C$.

Of particular interest is the [[counit]] of this adjunction at a morphism $f\colon x\to y$, which is a morphism $quot(ker(f)) \to f$ in $x/C$.  In other words, it is a *factorization* of $f$.  Often the factorizations produced in this way are familiar.

## Examples

1. With $V=Set$, let $H$ be the category
   $$ z \;\rightrightarrows\; 0 \to 1 $$
   with the two composites $z\to 1$ being equal.  That is, $H$ is the "[[walking structure|walking]] fork."  Then kernel data is a pair of parallel arrows, the kernel of a morphism is its [[kernel pair]], and the quotient of a parallel pair is their [[coequalizer]].  In a [[regular category]], the factorization produced in this way is the usual [[image]] factorization.

   Note that since coequalizers are epic, the full image of $quot\colon [J,C] \to [\mathbf{2},C]$ is a preorder, and thus in this case the adjunction is [[idempotent adjunction|idempotent]].  This yields the familiar fact that a morphism is a coequalizer iff it is the coequalizer of its kernel pair, and a parallel pair is a kernel pair iff it is the kernel pair of its coequalizer.

1. Again with $V=Set$, let $H$ be the category
   $$ \array{ 0 \\ & \searrow \\ z & \to & 1 } $$
   Then kernel data is just a pair of objects, the kernel of a morphism $f\colon x\to y$ is the pair $(y,x)$, and the quotient of a pair $(y,x)$ is the [[coproduct]] injection $x \to (x\sqcup y)$.  The factorization of $f\colon x\to y$ obtained in this way is $x \to (x\sqcup y) \overset{[f,id]}{\to} y$.

1. With $V=Ab$, let $H$ be the [[Ab-enriched category]] with three objects $z,0,1$ and $H(z,z) = H(0,0) = H(1,1) = H(z,0) = H(0,1) = \mathbb{Z}$ and all other hom-groups being $0$.  Then a diagram of shape $H$ in an Ab-category consists of a pair of composable arrows whose composite is zero.  Then kernel data is also just a single morphism, the kernel of a morphism is its [[kernel]] in the usual sense, and the quotient of a morphism (considered as kernel data) is its [[cokernel]] in the usual sense.  The resulting factorization is again the usual image factorization, and the adjunction is again idempotent.

1. With $V=Cat$, let $H$ be the (strict) [[2-category]] with three objects $z,0,1$ and $H(0,1) = *$, $H(z,1) = \mathbf{2}$, and $H(z,0)$ the "[[walking structure|walking]] [[parallel pair]]" $(\cdot \rightrightarrows \cdot)$.  Then kernel data in a 2-category consists of a pair of parallel 2-cells, and the quotient of such a pair is their [[coequifier]].  In $Cat$ at least, the kernel $f\colon x\to y$ of a functor is the category of parallel pairs in $x$ which become equal in $y$, and the resulting factorization of $f$ consists of an bijective-on-objects-and-full functor followed by a faithful one.  Since bijective-on-objects-and-full functors are epic in $Cat$, the adjunction is again idempotent.

1. Again with $V=Cat$, let $H$ be the 2-category with three objects $y,0,1$ and $H(0,1)=*$, $H(y,0)$ the discrete category with two objects, and $H(y,1)=\mathbf{2}$.  Then kernel data is a pair of parallel 1-morphisms, and the quotient of such a pair is their [[coinserter]].  The kernel of a morphism $f\colon x\to y$ is the [[comma object]] $(f/f)$.

1. Once again with $V=Cat$, let $H$ be the 2-category
   $$ \array{ && 0\\ & \nearrow & \cong & \searrow \\ z && \to && 1} $$
   Then kernel data is again a single arrow, its quotient is its pseudocolimit, and the kernel of a morphism is its pseudolimit.  The resulting factorization is one of the ones arising in the [[canonical model structure]] on any 2-category.

1. Continuing with $V=Cat$, let $H$ be the 2-category on $z,0,1$ with $H(0,1)=*$, $H(z,0) = \mathbf{2}$, and $H(z,1)$ the [[walking structure|walking]] [[isomorphism]].  Then kernel data is a 2-cell, and its quotient is its [[coinverter]].  The kernel of a morphism is sometimes called its [[invertee]].  Since coinverters are epic, the adjunction is idempotent: thus a morphism is a coinverter iff it is the coinverter of its invertee, and a 2-cell is an invertee iff it is the invertee of its coinverter.

1. Finally with $V=Cat$, let $H$ be the 3-truncated augmented [[simplex category]] with additional 2-cells added, in such a way that kernel data is *codescent data* and the quotient of such data is a [[codescent object]].  Then the factorization produced in $Cat$ consists of a bijective-on-objects functor followed by a fully faithful one.

## Factorization systems

In some of the above examples, the resulting factorization of a morphism is not what one would hope for.  In particular, it need not give a [[factorization system]] of any kind.  However, the adjunction does induce a [[comonad]] $Q_1 = quot \circ ker$ on $[\mathbf{2},C]$ such that $dom(Q_1(f)) = dom(f)$.  This is the "left half" of the factorization obtained in this way; the "right half" $R_1$ is merely a [[pointed endofunctor]] such that $cod(R_1(f)) = cod(f)$.  The [[algebra for an endofunctor|algebras]] for $R_1$ are usually what one would hope for as the "right class" of the (weak) factorization systems intuitively associated to the above examples.

