manifold-diagrammatic n-category

[[!redirects geometric n-category]]

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

#Contents#
* table of contents
{:toc}

## Idea

The notion of manifold-diagrammatic higher categories (in some [places](#geocats) also referred to as a 'geometric' or 'geometrical' higher categories) broadly refers to [[higher categories]] whose [[composition]] [[operation]] is modelled on the [[stratified space|stratified-geometric]] notion of [[manifold diagram|manifold diagrams]].

## Overview

Models of [[higher categories]] are usually defined as certain collections of [[k-morphisms]] that are parametrized by (that is, [[presheaf|presheafs]] on) some category of 'combinatorial shapes': examples include presheafs of [[globular set|globular]], [[simplicial set|simplicial]] or [[opetope|opetopic]] shapes. [[manifold diagram|Manifold diagrams]], while defined in geometric terms, admit a combinatorialization in terms of [[n-truss|trusses]] (or, more precisely, [[trusses#comb_mdiag|combinatorial manifold diagrams]]) and can therefore be used as combinatorial shapes for defining higher categories as presheafs on them. Importantly, combinatorial manifold diagrams come with a rich class of combinatorial mappings which (in contrast to most other classes of shapes) include internal representations of both 'embeddings' (='face and degeneracy' maps, up to geometric dualization) as well as 'quotients' (='subdivisions', up to geometric dualization). We give geometric insight into these classes below, but ultimately only care about their combinatorial definitions.

### Embeddings of manifold diagrams

#### Geometric idea

*Embeddings* of manifold $n$-diagrams are [[manifold diagram#framed_map_euclidean_case|framed]] [[stratified space|stratified maps]] whose underlying maps are [[subspace|embeddings]] of spaces (this is different from the notion of [[stratified space#stratified_map_types|substratifications]]). The next figure illustrates mappings in this class (in dimension $n = 2$).

Figure 1: Embeddings of manifold diagrams (for which substratifications and coarsenings form a factorization system).

Geometric dualization unveils that embeddings of manifold diagrams are precisely 'cellular' maps of their dual framed cell complexes ('cellular' is usually meant to imply that maps map closed cells to closed cells), which can be seen to have an [[(epi,mono) factorization system]] by degeneracy and face maps. The next figure illustrates this: the shown maps are precisely the the geometric duals of the maps in the previous figure.

Figure 2: Geometric duals of embeddings are cellular maps (for which face and degeneracy maps form a factorization system)

#### Combinatorial definition

Combinatorially, we define the following.

\begin{defn} \label{def_embeddings}
*Embeddings of combinatorial manifold diagrams* $T,S$ are described by [[spans]] of the form

\begin{center}
\begin{tikzcd}
        T & {T'} & S
        \arrow["{\mathsf{red}}"', from=1-2, to=1-1]
        \arrow[hook, from=1-2, to=1-3]
\end{tikzcd}
\end{center}

where $T' \xrightarrow{\mathsf{red}} T$ is a [[n-truss#reduction|reduction]] and $T' \hookrightarrow S$ is map of labeled trusses whose underlying map is [[n-truss#n-truss_bundle_map|regular]] and injective.
\end{defn}

Such spans can be composed by [[pullback]].

### Quotients of manifold diagrams

#### Geometric idea

*Quotients* of manifold diagrams are [[manifold diagram#framed_map_euclidean_case|framed]] stratified maps whose underlying maps are [[quotient space|quotient maps]]. The next figure illustrates mappings in this class (in dimension $n = 2$).

Figure 3: Quotients of manifold diagrams are the categorically dual notion to embeddings.

Geometric dualization unveils that quotients of manifold diagrams correspond to subdivisions of framed cell complexes. The next figure illustrates the geometric duals of the quotient maps in the preceding figure.

Figure 4: The geometric duals of quotients are subdivisions.

#### Combinatorial definition

Combinatorially, we define the following.

\begin{defn} \label{def_quotients}
*Quotients of combinatorial manifold diagrams* $T, S$ are maps of labeled trusses

$$
    T \twoheadrightarrow S
$$

whose underlying map of trusses is [[n-truss#n-truss_bundle_map|singular]] and surjective.
\end{defn}

\begin{rmk} To summarize: while the notion embeddings of manifold diagrams recovers (after geometric dualization) face and degeneracy maps often found in the combinatorics of shapes, in contrast, the notion of quotients dual to that of embeddings captures (after geometric dualization) something new: namely, subdivisions of cells. This is quite powerful e.g. for describing compositions as we will exploit below.(Moreover, it turns out that classifying subdivisions is generally a difficult (undecidable) problem, see e.g. [Dorn-Douglas 2021](#DornDouglas21), so it is convenient that it can be easily solved in the setting of manifold diagrams!) \end{rmk}

### Sketch Definition

We first organize embeddings and quotients (in the sense of Def. \ref{def_embeddings} resp. Def. \ref{def_quotients}) into categories as follows.

\begin{terminology}
Together, embeddings and quotients organize into the *[[double category]] $\mathbb{M}\mathsf{Diag}_n$ of combinatorial manifold $n$-diagrams*, with horizonal morphisms being embeddings, vertical morphisms being quotients, squares being commuting diagrams of the following form:

\begin{center}
    \begin{tikzcd}
        {T_1} & {T'_1} & {S_1} \\
        {T_2} & {T'_2} & {S_2}
        \arrow["{\mathsf{norm}}"', from=1-2, to=1-1]
        \arrow[hook, from=1-2, to=1-3]
        \arrow["{\mathsf{norm}}"', from=2-2, to=2-1]
        \arrow[hook, from=2-2, to=2-3]
        \arrow[two heads, from=1-1, to=2-1]
        \arrow[two heads, from=1-3, to=2-3]
        \arrow[dashed, two heads, from=1-2, to=2-2]
    \end{tikzcd}
\end{center}

(note that the dashed arrow is unique if it exists.) The *category of manifold $n$-diagrams* $\mathsf{MDiag}_n$ will refer to just the horizontal part of this double category.
\end{terminology}

When defining higher categories parametrized by manifold $n$-diagrams, we would want higher categories to be compatible both with embeddings ('morphism attachment') and quotients ('morphism composition') in the following sense.

{#SketchDefinition} ***Sketch definition***. A *manifold diagrammatic $n$-category* $\mathcal{C}$ should be a presheaf on $\mathsf{MDiag}_n$ subject to the following two conditions.

1. The collection of morphisms parametrized by a given diagram should be determined by subdiagrams that [[cover]] it. Formally, this can be enforced by an appropriate [[sheaf]] condition.
2. The composition of a given diagram should be compatible with the composition of its subdiagrams. Formally, this can be enforced by requiring that $\mathcal{C}$ extends to a double-(co)presheaf $\mathbb{M}\mathsf{Diag}^{\mathrm{op},\mathrm{co}}_n \to \mathbb{S}\mathbf{et}$ (covariant in the vertical part, where $\mathbb{S}\mathbf{et}$ is the double category of [[double category#examples|squares]] $\mathrm{Sq}(\mathbf{Set})$ in [[Set]])).

\begin{rmk} (Coherence conditions). What's worth pointing out about the above definition sketch is that there are no explicit higher-categorical coherence conditions (like [[Kan complex#KanComplexes|filler]] or [[Segal condition|Segal]] conditions) beyond ordinary sheaf conditions. This is because coherences can be expressed internally to the category of manifold diagrams (namely, as so-called 'manifold-diagram [[isotopies]]' including, for instance, a [[braiding]]).
\end{rmk}

{#mdiag_infty}
\begin{rmk} \label{mdiag_infty} (The $n = \infty$ case). So far we worked with $n$-diagrams for fixed finite $n \in \mathbb{N}$ which work well as parametrizing objects for $n$-categories. However, we can also address the case $n = \infty$ as follows. Given a combinatorial manifold $n$-diagram $T$, note its [[n-truss#cylinder|cylinder]] $\mathbb{I} \times T$ is a combinatorial manifold $(n+1)$-diagram. This defines a chain of inclusions of (ordinary resp. double) categories, the colimit of which is a category of manifold diagrams $\mathsf{MDiag}$ (resp. the double category $\mathbb{M}\mathsf{Diag}$) which may then be used to parametrized $(\infty,\infty)$-categories via the sketch definition above.
\end{rmk}

### A formal definition

To make the preceding sketch precise, one would need to formalize the sheaf condition. We now mention one way of how this could be achieved.

\begin{terminology} The category $\mathrm{MDiag}_n$ (and similarly, $\mathrm{MDiag}$ from Rmk. \ref{mdiag_infty}) contains [[wide subcategory|wide]] subcategories $\mathrm{MDiag}^L_n$ resp. $\mathrm{MDiag}^R_n$ consisting of spans
\begin{center}
\begin{tikzcd}
	T & S & S & {\mathrm{resp.}} & T & T & {S.}
	\arrow[equal, from=1-6, to=1-5]
	\arrow[from=1-6, to=1-7]
	\arrow[equal, from=1-2, to=1-3]
	\arrow["{\mathsf{norm}}"', from=1-2, to=1-1]
\end{tikzcd}
\end{center}
\end{terminology}

We define a [[coverage]] for $\mathrm{MDiag}^R_n$. (Note that $\mathrm{MDiag}^R_n$ does not have all pullbacks since, e.g., subdiagrams can intersect in non-diagrams.)

\begin{defn} The *neighborhood coverage* $J_n$ for $\mathrm{MDiag}^R_n$ is the coverage that assigns to $T \in \mathrm{MDiag}$ the single family $\{f_x : T^{\leq x} \to T \}_{x \in T}$ comprised of the [[n-truss#stratified_neighborhood|stratified neighborhoods]] of $T$.
\end{defn}

We may now formalize our earlier sketch definition as follows.

\begin{defn}
A *manifold-diagrammatic $n$-category* $C$ is a presheaf on $\mathrm{MDiag}_n$ such that:

1. $C$ is a [[sheaf]] on $(\mathrm{MDiag}^R_n,J_n)$ and [[locally constant sheaf|locally constant]] on (i.e. constant on the connected components of) $\mathrm{MDiag}^L_n$,
2. $C$ extends to a double-(co)presheaf $\mathbb{M}\mathsf{Diag}^{\mathrm{op},\mathrm{co}}_n \to \mathbb{S}\mathbf{et}$.

\end{defn}

(Note, however, that the practicality of this definition has not yet been established, and several variants are under consideration, cf. next section.)

### Variants

Instead of considering presheafs on combinatorial manifold diagrams, one may also consider

* presheafs on (unlabeled) [[n-truss|trusses]]
* presheafs on (unlabeled) [[n-truss##atomic_trusses_terminology|atomic trusses]]

with various sheaf conditions in place. In both cases, faces, degeneracies and compositions may be understood via mappings as outlined above. The potential advantage of dropping labelings from our parametrizing objects in this way (i.e. considering trusses in place of labeled trusses) is simplicity. For instance, an immediate simplification is that we can work with truss maps instead of spans of truss maps when combinatorially modelling embeddings. This could simplify the comparison to other presheaf models of higher categories.

Another related, potentially useful observation is that the spans occurring in the definition of the category of manifold diagrams may be expressed concisely in terms of bundles (slightly generalizing the notion of labeled [[trusses|truss bordism]]).

Several other approaches to and variations of the idea of manifold-diagrammatic higher categories are currently under consideration. The categories of combinatorial shapes considered here (whether combinatorial manifold diagrams, or (atomic) trusses, or similar) often have peculiarities that one doesn't encounter in other settings (e.g. failure to be a [[Reedy category]] due to the novel concept of reduction). The precise nature of these shape categories remains to be fully understood.

## Examples

* [[geometric computad|Geometric computads]] are [[free object|free]] instances of manifold-diagrammatic $n$-categories.

* [[associative n-category|Associative $n$-categories]] are, in an extended sense, instances of manifold diagrammatic $n$-categories (and, in fact, their combinatorics precedes some of the stratified-geometric ideas of manifold-diagrammatic $n$-categories).

\begin{rmk} (Globular sets in associative $n$-categories). Note that associative $n$-categories use [[globe|globular shapes]] as parametrizing objects rather than manifold diagrams. This can be related to the above discussion as follows. Using Rmk. \ref{mdiag_infty}, denote by $\mathsf{MDiag^{st}}$ the category of ('strictly globular') manifold diagrams with source/target inclusions as maps, and by $\mathsf{Glob}$ the [[globe category]]. Then there is a retraction $\mathsf{Glob} \hookrightarrow \mathsf{MDiag^{st}}$ (which maps $n$ as the 'terminal' manifold $n$-diagram, namely, the geometric dual of the $n$-globe) and $\mathsf{MDiag^{st}} \to \mathsf{Glob}$ (which maps an $n$-manifold diagram to the $n$-globe). This retraction can be used to '*work with [[globular sets]] whose composition is defined via manifold diagrams*'.
\end{rmk}

## Remarks

* {#coherence_paradigm} As mentioned above, [[coherence law|coherences]] in manifold-diagrammatic categories derive from isotopies of manifold diagrams. This provides a different mechanism for coherence construction than the mechanism found in many existing, spatially-inspired models of [[higher categories]] (which create coherences by enforcing contractibility of 'spaces of composites'). The conceptual difference between the two mechanisms is discussed in more detail in this [blog post](https://cxdorn.github.io/research/paradigms-of-higher-cats/).

* {#geocats} Manifold-diagrammatic categories have also been referred to 'geometric higher categories' or 'geometrical higher categories', based on a distinction of 'geometric', 'topological' and 'combinatorial' models of higher structures as explained [here](https://golem.ph.utexas.edu/category/2023/03/an_invitation_to_geometric_hig.html). See also [here](https://nforum.ncatlab.org/discussion/16159/geometric-ncategory/#Item_10) for a relevant discussion about terminology.

## Related concepts

* [[manifold diagram|manifold $n$-diagrams]]

* [[n-mesh|meshes]]

* [[n-truss|trusses]]

* [[directed topological space|directed topological spaces]]

* [[stratified space|stratified spaces]]

* [[higher category theory]]

## References

A condensed write-up of the mathematics needed for stating the above formal definition of manifold-diagrammatic higher categories was compiled in

* {#Zero23} [[Christoph Dorn]], _From zero to manifold-diagrammatric categories_, ([pdf](https://cxdorn.github.io/assets/pdfs/zero-to-mdhc.pdf))

and explained in

* From zero to manifold diagrams, TallCat Seminar Talk, ([video](https://www.youtube.com/watch?v=67RAzY-13r4))

A general introduction to some of the relevant ideas in the area can be found in

* {#NCC} [An Invitation to Geometric Higher Categories](https://golem.ph.utexas.edu/category/2023/03/an_invitation_to_geometric_hig.html), $n$-Category Café post,

with further details spelled out in

* {#Dorn23} [[Christoph Dorn]], _Nine short stories about geometric higher categories_, 2023 ([pdf](https://cxdorn.github.io/assets/pdfs/nine-stories.pdf)).