Now this data is exactly the required input for the construction of a *free* [[algebraic weak factorization system]].  If $C$ is sufficiently well-behaved (such as [[locally presentable category|locally presentable]]), then this free nwfs exists, and gives the factorizations we would hope for in most cases.  It is constructed by (possibly transfinite) iteration, i.e. we take the quotient of the kernel, then take the quotient of the kernel of the right half of the resulting factorization, and so on.  The resulting monad $R$ is usually [[free monad|algebraically-free]] on $R_1$, and thus the $R$-[[algebra for a monad|algebras]] are again just what we would expect.

For example, in the coinverter-invertee example above, the right clas of the factorization system produced in this way consists of [[conservative functor|conservative]] functors.

## Related concepts

* [[kernel]], [[cokernel]]

* **generalized kernel**

## References

This sort of idea has been thought of by various people at various times.  The above version, especially the view of factorization systems, is a slight modification of one presented by [[Richard Garner]] to the Sydney category seminar in February 2010.  Other references include

* Betti, Schumacher, and [[Ross Street|Street]], "Factorizations in bicategories" (unpublished)

* Betti, "Adjointness in descent theory", JPAA

* [[Mathieu Dupont]], *Abelian categories in dimension 2*, [arXiv:0809.1760](http://arxiv.org/abs/0809.1760), section 1.2.2

[[!redirects generalized kernels]]
[[!redirects generalized cokernel]]
[[!redirects generalized cokernels]]


generalized lambda-structure

[[!redirects Generalized Lambda-structure]]

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## Overview

The usual notion of [[Lambda-ring]] is directly related to the Banach ring
$(\mathbb{Z},|\cdot|_0)$ of integers equipped with their trivial norm in
the following way: a [[Lambda-ring]] is a usual ring equipped with
an action of the monoid $\mathbb{N}=\mathbb{Z}$-$\{0\}/\{\pm 1\}$.
Remark that some important Lambda-rings, such as K-theory, are
actually equipped with an additional $\mathbb{Z}/2$-grading,
that may be combined with the Lambda-structure to get an action
of the full monoid $\mathbb{Z}$-$\{0\}$. It is important to remark here
that the $\Lambda$-structure on $K$-theory allows one to get back
(as the spectrum of the Lambda-operations) the full $\mathbb{Z}$-grading
on Betti cohomology.

From the perspective of [[global analytic geometry]], one thinks of Lambda-structures
as related to the Banach ring of integers with their trivial norm,
so that one may seek for various generalizations, that will be called
"generalized Lambda-structure", associated to more general Banach or ind-Banach rings.

## Definition

Let $(R,|\cdot|)$ be an integral Banach ring equipped with a multiplicative norm.
We will denote $\Lambda(R,|\cdot|)$ the monoid given by
$$\Lambda(R,|\cdot|):=\mathrm{Frac}(R)\cap \{a\in R,\;|a|\leq 1\}.$$

## Examples

There is not yet a precise notion of generalized Lambda-structure, but one may easily give various of its concrete incarnations.

1. The classical notion of ($\mathbb{Z}/2$-graded) Lambda-ring may be seen as a $\Lambda(\mathbb{Z},|\cdot|_0)=\mathbb{Z}$-$\{0\}$-structure. The (geometric/Weil) cohomology theories of arithmetic geometry (i.e. for schemes over $\mathbb{Z}$) are often equipped with an $\mathbb{N}$-grading, that one may interpret as a classical Lambda-ring structure.

1. The [[absolute cohomology]] theories in arithmetic geometry such as Beilinson-[[Deligne cohomology]] or [[motivic cohomology]] are equipped with a natural bi-graduation, related to the fact that they are defined by the (homotopical: think of a Leray spectral sequence) combination of geometric methods (Lambda-structures/Frobenii) and of differential methods (Hodge filtration of de Rham cohomology or $\mathbb{G}_m$-stabilization in motivic homotopy theory, that corresponds to the "[[Tate twist]]" grading).
The corresponding Banach ring may be simply given by the Banach ring
$$(R,|\cdot|):=(\mathbb{Z}[T],|\cdot|_0)$$
of polynomials equipped with their trivial norm (analytic functions on the non-archimedean global analytic unit disc). The bigrading is given by the action of the monoid
$$
\Lambda(R,|\cdot|):=
\mathbb{Q}(T)^\times\cap \mathbb{Z}[T]=\mathbb{Z}[T]-\{0\}.
$$
The ([[Tate twist]]) motivic and cohomological gradings are given respectively by the actions of the monoid $\mathbb{Z}-\{0\}$ and the monoid of powers of $T$.