The type of 'directed combinatorial topology' that governs the stratified topology of manifold diagrams was described in

* {#DornDouglas21} [[Christoph Dorn]] and [[Christopher Douglas]], _Framed combinatorial topology_, 2021 ([arXiv](https://arxiv.org/abs/2112.14700), [latest](https://cxdorn.github.io/fct-book/))

[[!redirects manifold-diagrammatic n-categories]]
[[!redirects manifold-diagrammatic higher category]]
[[!redirects manifold-diagrammatic higher categories]]
[[!redirects manifold-diagrammatic higher category theory]]

[[!redirects geometric n-category]]
[[!redirects geometric n-categories]]
[[!redirects geometric higher category]]
[[!redirects geometric higher categories]]
[[!redirects geometric higher category theory]]
[[!redirects geometrical n-category]]
[[!redirects geometrical n-categories]]
[[!redirects geometrical higher category]]
[[!redirects geometrical higher categories]]
[[!redirects geometrical higher category theory]]

[[!redirects geometric category]]
[[!redirects geometrical category]]


manifolds and cobordisms - contents

**[[manifolds]]** and **[[cobordisms]]**

[[cobordism theory]], _[[Introduction to Cobordism and Complex Oriented Cohomology|Introduction]]_

**Definitions**

* [[locally Euclidean space]]

  * [[coordinate chart]], [[coordinate transformation]]

  * [[atlas]],

  * [[smooth structure]]

* [[manifold]]

  * [[topological manifold]]

  * [[differentiable manifold]], ,[[smooth manifold]]

* [[infinite dimensional manifold]]

  * [[Banach manifold]], [[Hilbert manifold]], [[ILH manifold]], [[Frechet manifold]], [[convenient manifold]]

* [[tangent bundle]]

* [[normal bundle]]

* [[G-structure]], [[torsion of a G-structure]]

  * [[orientation]], [[spin structure]], [[string structure]], [[fivebrane structure]]

* [[Cartan geometry]]:

  * [[Riemannian manifold]]

  * [[complex manifold]]

  * [[symplectic manifold]]

* [[cobordism]]

  * [[B-bordism]]

  * [[extended cobordism]]

  * [[cobordism category]]

  * [[(∞,n)-category of cobordisms]]

    * [[FQFT|functorial quantum field theory]]

  * [[Thom spectrum]]

    * [[cobordism ring]]

    * [[genus]]

**Genera and invariants**

* [[signature genus]], [[Kervaire invariant]]

* [[A-hat genus]], [[Witten genus]]

**Classification**

* [[2-manifolds]]/[[surfaces]]

  * [[genus of a surface]]

* [[3-manifolds]]

  * [[Kirby calculus]]

* [[4-manifolds]]

  * [[Dehn surgery]]

  * [[exotic smooth structure]]

**Theorems**

* [[Whitney embedding theorem]]

* [[Thom's transversality theorem]]

* [[Pontrjagin-Thom construction]]

* [[Galatius-Tillmann-Madsen-Weiss theorem]]

* [[geometrization conjecture]],

  * [[Poincaré conjecture]]

  * [[elliptization conjecture]]

* [[cobordism hypothesis]]-theorem


Mannque Rho

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

## Selected writings

On the [[bag model of quark confinement]] in relation to the [[Skyrmion model]]:

* {#BrownRho88} G. E. Brown, [[Mannque Rho]], _The Chiral Bag_, Comments Nucl. Part. Phys. 18 (1988) no.1, 1-29 ([spire:18025](http://inspirehep.net/record/18025))

Inclusion of further heavy [[flavor (particle physics)|flavors]], namle [[strange quark]]/[[kaons]], [[charm quarks]]/[[D-mesons]] and [[bottom quarks]]/[[B-mesons]], into the [[Skyrme model]]:

* [[Norberto Scoccola]], D. P. Min, H. Nadeau, [[Mannque Rho]], _The strangeness problem: An $SU(3)$ skyrmion with vector mesons_, Nuclear Physics A Volume 505, Issues 3–4, 25 December 1989, Pages 497-524 (doi:10.1016/0375-9474(89)90029-8)

* [[Mannque Rho]], D. O. Riska, [[Norberto Scoccola]], _Charmed baryons as soliton - D meson bound states_, Phys. Lett. B 251 (1990) 597-602 ([spire:297771](https://inspirehep.net/literature/297771), doi:10.1016/0370-2693(90)90802-D)

* [[Mannque Rho]], _Massive-quark baryons as Skyrmions_,  Modern Physics Letters A, Vol. 06, No. 23 (1991) ([doi:10.1142/S0217732391002268](https://doi.org/10.1142/S0217732391002268))

* Yongseok Oh, Dong-Pil Min, [[Mannque Rho]], [[Norberto Scoccola]], _Massive-quark baryons as skyrmions: Magnetic moments_, Nuclear Physics A
Volume 534, Issues 3–4 (1991) Pages 493-512 (doi:10.1016/0375-9474(91)90458-I)

* [[Mannque Rho]], D. O. Riska, [[Norberto Scoccola]], _The energy levels of the heavy flavour baryons in the topological soliton model_, Zeitschrift für Physik A Hadrons and Nuclei volume 341, pages343–352 (1992) ([doi:10.1007/BF01283544](https://doi.org/10.1007/BF01283544))

On the [[Cheshire cat principle]] for the [[bag model of quark confinement]] in [[hadrons]]:

* [[Mannque Rho]], _Cheshire Cat Hadrons_, Phys. Rept. 240 (1994) 1-142 ([arXiv:hep-ph/9310300](https://arxiv.org/abs/hep-ph/9310300), doi:10.1016/0370-1573(94)90002-7)

  updated as: _The Cheshire Cat Hadrons Revisited_ ([arXiv:hep-ph/0206003](https://arxiv.org/abs/hep-ph/0206003))

On [[effective field theory]] in [[nuclear physics]]:

* [[Mannque Rho]], _Lectures on Effective Field Theories for Nuclei, Nuclear Matter and Dense Matter_, 2002 ([arXiv:nucl-th/0202078](https://arxiv.org/abs/nucl-th/0202078), [cds:539674](https://cds.cern.ch/record/539674))

On [[nuclear physics]] via [[chiral perturbation theory]]/[[quantum hadrodynamics]]:

* [[Maciej Nowak]], [[Mannque Rho]], [[Ismail Zahed]], _[[Chiral Nuclear Dynamics]]_,  World Scientific 1996 ([doi:10.1142/1681](https://doi.org/10.1142/1681))

On [[Skyrmions]]:

* {#RhoZahed16} [[Mannque Rho]], [[Ismail Zahed]] (eds.) _The Multifaceted Skyrmion_, World Scientific 2016 ([doi:10.1142/9710](https://doi.org/10.1142/9710))

On [[skyrmion]] models for [[neutron stars]]:

* Xiang-Hai Liu, Yong-Liang Ma, [[Mannque Rho]], _Topology change and nuclear symmetry energy in compact-star matter_, Phys. Rev. C 99, 055808 (2019) ([arXiv:1811.10012](https://arxiv.org/abs/1811.10012))

On [[nuclear physics]] and [[quark-hadron duality]] at high density such as in [[neutron stars]]:

* [[Mannque Rho]], _Fractionalized Quasiparticles and the Hadron-Quark Duality in Dense Baryonic Matter_ ([arxiv:2004.09082](https://arxiv.org/abs/2004.09082))

On [[Skyrmions]] with [[vector mesons]] in relation to [[hidden local symmetry]]:

* [[Mannque Rho]], *Skyrmions and Fractional Quantum Hall Droplets Unified by Hidden Symmetries in Dense Matter* ([arXiv:2109.10059](https://arxiv.org/abs/2109.10059))

category: people


Manos Lydakis

* [website](http://www.math.uoc.gr/~mlydakis/)

## Selected writings

On a [[model structure for excisive functors]] on [[simplicial sets]] (hence a [[model structure for spectra]]):

* {#Lydakis98} [[Manos Lydakis]], _Simplicial functors and stable homotopy theory_ Preprint, 1998 (Hopf archive [pdf](http://hopf.math.purdue.edu/Lydakis/s_functors.pdf), [[LydakisSimplicialFunctors.pdf:file]])

category: people


Mans Henningson

* [webpage](http://fy.chalmers.se/~mans/)

## related $n$Lab entries

* [[super p-brane]]

* [[self-dual higher gauge theory]]

category:people


Manuel Amann

* [Institute page](https://www.uni-augsburg.de/de/fakultaet/mntf/math/prof/diff/team/dr-habil-manuel-amann/)

## Selected writings

On [[nilpotent group|nilpotent]] [[Lie groups]]:

* [[Manuel Amann]], pp. 4 of: *On quasi-isometric nilpotent Lie groups* [[arXiv:1710.04542](https://arxiv.org/abs/1710.04542)]

category: people


Manuel Bullejos

Manuel Bullejos is a Spanish mathematician based in the [Departamento de Álgebra](http://www.ugr.es/~algebra/) in Granada.