1. The (yet to be properly defined) cohomology theories in [[global analytic geometry]] have a different type of bigrading (that is related to the idea of the algebra of polynomials over the [[field with one element]], formulated precisely, e.g., in Durov's setting of [[generalized rings]], i.e., commutative algebraic monads).
We will now extend the above definition of the monoid $\Lambda$ to the setting of ind-Banach ring, since this operation seems necessary to understand absolute cohomologies.
The corresponding (ind-)Banach ring may be simply given by the ind-Banach ring
$$R:=\mathbb{Z}\{T\}^\dagger$$
of overconvergent power series on the unit disc with coefficients in the Banach ring $(\mathbb{Z},|\cdot|_\infty)$: the `geometric` classical Lambda-structure is given by the base Banach ring, and the differential/absolute graduation is given by the $T$-part of the monoid (we may need to make a completion here)
$$
\Lambda(R):=
\mathrm{Frac}(\mathbb{R}\{T\}^\dagger)^\times\cap \{P\in \mathbb{Z}\{T\}^\dagger,|P|_{\infty,1}\leq 1\}=\mathbb{F}_{\{\pm 1\}}[T]-\{0\}.
$$
This is the monoid of polynomials whose terms are all equal to zero except exactly one, that is equal to $\{\pm 1\}$. It contains and extends the monoid $\Lambda(\mathbb{Z},|\cdot|_\infty)=\{\pm 1\}=(\mathbb{F}_{\{\pm 1\}})^\times$ in degree zero. This will be the natural grading monoid (generalized Lambda-structure) for absolute motives, i.e., motivic cohomology theories over $(\mathbb{Z},|\cdot|_\infty)$. Remark that the recent work of [[Peter Scholze]] on local Schtukas in mixed characteristic also uses in an essential way objects such as the unit disc over the given base Banach ring.

1. The notion of $\Lambda(\mathbb{Z},|\cdot|_\infty)=\{\pm 1\}$-structure is simply given by the notion of $\mathbb{Z}/2$-grading. Many cohomological invariants, such as [[K-theory]], negative [[cyclic homology]] and the [[Chern character]] are equipped with a natural $\mathbb{Z}/2$-grading. It is quite probable that one can't hope to get something more that a $\mathbb{Z}/2$-grading on a "really natural" cohomology theory in [[global analytic geometry]].

1. In the theory of $(\Phi,\Gamma)$-modules, the monoid $\Lambda(\mathbb{Z}_p,|\cdot|_p)=\mathbb{Z}_p$-$\{0\}$ plays a central role. It looks like a not so hard but important task to clarify the relation of this theory with the classical notion of Lambda-ring.

1. It is an interesting question to try to understand the relation of classical Hodge theory (over $\mathbb{R}$ or $\mathbb{C}$) with the notion of Lambda-structure on the corresponding Banach ring. This may show interesting limits to the idea of generalizing Lambda-structures to other Banach rings. The case of $\mathbb{R}$ may (or may not) be treated using $\mathbb{Z}/2$-equivariant methods. An important point, in this perspective, is that the naive archimedean generalization of the notion of $(\Phi,\Gamma)$-module does not work, because $S^1$ does not act directly on the open complex unit disc $D^\circ(1,1)$. One only has an infinitesimal action (connection $\nabla$), whose combination with the infinitesimal generator $\Phi$ of $\mathbb{R}_+^*$ may be seen as an archimedean analog of the $p$-adic differential equations used in Berger's thesis to prove the monodromy theorem of $p$-adic Hodge theory. An important drawback of this infinitesimal approach (in the $p$-adic setting) is that the functor from $p$-adic Hodge structures (i.e., $(\Phi,\Gamma)$-modules) to $p$-adic Frobenius-equivariant differential equations is `not fully faithful`: making the action of $U(1)=\mathbb{Z}_p^*$ infinitesimal kills an important part of the information (essentially, the Hodge filtration on de Rham cohomology). A possible solution to this problem may be to work with a multiplicative theory over $[\mathbb{A}^1/\mathbb{G}_m]$ instead of an additive one over $[D^\circ(1,1)/\Lambda]$ or $[D^\circ(1,1)/(\Phi,\nabla)]$, or to use a combination of the additive and multiplicative theory.

1. The monoid that should come in play into the theory of [[spectral interpretation]] for zeroes and poles of global arithmetic and automorphic [[L-functions]] may be given by the monoid $\Lambda(\mathbb{A})$, where $\mathbb{A}$ is the ind-Banach ring of adèles.

## Geometric interpretation (to be checked very carefully: may be problematic)

One may take inspiration from the theory of $(\Phi,\Gamma)$-modules ($p$-adic Hodge structures) to define a natural notion of $\Lambda$-module in [[global analytic geometry]]. This gives a version of the notion of a "Hodge structure" that works over an integral base, which makes it quite well adapted to the global analytic situation.

Let $R$ be a Banach ring, and $\overset{\circ}{D}(1,1)$ be the open unit disc on $R$. We denote (be careful, this differs from the previously used notation, because it is a different kind of object)
$$\Lambda\subset \overset{\circ}{D}(1,1)\times \overset{\circ}{D}(1,1)$$
the (non-strict) analytic subgroupoid of the groupoid of pairs acting on $\overset{\circ}{D}(1,1)$ given by pairs of the form $(1+x,(1+x)^a)$ where $a\in D(0,1)\cap \GL_1\subset \mathbb{A}^1$,  and
$$(1+x)^a:=\sum_{k\geq 0}\binom{a}{k}x^k.$$
If $R=(\mathbb{Q}_p,|\cdot|_p)$, then we have actually that $\mathbb{Z}_p-\{0\}\subset D(0,1)(\mathbb{Q}_p)$.

A $\Lambda$-module over $R$ is a module over the analytic stack $B\Lambda$ that one may denote as a quotient stack $[\overset{\circ}{D}(1,1)/\Lambda]$. We then have, if $R$ contains the rational numbers, a natural logarithm map
$$log:[\overset{\circ}{D}(1,1)/\Lambda]\to [\mathbb{A}^1/\GL_1]$$
that allows us to give a relation between the classical Hodge filtration of a (say) proper or logarithmically proper analytic space over $R$ to its $R$-Hodge structure, that should be a module over $[\overset{\circ}{D}(1,1)/\Lambda]$.

If we suppose given a (say) strict analytic space over $R$, and one wants to define the associated $R$-Hodge structure, one may simply try to adapt Simpson's construction of the deformation to the normal bundle, to get what one wants.
Actually, one needs a loop space analog of this construction, that is due to Vezzosi for a derived scheme. Recall that in this derived scheme case, we have
$$LX=\Hom_{dSt}(B\mathbb{Z},X)\cong \Hom_{dSt}(B\mathbb{G}_a,X).$$
We may use the action by multiplication of $\mathbb{A}^1$ on $B\mathbb{G}_a$ to define a family
of actions of $B\mathbb{G}_a$ on $LX$, parametrized by $\mathbb{A}^1$. This gives a $\mathbb{G}_m$-equivariant family
$$Hod(LX)\to \mathbb{A}^1$$
whose fiber at $0$ is $LX$ with the trivial action of $B\mathbb{G}_a$ and whose fiber at $1$ is $LX$ equipped with the usual action of $B\mathbb{G}_a$.

If we want to define a construction that is related to the loop space Hodge filtration through the logarithm map
$$log:[\overset{\circ}{D}(1,1)/\Lambda]\to [\mathbb{A}^1/\mathbb{G}_m],$$
we need to replace $B\mathbb{G}_a$ by $BD^\circ:=B\overset{\circ}{D}(1,1)$,
and the multiplicative action of $\mathbb{A}^1$ on $B\mathbb{G}_a$ by the (partial) action of $D(0,1)$ on $BD^\circ$ through the power map
$$(1+x,a)\mapsto (1+x)^a.$$
We thus replace the derived loop space by the space
$$L^D X:=Hom_{dSt}(B\overset{\circ}{D}(1,1),X),$$
together with its (partial) action of $B\overset{\circ}{D}(1,1)$ given by the (partial) multiplication of $\overset{\circ}{D}(1,1)\subset \mathbb{G}_m$. There is actually a family of such actions parametrized by $D(0,1)$ through the (partial) map
$$D(0,1)\times B\overset{\circ}{D}(1,1)\times L^D X\to L^D X$$
given by $(a,d,\gamma)\mapsto \gamma\circ m_{d^a}$. This family of actions gives a space
$$Hod(L^D X)\to D(0,1)$$
whose fiber at $0$ is the trivial action of $B\overset{\circ}{D}(1,1)$ on $L^D X$
and whose fiber at $1$ is the standard action.

## Related notion

[[generalized higher loop spaces]]


generalized multicategory

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## Idea

An ordinary [[category]] consists of a set of objects and a set of arrows, each with a single input or domain object and a single output or codomain object. Arrows are composed by plugging outputs into inputs, as when one composes unary operations.

An ordinary [[multicategory]] consists of a set of objects and a set of "multi-arrows", each with a finite list of inputs or domain objects and a single output or codomain object. These multi-arrows are composed by means of multiple plug-ins or substitutions, as when one substitutes a list of $m$ operations of varying arities into an $m$-ary operation.

Both categories and multicategories can be seen as [[monads]] in an appropriate [[bicategory]] of [[span]]-like objects. Categories are monads in the bicategory of ordinary spans (of sets).  Multicategories are monads in a bicategory of spans of shape

$$X \leftarrow R \to Y^*$$

where $Y^*$ is the set of finite lists of elements of $Y$.  In order to compose spans of this type, one uses the fact that $(-)^*$ has itself the structure of a monad on $Set$, namely the [[list monad]] or "free monoid" monad.  The idea of a *generalized multicategory* is to replace this monad by some more general monad $T$.

There are a lot of ways of making this precise, ranging in abstractness from the fairly concrete (cartesian monads on a cartesian category) to the most general and abstract (monads on a [[virtual equipment]]).  We will start from the former and gradually generalize to the latter.

## Definitions

### Cartesian monads

Let $V$ be a [[finitely complete category]], and let $T$ be a [[cartesian monad]] on $V$, i.e. a monad whose functor part $T$ preserves pullbacks, and for which the naturality squares of its unit and multiplication are pullbacks.  Then there is a [[bicategory]] $Span(V,T)$ of **$T$-spans** in $V$, whose objects $X$ are objects of $V$, whose morphisms $X \to Y$ are spans from $X$ to $T Y$, and whose 2-cells are morphisms of such spans. The identity $X \to X$ is the span

$$X \overset{id}{\leftarrow} X \overset{\eta X}{\to} T X$$

where $\eta: 1_V \to T$ is the unit of $T$.  The composite of $T$-spans $R\colon X \to Y$, $S\colon Y \to Z$ is given by taking the following pullback:

$$\array{
& & & & R \circ S & & & &\\
& & & \swarrow & & \searrow & & & \\
& & R & & & & T S & &\\
& ^f \swarrow & & \searrow ^g & & ^{T h} \swarrow & & \searrow ^{T k} & \\
X & & & & T Y & & & & T T Z & \overset{\mu Z}{\rightarrow} & TZ}$$

A $T$**-multicategory** in $V$ is by definition a [[monad]] in the bicategory described above.  However, it is convenient to use the alternate word *monoid* rather than *monad* here, to avoid confusion with the monad $T$ which lives *on* $V$, and because the relevant morphisms of monoids (see below) are rather different from any of the usual sort of morphisms of monads.