* [Home page](http://www.ugr.es/~bullejos/)

## Selected writings

On [[cat-n-groups]]:

* [[Manuel Bullejos]], [[Antonio M. Cegarra]], [[John W. Duskin]], *On cat$^n$ -groups and homotopy types*, J. Pure
Appl. Alg. **86** (1993) 135-154 \[doi:10.1016/0022-4049(93)90099-F]

On "classifying spaces" (ie: [[topological realization]] of [[simplicial nerves]]) of [[2-categories]]:

* [[Manuel Bullejos]], [[Antonio M. Cegarra]],  *On the geometry of 2-categories and their classifying spaces*, K-Theory **29** 3 (2003) 211-229  [[doi:10.1023/B:KTHE.0000006921.50151.00](http://dx.doi.org/10.1023/B:KTHE.0000006921.50151.00), [pdf](http://www.ugr.es/\%7Ebullejos/geometryampl.pdf)]

On [[non-abelian cohomology]] with [[coefficients]] in [[braided 2-groups]] and [[symmetric 2-groups]]:

* [[Manuel Bullejos]], [[Pilar Carrasco]], [[Antonio M. Cegarra]], *Cohomology with coefficients in symmetric cat-groups. An extension of Eilenberg–MacLane's classification theorem*, Mathematical Proceedings of the Cambridge Philosophical Society , Volume 114 , Issue 1 , July 1993 , pp. 163 - 189 ([doi:10.1017/S0305004100071498](https://doi.org/10.1017/S0305004100071498))

category:people

[[!redirects M. Bullejos]]


Manuel Bärenz

[[!redirects Manuel Baerenz]]
[[!redirects ManuelBaerenz]]
[[!redirects ManuelBärenz]]
## Personal ##
* Bachelor in Heidelberg
* Bachelor's Thesis with Derek Wise
* Master in Cambridge
* PhD in Nottingham with [[John Barrett]]
* [Website](http://manuelbaerenz.de/)

## Interests ##
* Quantum gravity
* [[higher category theory and physics]]
* [[state sum model]]s / [[extended topological quantum field theory]]
* Cartan geometry
* 4d TQFTs
* Type theory


Manuel Rivera

* [webpage](https://riveramanuel.com/)

## Selected writings

Identifying the [[cobar construction]] on [[singular homology|singular chains]] on a [[topological space]] with the dg-algebra of chains on its [[loop space]]:

* [[Manuel Rivera]], [[Mahmoud Zeinalian]], *Cubical rigidification, the cobar construction, and the based loop space*, Algebr. Geom. Topol. **18** (2018) 3789-3820 [[arXiv:1612.04801](https://arxiv.org/abs/1612.04801), [doi:10.2140/agt.2018.18.3789](https://doi.org/10.2140/agt.2018.18.3789)]

On [[infinity-local systems|$\infty$-local systems]] in the sense of [[(infinity,1)-vector bundles|$(\infty,1)$-vector bundles]] with [[flat infinity-connections|flat $\infty$-connections]]:

* [[Manuel Rivera]], [[Mahmoud Zeinalian]], §5 of: *The colimit of an $\infty$-local system as a twisted tensor product*, Higher Structures **4** 1 (2020) 33-56  [[arXiv:1805.01264](https://arxiv.org/abs/1805.01264), [higher-structures:Vol4Iss1](https://higher-structures.math.cas.cz/api/files/issues/Vol4Iss1/RiveraZeinalian)]

See also:

* M. Rivera,  and [[S. Saneblidze]], _A combinatorial model for the free loop space fibration_, Bull. London Math. Soc. (2018) doi:10.1112/blms.12202,
[arXiv:1712.02644](https://arxiv.org/abs/1706.00983)

* M. Rivera, F. Wierstra, [[M. Zeinalian]], _The functor of singular chains detects weak homotopy equivalences_,
Proc. Amer. Math. Soc. (2019) [arXiv:1808.10237](https://arxiv.org/abs/1808.10237)

* M. Rivera and Z. Wang, _Singular Hochschild cohomology and algebraic string operations_,  J. Noncommut. Geom. 13 (2019), 297-361.[arXiv:1703.03899](https://arxiv.org/abs/1703.03899)

On [[rational homotopy theory]] for possibly non-[[nilpotent spaces]] via [[deck transformation|deck]]-[[Borel-equivariant rational homotopy theory]]:

* [[Manuel Rivera]], [[Felix Wierstra]], [[Mahmoud Zeinalian]], *Rational homotopy equivalences and singular chains*, Algebr. Geom. Topol. **21** (2021) 1535-1552 [[arXiv:1906.03655](https://arxiv.org/abs/1906.03655). [doi:10.2140/agt.2021.21.1535](https://doi.org/10.2140/agt.2021.21.1535)]

Relating the [[Dwyer-Kan loop groupoid]]-construction to the [[homotopy coherent nerve]]-construction:

* {#MinichielloRiveraZeinalian23} [[Emilio Minichiello]], [[Manuel Rivera]], [[Mahmoud Zeinalian]], *Categorical models for path spaces*, Advances in Mathematics **415** (2023) 108898 [[arXiv:2201.03046](https://arxiv.org/abs/2201.03046), [doi:10.1016/j.aim.2023.108898](https://doi.org/10.1016/j.aim.2023.108898)]

category:people


map

# Contents
* table of contents
{: toc}

## Idea

In the original use of the term, a _map_ or _mapping_ is just a _[[function]]_ (between [[sets]]).  The term is often used for [[continuous functions]] between [[topological spaces]] and indeed also generally for any [[morphism]] in any [[category]].  Accordingly, for instance the term _[[mapping space]]_ is often used synonymously for _[[internal hom]]_.

However, in the [[category theory]] literature one should be aware of a different usage; see [[bicategory of maps]].

## Related concepts

* [[function]]
* [[homomorphism]]
* [[morphism]]

[[!redirects map]]
[[!redirects maps]]
[[!redirects mapping]]
[[!redirects mappings]]


map in a dagger 2-poset

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

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

## Definition ##

A [[morphism]] $f:hom_A(a,b)$ of a [[dagger 2-poset]] $A$ is a **map** if it is [[functional morphism in a dagger 2-poset|functional]] and [[entire morphism in a dagger 2-poset|entire]].

The [[set]] of all maps in $hom_A(a,b)$ is defined as

$$Map_A(a, b) \coloneqq \{f \in hom_A(a,b) \vert isFunctional(f) \wedge isEntire(f)\}$$

## See also ##

* [[functional morphism in a dagger 2-poset]]
* [[entire morphism in a dagger 2-poset]]
* [[monic map in a dagger 2-poset]]
* [[category of maps]]


mapping class group

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Manifolds and cobordisms
+--{: .hide}
[[!include manifolds and cobordisms - contents]]
=--
#### Group Theory
+-- {: .hide}
[[!include group theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

Given a (oriented) [[topological manifold]] $X$, its __mapping class group__ $MCG(X)$ is the [[group]] of [[isotopy]] classes of (orientation preserving) [[homeomorphisms]] $X\to X$.

Often this is considered specifically for $X$ a [[Riemann surface]] with punctures in which case
a central role is played by [[Dehn twists]].