If we spell out what this means more explicitly, a $T$-multicategory consists of an object $C_0$ of $V$, along with a span $C_0 \leftarrow C_1 \to T C_0$, and suitable identity-assigning and composition morphisms.

If $C_0$ is the [[terminal object]], then we speak instead of a $T$**-operad**.

#### Examples

* When $T$ is the "free monoid" monad on $Set$, a $T$-multicategory is precisely an ordinary multicategory.

* When $T$ is the identity monad on $V$, a $T$-multicategory is simply an [[internal category]] in $V$.

* When $T$ is the "free [[strict ∞-category]]" monad on [[globular sets]], a $T$-multicategory with $C_0 = 1$ is a [[globular operad]] in the sense of Batanin (this reformulation was found by Leinster).

* If $M$ is a monoid in $Set$ and $T = M\times -$ is the "free $M$-set" monad, then a $T$-multicategory is an "$M$-graded category."

* When $T$ is the [[free category]] monad on [[quivers]], a $T$-multicategory is a [[virtual double category]] (also called an *fc-multicategory*, coming from the name "fc" for this monad).

* If $A$ is a small category and $T$ is the monad on $Set^{ob(A)}$ whose algebras are functors $A\to Set$, then a $T$-multicategory can be identified with an object of $Cat/A$.

### Kleisli bicategories

Note that when $T$ is a cartesian monad on a finitely complete category $V$, then it extends to a (pseudo) [[2-monad]] on the bicategory $Span(V)$.  The functor part of this 2-monad is given simply by applying $T$; this is a (pseudo) [[2-functor]] since $T$ preserves pullbacks.  The unit of this 2-monad is given by the spans
$$ X \overset{id}{\leftarrow} X \overset{\eta X}{\to} T X$$
and its multiplication by
$$ T T X \overset{id}{\leftarrow} T T X \overset{\mu X}{\to} T X;$$
these are [[pseudonatural transformations]] since $\eta$ and $\mu$ are cartesian natural transformations.

Moreover, the bicategory $Span(V,T)$ defined above is easily seen to be precisely the [[Kleisli category|Kleisli bicategory]] of this extended 2-monad $T$ on $Span(V)$.  Thus, a more general notion of "generalized multicategory" would be a monad (monoid) in some Kleisli bicategory.

#### Examples

Are ordinary *symmetric* multicategories the $T$-multicategories relative to any monad $T$?  Recall that in a symmetric multicategory, the source of a morphism $f\colon (x_1,\dots,x_n) \to y$ is still an *ordered* list, but we have an action of the symmetric groups on the morphisms, in such a way that for $\sigma\in S_n$ we have $\sigma \cdot f \colon (x_{\sigma 1}, \dots, x_{\sigma n}) \to y$, and composition is equivariant for this action.

It doesn't work to let $T$ be the "free commutative monoid" monad, since that would destroy all ordering on the lists.  (Also, that monad is not cartesian.)  Really what we want is to let $T$ be the "free symmetric strict monoidal category" monad, since for a discrete set $A$ we have $T A = $ the set of finite ordered lists of elements of $A$, with isomorphisms imposed for all permutations.

However, of course this monad does not live on $Set$, but rather on $Cat$.  But there is a natural bicategory which extends $Cat$ analogously to how $Span$ extends $Set$, namely the bicategory $Prof$ of categories and [[profunctors]].  The "free symmetric strict monoidal category" monad $T$ does extend to a pseudomonad on $Prof$, and symmetric multicategories can be identified with monads $C_0 ⇸ T C_0$ in the Kleisli bicategory of this $T$ such that $C_0$ is a [[discrete category]].

### Double categories

While the bicategorical framework is very nice, it does not provide a good definition of the *functors* between generalized multicategories.  From the explicit point of view, a functor $f\colon C\to D$ between $T$-multicategories should consist of morphisms

$$
\array { C_0 & \leftarrow & C_1 & \to & T C_0 \\
  ^{f_0} \downarrow & & ^{f_1}\downarrow & & \downarrow^{T f_0}\\
  D_0 & \leftarrow & D_1 & \to & T D_0 }
$$

which respect the identity and structure maps.  This reduces to the natural notion of functor in all the examples mentioned above.  However, such morphisms cannot be identified with any of the usual notions of "morphisms of monads" in the bicategory $Span(V,T)$.  The closest thing there is would be a "colax morphism of monads," but that would allow $f_0$ to be an arbitrary *span* rather than a morphism in $V$.

We can remedy this problem if instead of the bicategory $Span(V,T)$, we consider a [[pseudo double category]] whose horizontal bicategory is $Span(V,T)$, and whose vertical arrows are the morphisms of $V$.  In particular, a square in this double category will be precisely a diagram of the above sort.  Thus, if we define a *monoid* in a double category to be the same as monad in its horizontal bicategory, we can also define a *monoid homomorphism* in a double category to consist of a vertical arrow and a square respecting the multiplication and identities, in such a way that monoid homomorphisms in $Span(V,T)$ are precisely the correct functors of $T$-multicategories mentioned above.

Thus, a better definition of $T$-multicategory is "a monoid in the double category $Span(V,T)$," since this definition recovers the correct morphisms immediately.  And a better generalized definition, which works for symmetric multicategories as well (using the double category of categories, functors, and profunctors), is "a monoid in a Kleisli double category."

### Equipments

When we try to make the notion of "Kleisli double category" precise, however, we run into some issues.  We want the "Kleisli-ness" to happen *horizontally*, i.e. in the span/profunctor direction.  However, in the examples, the monads in question live most naturally in in the 2-category of double categories, functors, and *vertical* natural transformations.  For example, when $T$ is a monad on $V$, its multiplication and unit transformations naturally induce vertical transformations $Id_{Span(V)} \to Span(T)$ and $Span(T)^2 \to Span(T)$, where $Span(T)$ denotes the induced functor on $Span(V)$.  Similarly, for the "free symmetric strict monoidal category" monad on the double category $Prof$, the unit and multiplication are naturally *functors*, i.e. vertical arrows, not profunctors.

However, in general, if all we know is that $T$ is a monad on a double category $X$ in the "vertical" sense, then there is no way to define a "horizontally Kleisli" double category of $T$.  The composite of horizontal arrows $A\to T B$ and $B \to T C$ in such a double category would have to be the composite $A\to T B \to T T C \overset{\mu}{\to} T C$, but in a double category there is no way to compose the horizontal arrow $A\to T T C$ with the vertical arrow $\mu\colon T T C \to T C$.  Specifically, the problem is that while any (pseudo) double functor between pseudo double categories induces a (pseudo) 2-functor between horizontal bicategories, a vertical transformation does not necessarily induce a pseudonatural one.

There are two solutions to this problem, and it turns out that the best approach is to use them both.  The first, and most obvious, is to generalize the way in which the monads in these two examples do, in fact, induce pseudomonads on the horizontal bicategory.  Namely, in both of the double categories $Span$ and $Prof$, every vertical arrow $f\colon A\to B$ gives rise to a horizontal arrow $B(1,f)\colon A\to B$ in a universal way.  This construction, along with its dual, makes $Span$ and $Prof$ into [[framed bicategories]], or equivalently [[proarrow equipments]].

Now in general, a vertical transformation between double categories that are proarrow equipments does not quite induce a *pseudo* natural transformation on horizonal bicategories, but only an *oplax* natural transformation.  However, in the examples we have considered so far, the oplax transformations are in fact pseudo, and so there is a horizontally-Kleisli double category (which is easily seen to be, itself, a proarrow equipment).

### Virtual double categories

From the double-category point of view, it seems unnatural to require that the unit and multiplication of the monad induce "horizontally pseudo" transformations; the vertical transformation of double categories is surely the more basic notion.  Moreover, it is also unnatural to require the functor $T$ to be pseudo; there are interesting examples where $T$ is only a (horizontally) *lax* functor.  Note that unlike the case for bicategories, there is a 2-category of double categories, lax functors, and vertical transformations, so we can talk about monads in such a 2-category.

However, at this level of generality, the horizontal Kleisli construction does not yield a double category.  It does, however, yield a [[virtual double category]].  Moreover, it also suffices to take as *input* a monad $T$ on a virtual double category $X$; we write the result of this construction as $HKl(X,T)$.  (Recall that for pseudo double categories regarded as virtual ones, functors of virtual double categories can be identified with *lax* functors of pseudo double categories.)  Thus a natural and even more general notion of generalized multicategory is "a monoid in the horizontal-Kleisli virtual double category for some monad $T$ on a virtual double category;" we call such a thing a **$T$-monoid**.

Note also that for any virtual double category $X$, there is another virtual double category $Mod(X)$ whose objects are monoids in $X$, whose vertical arrows are monoid homomorphisms, and whose horizontal arrows are "bimodules."  Therefore, for any monad $T$ on a virtual double category $X$, there is another virtual double category $KMod(X,T) = Mod(HKl(X,T))$ whose objects are $T$-monoids, and whose vertical arrows are $T$-monoid homomorphisms.

#### Examples

* If $V$ has pullbacks and $T$ is a monad preserving pullbacks (no condition on $\eta$ and $\mu$), then $T$ induces a monad on the virtual double category (which is actually, of course, a pseudo double category) $Span(V)$.  The $T$-monoids in $Span(V)$ are then precisely the $T$-multicategories, as defined above.

* The "free symmetric strict monoidal category" monad on $Prof$ can also be regarded as a monad on the virtual double category $Prof$, and its $T$-monoids on a discrete category are [[symmetric multicategories]].

* If we use instead the "free category with finite products" monad on the virtual double category $Prof$, then $T$-monoids on discrete categories can be identified with [[cartesian multicategories]], which include a sub-collection equivalent to that of [[Lawvere theories]].

* More basically, we also have a "free strict monoidal category" monad on $Prof$, whose $T$-monoids on discrete categories are ordinary non-symmetric multicategories.  Noting that $Prof = Mod(Span)$, this $T$ is also $Mod(S)$, where $S$ is the "free monoid" monad on $Span = Span(Set)$.  In fact, quite generally for any monad $S$ on a virtual double category $X$, we can identify $S$-monoids in $X$ with $Mod(S)$-monoids in $Mod(X)$ on "discrete objects."