The mapping class group is of importance in many areas of geometry including study of [[Teichmüller spaces]], of [[moduli spaces]] of surfaces, of [[automorphisms]] of [[free groups]] and in geometric and [[combinatorial group theory]], hyperbolic geometry and so on. Some of the key contributors were [[Max Dehn]], [[Jakob Nielsen]], [[William Thurston]], [[David Mumford]]. Recent proof of the related Mumford conjecture has been accomplished by Madsen and Weiss.

## Definition

For $\mathbf{Aut}(X)$ the [[automorphism group]] of the [[manifold]] formed in [[Euclidean-topological infinity-groupoids|Euclidean topological geometry]], hence equipped with its canonical structure of a [[topological group]]. Let furthermore $\mathbf{Aut}_0(X) \hookrightarrow \mathbf{Aut}(X)$ be the inclusion of the connected component of the identity.

Then

$$
  \mathbf{MCG}(X) \coloneqq \mathbf{Aut}(X)/\mathbf{Aut}_0(X)
$$

is the corresponding [[coset space]]/[[quotient group]]. In other words, the mapping class group is the group of [[homeomorphism]] of $X$ onto itself, modulo [[isotopy]].

This is a [[discrete group]]. Equivalently it is the group of [[connected components]] of $\mathbf{Aut}(X)$. If $X$ is a [[smooth manifold]], then the mapping class group is the group of connected components of the [[diffeomorphism group]]

$$
  MCG(X) = \pi_0(Diff(X))
 \,.
$$

If $X$ is a manifold with boundary $\partial X$, then it is usual to consider automorphisms which restrict to the identity on the boundary.

## Properties

### For 2-dimensional surfaces

The mapping class group for 2-dimensional manifolds controls the [[moduli stack of complex curves]].

The [[classifying spaces]] of mapping class groups for 2-[[dimension]]al [[manifolds]] may also be encoded combinatorially in the [[geometric realization]] of a [[category]] of [[ribbon graphs]]. See there for details.

One of the classical results is that the (oriented) mapping class group of the [[torus]] $\mathbb{R}^2/\mathbb{Z}^2 \cong (S^1)^2$ is isomorphic to the [[special linear group]] [[SL(2,Z)|$SL_2(\mathbb{Z})$]], and generally $MCG(\mathbb{R}^n/\mathbb{Z}^n) \cong SL_n(\mathbb{Z})$).

Certain generators called [[Dehn twists]] may be visualized as cutting a torus along a circle $\{a\} \times S^1$ (or $S^1 \times \{b\}$), thus producing a cylinder, then twisting one of the ends of the cylinder through $2\pi$ and reattaching the two ends.

Another example is a 2-disk with $n$ punctures. The group of diffeomorphisms (fixing the boundary pointwise) modulo isotopy is the [[braid group]] $B_n$.

The relation to the [[homotopy type]] of the [[diffeomorphism group]] is as follows:

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

For $\Sigma$ a [[closed manifold|closed]] [[orientation|orientable]] [[surface]], then the [[fundamental ∞-groupoid|bare homotopy type]] of its diffeomorphism group is

1. if $\Sigma$ is the [[sphere]] then

   $$
     \begin{aligned}
        \Pi(Diff(S^2)) & \simeq \Pi(O(3))
        \\
        & \simeq MCG(S^2)\times \Pi(SO(3))
        \\
        & \simeq \mathbb{Z}_2 \times \Pi(SO(3))
     \end{aligned}
   $$

1. if $\Sigma$ is the [[torus]] then

   $$
     \begin{aligned}
       \Pi(Diff(S^1 \times S^1))
       & \simeq
       MCG(S^1 \times S^1)\times \Pi(S^1 \times S^1 )
       \\
       & \simeq GL_2(\mathbb{Z}) \times B(\mathbb{Z} \times\mathbb{Z})
     \end{aligned}
   $$

1. in all other cases all higher [[homotopy groups]] vanish:

   $$
     \Pi(Diff(\Sigma)) \simeq MCG(\Sigma)
   $$

=--

The first statement is due to ([Smale 58](#Smale58)), see also at _[[sphere eversion]]_. The second and third are due to ([Earle-Eells 67](#EarleEells67), [Gramain 73](#Gramain73)).

See ([Hatcher 12](#Hatcher12)) for review.

### Rational cohomology

The [[ordinary cohomology]] with rational coefficients of the [[delooping]] of the stable mapping class group of 2-dimensional manifolds (hence essentially the [[orbifold cohomology]] of the [[moduli stack of complex curves]]) is the content of [[Mumford's conjecture]], proven in ([Madsen-Weiss 02](#MadsenWeiss02)).

## Related concepts

* [[diffeomorphism group]]

* [[Alexander's trick]]

* [[braid group]]

## References

### General

Monographs:

* {#Birman75} [[Joan S. Birman]], _Braids, links, and mapping class groups_, Princeton Univ Press (1975) [[ISBN:9780691081496](https://press.princeton.edu/books/paperback/9780691081496/braids-links-and-mapping-class-groups-am-82-volume-82), [preview pdf](https://api.pageplace.de/preview/DT0400.9781400881420_A26691398/preview-9781400881420_A26691398.pdf)]

Surveys:

* Massyuyeau, _A short introduction to mapping class groups_ ([pdf](http://www-irma.u-strasbg.fr/~massuyea/talks/MCG.pdf))

* Benson Farb, Dan Margalit, _A primer on mapping class
groups_, draft, [web]()

* Nikolai V. Ivanov. Mapping class groups. In Handbook of geometric
topology, pages 523–633. North-Holland, Amsterdam, 2002; 1998 draft: [ps](http://www.mth.msu.edu/~ivanov/m99.ps)

* {#BakalovKirillov} [[Bojko Bakalov]], [[Alexander Kirillov]], chapter 5 of _Lectures on tensor categories and modular functor_ ([web](http://www.math.sunysb.edu/~kirillov/tensor/tensor.html), [[BakalovKirillovChapter5.pdf:file]])

See also:

* Wikipedia: [mapping class group](http://en.wikipedia.org/wiki/Mapping_class_group)

In relation to [[motion groupoids]]:

* [[Fiona Torzewska]], [[João Faria Martins]], [[Paul Purdon Martin]], *Motion groupoids and mapping class groupoids*, Comm. Math. Phys. **402** (2023) 1621-1705 [[arXiv:2103.10377](https://arxiv.org/abs/2103.10377), [doi:10.1007/s00220-023-04755-0](https://doi.org/10.1007/s00220-023-04755-0)]

### Braid groups

Understanding of [[braid groups]] as mapping class group of punctured [[surfaces]] with [[boundary]]:

* [[Wilhelm Magnus]], *Über Automorphismen von Fundamentalgruppen berandeter Flächen*, Mathematische Annalen **109** (1934) 617–646 [[doi:10.1007/BF01449158](https://doi.org/10.1007/BF01449158)]

* {#MargalitWinarski21} [[Dan Margalit]], [[Rebecca R. Winarski]], *Braid groups and mapping class groups: The Birman–Hilden theory*, Bull. London Math. Soc. **53** 3 (2021) 643-659 [[arXiv:1703.03448](https://arxiv.org/abs/1703.03448), [doi:10.1112/blms.12456](https://doi.org/10.1112/blms.12456)]

Algebraic descriptions of such mapping class groups/[[braid groups]]:

* [[Warren Dicks]], Edward Formanek, *Algebraic Mapping-Class Groups of Orientable Surfaces with Boundaries*, in: *Infinite Groups: Geometric, Combinatorial and Dynamical Aspects*, Progress in Mathematics **248** Birkhäuser (2005) [[doi:10.1007/3-7643-7447-0_4](https://doi.org/10.1007/3-7643-7447-0_4)]

See also:

* {#MadsenWeiss02} [[Ib Madsen]], [[Michael Weiss]], _The stable moduli space of Riemann surfaces: [[Mumford's conjecture]]_,  Ann. of Math. (2) __165__ (2007), no. 3, 843--941, [MR2009b:14051](http://www.ams.org/mathscinet-getitem?mr=2009b:14051), [doi](http://dx.doi.org/10.4007/annals.2007.165.843), [math.AT/0212321](http://arxiv.org/abs/math.AT/0212321)

* John Harer, _The second homology group of the mapping class group of an orientable surface_, Invent. Math., **72**  2  221-239 (1983) _The cohomology of the moduli space of curves_ in: Theory of moduli (Montecatini Terme, 1985), Lecture Notes in Math. __1337__, p. 138–221. Springer, Berlin, 1988.