* The [[ultrafilter monad]] $U$ on $Set$ has a "canonical" extension to the virtual double category $Rel$ of sets, functions, and binary relations.  Although $Rel$ is a pseudo double category and this monad is a strict functor, $\eta$ is only oplax.  The $U$-monoids in $Rel$ can be identified with [[topological spaces]], by an observation originally due to Barr.  In this context they are often referred to as [[relational beta-modules]], where $\beta$ is another name for $U$.

* Similarly, the powerset monad $P$ on $Set$ extends to $Rel$ in a canonical way, and the $P$-monoids in $Rel$ are [[closure space]]s.  Topological examples of this sort are sometimes called **$(T,V)$-algebras**, where $T$ denotes a Set-monad like $U$ or $P$, and $V$ a category (usually a [[quantale]]) of enrichment.

* There is a 2-monad on $Cat-Prof$ (whose objects are [[2-categories]]) whose generalized multicategories are [[2-categories with contravariance]].

### Units and transformations

...

### Equipments and normalization

...

## Obtaining the relevant monads

Each of the above contexts for generalized multicategories requires a monad of a certain sort.  Various technologies exist for producing examples of such monads, for instance:

* [[clubs]] and [[polynomial monads]] give ways to construct cartesian monads.
* [[relative pseudomonads]] give a way to construct pseudomonads on bicategories like $Prof$.
* [[double clubs]] and [[double polynomials]] give ways to construct monads on double categories.

## References

* [[Tom Leinster]], _Higher Operads, Higher Categories_, London Math. Soc. Lec. Note Series __298__, [math.CT/0305049](http://arxiv.org/abs/math.CT/0305049), [another link](http://www.maths.gla.ac.uk/~tl/book.html)

* {#CruttwellShulman10} [[Geoff Cruttwell]] and [[Mike Shulman]], _A unified framework for generalized multicategories_, Theory Appl. Categ. **24** 21 (2010) 580-655 [[arxiv/0907.2460](http://arxiv.org/abs/0907.2460), [TAC:24-21](http://www.tac.mta.ca/tac/volumes/24/21/24-21abs.html)]

* [[Tom Leinster]], _Generalized enrichment for categories and multicategories_,  [math.CT/9901139](http://arxiv.org/abs/math.CT/9901139)

* A blog post (of probably many; someone should find some others): [Generalized Operads in Classical Algebraic Topology](http://golem.ph.utexas.edu/category/2009/10/generalized_operads_in_classic.html)

* [[Martin Hyland]], _Elements of a theory of algebraic theories_, [arXiv:1311.7642](http://arxiv.org/abs/1311.7642).

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generalized polycategory

# Generalized polycategories

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## Idea

A *generalized polycategory* combines the notion of [[generalized multicategory]] with that of [[polycategory]] (or [[prop]]).  While a generalized multicategory is parametrized by a [[monad]] (on a [[double category]] or [[bicategory]] or [[virtual double category]]) that specifies the arities of domains of morphisms, a generalized polycategory is parametrized by *two* such monads, specifying the arities of domains and codomains, together with a suitable sort of [[distributive law]] between them that specifies the allowable composites.

The theory of generalized polycategories is very underdeveloped in the literature compared to generalized multicategories. A Burroni–Leinster style definition, using a double category of spans only, appears in [Koslowski 2005](#Koslowski), and is used to define planar (non-symmetric) polycategories.  A definition using profunctors is implicit in [Garner 2008](#Garner), and is used to define symmetric polycategories.  But no general framework, or other examples, appears to have been written down yet.

## References

A Burroni-style definition of generalized polycategory, called "D-categories" is defined in

* [[Elizabeth Burroni]]. _Algèbres non déterministiques et D-catégories_. Notebooks of Categorical Topology and Differential Geometry 14.4 (1973): 417-475. ([pdf](http://www.numdam.org/item/CTGDC_1973__14_4_417_0.pdf))

though has a weaker notion of associativity than might be expected. The notion of associativity is strengthened by

* {#Koslowski} Juergen Koslowski, *A monadic approach to polycategories*, [TAC](http://www.tac.mta.ca/tac/volumes/14/7/14-07abs.html) Vol. 14, 2005, No. 7, pp 125-156.

* {#Garner} [[Richard Garner]], *Polycategories via pseudo-distributive laws*, [arXiv](http://arxiv.org/abs/math/0606735), Adv. Math. 218 no. 3 (2008)

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generalized quantifier

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### Context
#### Model theory
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[[!include model theory - contents]]
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#### Linguistics
+-- {: .hide}
[[!include linguistics - contents]]
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# Contents
* table of contents
{:toc}

## Idea

**Generalized quantifiers** were introduced into [[model theory]] by the Polish logician A. Mostowski in order to supplement the usual [[existential quantifier|existential]] and [[universal quantifiers]] of [[first-order logic|first-order predicate logic]] with the aim to circumvent shortcomings concerning expressivity and categoricity.

Their subsequent use for the [[semantics|semantic]] analysis of natural language determiners in the footsteps of [[Richard Montague]] revolutionized [[linguistics]] in the early 1970s.