* Robert C. Penner, _The decorated Teichmüller space of punctured surfaces_, Commun. Math. Phys. __113__ (2) (1987) 299--339. [MR89h:32044](http://www.ams.org/mathscinet-getitem?mr=89h:32044); _A construction of pseudo-Anosov homeomorphisms_, Trans. Amer. Math. Soc. __310__ (1):179–197, 1988.

* {#EarleEells67} C.J. Earle,  J. Eells, _The diffeomorphism group of a compact Riemann surface,
Bulletin of the American Mathematical Society 73(4) 557--559, 1967

* [[Allen Hatcher]], [[William Thurston]], _A presentation for the mapping class group of a closed orientable surface_, Topology, 19(3):221–237, 1980.

* {#Hatcher12} [[Allen Hatcher]], _A 50-Year View of Diffeomorphism Groups_, talk at the 50th Cornell Topology Festival in May 2012 ([pdf](http://www.math.cornell.edu/~hatcher/Papers/Diff%28M%292012.pdf))

* [[Max Dehn]], Papers on group theory and topology. Springer-Verlag, New York, 1987. Transl. from German with intro. and appendix by John Stillwell, and appendix by [[Otto Schreier]].

* [[Jakob Nielsen]], _Untersuchungen zur Topologie der geschlossenen zweiseitigen Flächen. I-III_, Acta Math. 50 (1927), no. 1, 189--358, MR1555256, [doi](http://dx.doi.org/10.1007/BF02547775); Acta Math. 53 (1929), no. 1, 1--76, MR1555290, [doi](http://dx.doi.org/10.1007/BF02547566);  Acta Math. 58 (1932), no. 1, 87--167, MR1555345, [doi](http://dx.doi.org/10.1007/BF02547775)

* [[David Mumford]], _Abelian quotients of the Teichmüller modular group_, J. Analyse Math., 18:227–244, 1967.
* [[David Mumford]], _Towards an enumerative geometry of the moduli space of curves_, Arithmetic and geometry, Vol. II, Birkhäuser Boston, Boston, MA, 1983, pp. 271–328, [MR85j:14046](http://www.ams.org/mathscinet-getitem?mr=85j:14046)

* [[Ulrike Tillmann]], _On the homotopy of the stable mapping class group_,  Invent. Math. __130__ (1997), no. 2, 257--275, [MR99k:57036](http://www.ams.org/mathscinet-getitem?mr=99k:57036), [doi](http://dx.doi.org/10.1007/BF02421324)

* S. Morita, _Introduction to mapping class groups of surfaces and related groups_, in: Handbook of Teichmüller theory (A. Papadopoulos, editor), vol. I, EMS Publishing House, Zürich, 2007, 353–386.

* E. Y. Miller, _The homology of the mapping class group_, J. Differential Geom. __24__ (1986), no. 1, 1–14, [MR88b:32051](http://www.ams.org/mathscinet-getitem?mr=88b:32051), [euclid](http://projecteuclid.org/euclid.jdg/1214440254)

[[!redirects mapping class groups]]


mapping cocone

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Limits and colimits
+--{: .hide}
[[!include infinity-limits - contents]]
=--
#### Homotopy theory
+--{: .hide}
[[!include homotopy - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The _mapping cocone_ of a morphism $f : A \to C$, with $C$ a [[pointed object]], is a particular [[model category]] realization of the [[homotopy fiber]] of $f$, i.e. of the [[homotopy pullback]]

$$
  \array{
    Cocone(f) &\to& A
    \\
    \downarrow && \downarrow^{\mathrlap{f}}
    \\
    {*} &\to& C
  }
$$

of the point along $f$.

The concept is [[duality|dual]] to that of [[mapping cone]].

Mapping co-cones are realized in terms of [[mapping cocylinders]], as described below.  These typically play the role of [[generalized universal bundle]]s.

## Definition

Under sufficiently nice conditions (...spell out...), the [[homotopy pullback]]

$$
  \array{
    Cocone(f) &\to& A
    \\
    \downarrow && \downarrow^{\mathrlap{f}}
    \\
    {*} &\to& C
  }
$$

may be computed as the ordinary [[limit]] over the diagram

$$
  \array{
    && && A
    \\
    &&&& \downarrow^{\mathrlap{f}}
    \\
    &&C^I &\to& C
    \\
    && \downarrow
    \\
    {*} &\to& C
  }
  \,,
$$

where $C^I$ is a [[path space object]] of $C$. This, in turn, may be computed as two consecutive ordinary [[pullback]]s. The first one

$$
  \array{
    && \mathbf{E}_f C &\to& A
    \\
    && \downarrow && \downarrow^{\mathrlap{f}}
    \\
    &&C^I &\to& C
    \\
    && \downarrow
    \\
    {*} &\to& C
  }
$$

yields the [[mapping cocylinder]] $\mathbf{E}_f C$. This is the kind of object discussed at [[generalized universal bundle]]. The second pullback then produces the mapping cocone

$$
  \array{
    Cocone(f)  &\to& \mathbf{E}_f C &\to& A
    \\
    \downarrow && \downarrow && \downarrow^{\mathrlap{f}}
    \\
     &&C^I &\to& C
    \\
    \downarrow && \downarrow
    \\
    {*} &\to& C
  }
  \,.
$$

## Examples

### Principal bundles

In the category [[Top]] of [[topological space]]s, let $G$ be group and $\mathcal{B}G$ its [[classifying space]], and let $f : X \to \mathcal{B}G$ be a morphism. Then

* the mapping cocylinder of ${* }\to \mathcal{B}G$ is the universal $G$-bundle;

* the mapping cocone of $X \to \mathcal{B}G$ is the $G$-principal bundle $P \to X$ classified by $f$:

  $$
    \array{
      P  &\to& \mathcal{E} G &\to& {*}
      \\
      \downarrow && \downarrow && \downarrow^{}
      \\
       &&(\mathcal{B}G)^I &\to& \mathcal{B}G
      \\
      \downarrow && \downarrow
      \\
      X &\stackrel{f}{\to}& \mathcal{B}G
    }
    \,.
  $$

## Note on terminology {#Terminology}

Whitehead complained about the term _cocone_ back in the old days, because of the seeming (though false) double dualization, so he used _mapping path space_. This practice was followed by his school (in most of US for example). But he himself was not confident in that terminology. For example there is a table in his book where he lists the dual notions and at the place where mapping cocone/mapping path space should fit he puts just the symbol for the construction while on the dual side he puts the whole name. Similarily for the mapping cocylinder.

Somebody -- maybe [[Samuel Eilenberg]] -- (-- check --) suggested to Whitehead to use _ne_ insteaad of _cocone_ , jokingly cancelling one _co_ against the other, as if both expressed abstract [[duality]].

Postnikov uses the term _mapping cocylinder_ , while for on Whitehead's complaint he comments:

> we do not see a particular criminal in the cocone terminology, but will anyway not use it.

## Related concepts

[[!include universal constructions of topological spaces -- table]]

[[!redirects mapping cocones]]


mapping complex

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

#Contents#
* table of content
{:toc}

## Idea

_Mapping complexes_ of [[dg-categories]] are the direct analogue of [[mapping spaces]] of [[simplicial categories]].

## Related concepts

* [[function complex]], [[simplicial mapping complex]]

* [[mapping space]]

* [[internal hom]]

[[!redirects mapping complexes]]


mapping cone

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Limits and colimits
+--{: .hide}
[[!include infinity-limits - contents]]
=--
#### Homotopy
+--{: .hide}
[[!include homotopy - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The _mapping cone_ of a [[morphism]] $f : X \to Y$ in some [[homotopical category]] (precisely: a [[category of cofibrant objects]]) is, if it exists, a particular representative of the [[homotopy fiber|homotopy cofiber]] of $f$.

It is also called the _homotopy [[cokernel]]_ of $f$ or the _[[weak quotient]]_ of $Y$ by the [[image]] of $X$ in $Y$ under $f$.

The dual notion is that of [[mapping cocone]].

<img src="http://ncatlab.org/nlab/files/mappingcone.jpg" width="660" >

(graphics taken from [Muro 2010](#Muro10))

## Definition
 {#Definition}

The mapping cone construction is a means to _present_ in a [[category with weak equivalences]] the following canonical
construction in [[homotopy theory]]/[[(∞,1)-category theory]].

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

In an [[(∞,1)-category]] $\mathcal{C}$ with [[terminal object]] and [[(∞,1)-pushout]], the [[homotopy fiber|homotopy cofiber]] of a [[morphism]] $f : X \to Y$ is the [[homotopy pushout]]

$$
  coker(f) \coloneqq Y \coprod_X {*}
$$

hence the object [[universal construction]] sitting universally in a [[diagram]] of the form

$$
  \array{
     X &\stackrel{}{\to}& {*}
     \\
     \downarrow^{\mathrlap{f}} &\swArrow_{\simeq}& \downarrow
     \\
     Y &\to& coker(f)
  }
  \,.
$$

=--

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

If the [[(∞,1)-category]] $\mathcal{C}$ is presented by (is [[equivalence of (infinity,1)-categories|equivalent]] to the [[simplicial localization]] of) a [[category of cofibrant objects]] $C$ (for instance given by the [[cofibrant objects]] in a [[model category]]) then this homotopy cofiber is presented by the ordinary [[colimit]]

$$
  \array{
    && X &\stackrel{f}{\to}& Y
    \\
    && \downarrow^{\mathrlap{i_1}} && \downarrow^{\mathrlap{i}}
    \\
    X &\stackrel{i_0}{\to}& cyl(X)
    \\
    \downarrow && &\searrow & \downarrow
    \\
    {*} &\to& &\to& cone(f)
  }
$$

in $C$ using any [[cylinder object]] $cyl(X)$ for $X$.

=--

This is discussed in detail at [[factorization lemma]] and at [[homotopy pullback]].

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

Intuitively this says that $cone(f)$ is the object obtained by

1. forming the cylinder over $X$;

1. gluing to one end of that the object $Y$ as specified by the map $f$.

1. shrinking the other end of the cylinder to the point.

Intuitively it is clear that this way every [[cycle]] in $Y$ that happens to be in the image of $X$ can be "continuously" translated in the cylinder-direction, keeping it constant in $Y$, to the other end of the cylinder, where it becomes the point. This means that every [[homotopy group]] of $Y$ in the image of $f$ vanishes in the mapping cone. Hence in the mapping cone **the image of $X$ under $f$ in $Y$ is removed up to homotopy**. This makes it clear how $cone(f)$ is a homotopy-version of the [[cokernel]] of $f$. And therefore the name "mapping cone".

=--

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

A morphism $\eta : cyl(X) \to Y$ out of a [[cylinder object]] is a [[left homotopy]] $\eta : g \Rightarrow h$ between its restrictions $g\coloneqq \eta(0)$ and $h \coloneqq \eta(1)$ to the cylinder boundaries

$$
  \array{
     X
     \\
     \downarrow^{\mathrlap{i_0}} & \searrow^{\mathrlap{g}}
     \\
     cyl(X) &\stackrel{\eta}{\to}& Y
     \\
     \uparrow^{\mathrlap{i_1}} & \nearrow_{\mathrlap{h}}
     \\
     X
  }
  \,.
$$

Therefore prop. \ref{HomotopyCofiberByFactorizationLemma} says that
the mapping cone is the the [[universal property|universal]] object with a morphism $i$ from $Y$ and a [[left homotopy]] from $i \circ f$ to the [[zero morphism]]. This is of course also precisely what def. \ref{InfinityCokernel} is saying.

=--

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

The colimit in prop. \ref{HomotopyCofiberByFactorizationLemma} may be computed in two stages by two consecutive [[pushouts]] in $C$, and in two ways by the following [[pasting diagram]]:

$$
  \array{
    && X &\stackrel{f}{\to}& Y
    \\
    && \downarrow^{i_1} && \downarrow
    \\
    X &\stackrel{i_0}{\to}& cyl(X) &\to & cyl(f)
    \\
    \downarrow && \downarrow && \downarrow
    \\
    {*} &\to& cone(X) &\to& cone(f)
  }
  \,.
$$

Here every square is a [[pushout]], (and so by the [[pasting law]] is every rectangular pasting composite).

=--

This now is a basic fact in ordinary [[category theory]]. The pushouts appearing here go by the following names:

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

The pushout

$$
  \array{
     X &\stackrel{i_0}{\to}& cyl(X)
     \\
     \downarrow && \downarrow
     \\
     {*} &\to& cone(X)
  }
$$

defines the **[[cone]]** $cone(X)$ over $X$ (with respect to the chosen [[cylinder object]]): the result of taking the [[cylinder object|cylinder]] over $X$ and identifying one $X$-shaped end with the [[point]].

The pushout

$$
  \array{
    X &\stackrel{f}{\to}& Y
    \\
    \downarrow && \downarrow
    \\
    cyl(X) &\to& cyl(f)
  }
$$

defines the **[[mapping cylinder]]** $cyl(f)$ of $f$, the result of identifying one end of the cylinder over $X$ with $Y$, using $f$ as the gluing map.

The pushout

$$
  \array{
    cyl(X) &\to& cyl(f)
    \\
    \downarrow && \downarrow
    \\
    cone(X) &\to& cone(f)
  }
$$

defines the **mapping cone** $cone(f)$ of $f$: the result of forming the cylinder over $X$ and then identifying one end with the point and the other with $Y$, via $f$.

=--

The geometric intuition behind this is best seen in the archetypical example of the [[classical model structure on topological spaces]]. See the example _[For topological spaces](#ForTopologicalSpaces)_ below. The example _[For chain complexes](InChainComplexes)_ can be understood similarly geometrically by thinking of all chain complexes as [[singular chain complex|singular chains]] on topological spaces.

## Examples

We discuss realizations of the general construction in various contexts.
Some of these examples are regarded in parts of the literature as the default examples, notably that [for topological spaces](#ForTopologicalSpaces) and that [for chain complexes](#InChainComplexes).

### Suspension

The mapping cone of the morphism $X \to {*}$ to the [[terminal object]] is the [[suspension object]] $\Sigma X$ of an object $X$. The dual notion of the [[loop space object]] of $X$.

### For topological spaces
 {#ForTopologicalSpaces}

The notion _mapping cone_  derives its name from its geometrical
interpretation in the category [[Top]] of [[topological spaces]].

For more details see also at _[[topological cofiber sequence]]_.

With respect to the standard [[model structure on topological spaces]] every [[CW-complex]] is a cofibrant object, and hence mapping cones on maps between CW-complexes have intrinsic meaning in [[homotopy theory]].

Write $I \coloneqq [0,1] \subset \mathbb{R} \in $ [[Top]] for the [[closed interval]] with its [[Euclidean space|Euclidean]] [[metric topology]]. This is an [[interval object]] for the standard model structure. We may therefore take the [[cylinder object]] of a topological space $X$ to be

$$
  cyl(X) \coloneqq X \times I
  \,,
$$

which is literally the cylinder over $X$.