## Definition

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

A _generalized quantifier_ or an _interpretation of a quantifier symbol_ $Q$ is a mapping $\mu_Q$ from triples of cardinal numbers $\langle \mathfrak{m},\mathfrak{n},\mathfrak{p}\rangle$ such that $\mathfrak{m}+\mathfrak{n}=\mathfrak{p}$ to $\Omega=\{0,1\}$. The satisfaction relation for $Q$ is defined for structures $\mathfrak{A}$ with carrier $A$ and valuation $x$ by

$$
\begin{aligned}
\mathfrak{A}\models_x(Qv_n)\phi\quad&{iff}\quad \mu_Q(\mathfrak{m},\mathfrak{n},\mathfrak{p})=1\;,\; with
\\
\mathfrak{m}&=card(\{a\in A:\mathfrak{A}\models_{x(n/a)}\phi\})
\\
\mathfrak{n}&=card(\{a\in A:\mathfrak{A}\models_{x(n/a)}\not\phi\})
\\
\mathfrak{p}&=card(A)\;.
\end{aligned}
$$

=--

## Examples

* The [[universal quantifier]] $\forall$ has $\mu_\forall(\mathfrak{m},\mathfrak{n},\mathfrak{p})=1$ iff $\mathfrak{n}=0$.

* The quantifier $Q_\alpha$ "there exist at least $\aleph_\alpha$" is given by $\mu_{Q_\alpha}(\mathfrak{m},\mathfrak{n},\mathfrak{p})=1$ iff $\mathfrak{m}\geq\aleph_{\alpha}\;$.

* The quantifier $\mathsf{W}$ "most" is given by $\mu_{\mathsf{W}}(\mathfrak{m},\mathfrak{n},\mathfrak{p})=1$ iff $\mathfrak{m}\gt\mathfrak{n}\;$. Note that this uses the second variable as well, whereas $\forall$ or $Q_\alpha$ depend only on $card(\{a\in A:\mathfrak{A}\models_{x(n/a)}\phi\})$.

## Related entries

* [[model theory]]

* [[second-order logic]]

* [[cardinal number]]

* [[Lawvere distribution]]

* [[Richard Montague]]

## References

Generalized quantifiers were introduced in

* A. Mostowski, _On a Generalization of Quantifiers_ , Fund. Math. **44** (1957). ([pdf](http://matwbn.icm.edu.pl/ksiazki/fm/fm44/fm4412.pdf))

An early textbook account is in ch.13 of

* {#BellSlomson}[[John Bell|J. L. Bell]], A. B. Slomson, _Models and Ultraproducts: An Introduction_ , North-Holland Amsterdam 1969. (Dover reprint)

Several chapters treat their model-theoretic role in

* J. Barwise, [[Solomon Feferman|S. Feferman]] (eds.), _Model-theoretic Logics_ , Springer Heidelberg 1985 (freely available online: [toc](http://projecteuclid.org/euclid.pl/1235417263#toc)) .

In the context of [[Martin-Löf type theory]] they are discussed by

* G. Sundholm, _Constructive Generalized Quantifiers_ , Synthese **79** no.1 (1989) pp.1-12.

For the use of generalized quantifiers in natural language semantics see

* J. Barwise, R. Cooper, _Generalized Quantifiers and Natural Language_ , Linguistics and Philosophy **4** no.2 (1981) pp.159-219.

* E. Keenan, D. Paperno (eds.), _Handbook of Quantifiers in Natural Language _ , Springer 2012.

* E. Keenan, D. Paperno (eds.), _Handbook of Quantifiers in Natural Language: vol.2 _ , Springer 2017.

* E. Keenan, J. Stavi, _A semantic characterization of natural language determiners_ , Linguistics & Philosophy **9** (1986) pp.253–326.

* D. Westerståhl, _Generalized quantifiers: linguistics meets model theory_ , in Aloni, Dekker (eds.), _The Cambridge Handbook of Semantics_ , Cambridge UP 2014. ([draft](https://www.philosophy.su.se/polopoly_fs/1.165898.1391710202!/menu/standard/file/2013%20CambridgeHandbookGQ.pdf))

Some problems with the standard approach in linguistics are discussed in

* H. Ben-Yami, _Generalized Quantifiers, and Beyond_ , Logique Et Analyse no.208 (2009) pp.309-326.

* M. Hackl, _On the Grammar and Processing of Proportional Quantifiers: Most Versus More Than Half_ , Natural Language Semantics **17** no.1 (2009) pp.63-98.

A dependent type theoretic analysis of natural language generalized quantifiers is in

* T. Fernando, _Conservative generalized quantifiers and presupposition_ , Semantics and Linguistic Theory **XI** (2001) pp.172–91. ([draft](http://www.scss.tcd.ie/Tim.Fernando/cgq.pdf))

They are modelled in compact closed categories in

* J. Hedges, [[Mehrnoosh Sadrzadeh|M. Sadrzadeh]], _A Generalised Quantifier Theory of Natural Language in Categorical Compositional Distributional Semantics with Bialgebras_  , Math. Struc. Theor. Comp. Sci. 2019. ([link](https://www.cambridge.org/core/journals/mathematical-structures-in-computer-science/article/generalised-quantifier-theory-of-natural-language-in-categorical-compositional-distributional-semantics-with-bialgebras/9738AE3A40B9A51AE2134D13E4FD3C5B))

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