Given a [[continuous function]] $f:X\to Y$, the
[[topological space]] $cone(f)$ is

$$
  cone(f) =  (X \times I) \cup_{f} Y
$$

This is the [[disjoint union]] of $X \times I$ with $Y$
followed by an identification under which for each
$x\in X$ a point $(x,1) \in X \times I$ is identified with
the point $f(x) \in Y$ and followed by the contraction of $X\times \{0\}$ to a point.

Of course the opposite convention is also possible: identify $(x,0)$ with $f(x)$ for all $x$ and then contract $X\times\{1\}$ to a point; the two constructions of cones are canonically homeomorphic; the first is sometimes called the "inverse mapping cone".

The [[singular chain complex]] [[functor]] from [[Top]] to the
[[category of chain complexes]] of [[abelian groups]] sends the mapping cone to a mapping cone in the sense of chain complexes (up to conventions on the orientation of the interval and vector order in the definition of mapping cone of chain complexes).

### For chain complexes
 {#InChainComplexes}

Let $Ch_\bullet = Ch_\bullet(R Mod)$ be the [[category of chain complexes]] in $R$[[Mod]] for some [[ring]] $R$.

(For instance if $R = \mathbb{Z}$ the [[integers]], then this is $Ch_\bullet(Ab)$, chain complexes of [[abelian groups]]. More generally $R Mod$ can be replaced by any [[abelian category]] in the following, with the evident changes in  the presentation here and there.)

We derive an explicit presentation of the mapping cone $cone(f)$ of a [[chain map]] $f$, according to the general definition \ref{CylindersAndCones}. The end result is prop. \ref{ComponentsOfMappingConeInChainComplexes} below,
reproducing the classical formula for the mapping cone.

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

Write $*_\bullet \in Ch_\bullet(\mathcal{A})$ for the chain complex
concentrated on $R$ in degree 0

$$
  *_\bullet 0
   =
  [\cdots \to 0 \to 0 \to R]
  \,.
$$

=--

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

This may be understood as the [[normalized chain complex]] of [[simplicial homology|chains of simplices]] on the terminal [[simplicial set]] $\Delta^0$, the 0-[[simplex]].

=--

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

Let $I_\bullet \in Ch_{\bullet}(\mathcal{A})$ be given by

$$
  I_\bullet
  =
  (\cdots 0 \to 0 \to R \stackrel{(-id,id)}{\to} R \oplus R)
  \,.
$$

Denote by

$$
  i_0 : *_\bullet \to I_\bullet
$$

the [[chain map]] which in degree 0 is the canonical inclusion into the second summand of a [[direct sum]] and by

$$
  i_1 : *_\bullet \to I_\bullet
$$

correspondingly the canonical inclusion into the first summand.

=--

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

This is the standard [[interval object in chain complexes]].

It is in fact the [[normalized chain complex]] of [[chains on a simplicial set]] for the canonical simplicial interval, the 1-[[simplex]]:

$$
  I_\bullet = C_\bullet(\Delta[1])
  \,.
$$

The [[differential]] $\partial^I = (-id, id)$ here expresses the [[alternating face map complex]] [[boundary]] operator, which in terms of the three non-degenerate [[basis]] elements is given by

$$
  \partial ( 0 \to 1 ) = (1) - (0)
  \,.
$$

=--

We decompose the proof of this statement is a sequence of substatements.

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

For $X_\bullet \in Ch_\bullet$ the
[[tensor product of chain complexes]]

$$
  (I \otimes X)_\bullet
  \in
  Ch_\bullet
$$

is a [[cylinder object]] of $X_\bullet$
for the structure of a [[category of cofibrant objects]] on $Ch_\bullet$
whose cofibrations are the [[monomorphisms]] and whose weak equivalences are the [[quasi-isomorphisms]] (the substructure of the standard [[injective model structure on chain complexes]]).

=--

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

The complex $(I \otimes X)_\bullet$ has components

$$
  (I \otimes X)_n
  =
  X_n \oplus X_n \oplus X_{n-1}
$$

and the [[differential]] is given by

$$
  \array{
    X_{n+1} \oplus X_{n+1} &\stackrel{\partial^X \oplus \partial^X}{\to}& X_n \oplus X_n
    \\
    \oplus &\nearrow_{(-id,id)}& \oplus
    \\
    X_{n} &\underset{-\partial^X}{\to}& X_{n-1}
  }
  \,,
$$

hence in [[matrix calculus]] by

$$
  \partial^{I \otimes X}
   =
  \left(
    \array{
      \partial^X \oplus \partial^X & (-id, id)
      \\
      0 & -\partial^X
    }
  \right)
  \;\;\colon\;\;
  (X_{n+1} \oplus X_{n+1}) \oplus X_{n}
  \to
  (X_{n} \oplus X_{n}) \oplus X_{n-1}
  \,.
$$

=--

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

By the formula discussed at [[tensor product of chain complexes]]
the components arise as the [[direct sum]]

$$
  (I \otimes X )_n
  =
  (R_{(0)} \otimes X_n )
  \oplus
  (R_{(1)} \otimes X_n )
  \oplus
  (R_{(0 \to 1)} \otimes X_{(n-1)} )
$$

and the [[differential]] picks up a sign when passed past the
degree-1 term $R_{(0 \to 1)}$:

$$
  \begin{aligned}
    \partial^{I \otimes X}
    (
      (0 \to 1), x
    )
    &=
    ( (\partial^I (0 \to 1)), x )
    -
    ( (0\to 1), \partial^X x )
    \\
    & =
    ( - (0) + (1), x )
    -
    ( (0 \to 1), \partial^X x )
    \\
    & = -((0), x) + ((1), x) -  ( (0 \to 1), \partial^X x )
  \end{aligned}
  \,.
$$

=--

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

The two boundary inclusions of $X_\bullet$ into the cylinder
are given in terms of def. \ref{BoundaryInclusionIntoChainComplexIntervalObject} by

$$
  i^X_0
   :
  X_\bullet \simeq *_\bullet \otimes X_\bullet
   \stackrel{i_0 \otimes id_X}{\to} (I\otimes X)_\bullet
$$

and

$$
  i^X_1
   :
  X_\bullet \simeq *_\bullet \otimes X_\bullet
   \stackrel{i_1 \otimes id_X}{\to} (I\otimes X)_\bullet
$$

which in components is the inclusion of the second or first
direct summand,
respectively

$$
  X_n \hookrightarrow X_n \oplus X_n \oplus X_{n-1}
  \,.
$$

=--

One part of definition \ref{CylindersAndCones} now reads:

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

For $f_\bullet : X_\bullet \to Y_\bullet $ a [[chain map]],
the [[mapping cylinder]] $cyl(f)$ is the [[pushout]]

$$
  \array{
    cyl(f)_\bullet &\leftarrow& Y_\bullet
    \\
    \uparrow && \uparrow^{\mathrlap{f}}
    \\
    I_\bullet \otimes X_\bullet  &\stackrel{i_0}{\leftarrow}& X_\bullet
  }
  \,.
$$

=--

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

The components of $cyl(f)_\bullet$ are

$$
  cyl(f)_n = X_n \oplus Y_n \oplus X_{n-1}
$$

and the [[differential]] is given by

$$
  \array{
    X_{n+1} \oplus Y_{n+1} &\stackrel{\partial^X \oplus \partial^Y}{\to}& X_n \oplus Y_n
    \\
    \oplus &\nearrow_{(-id,f)}& \oplus
    \\
    X_{n} &\underset{-\partial^X}{\to}& X_{n-1}
  }
  \,,
$$

hence in [[matrix calculus]] by

$$
  \partial^{cyl(f)}
   =
  \left(
    \array{
      \partial^X \oplus \partial^Y & (-id, f_n)
      \\
      0 & -\partial^X
    }
  \right)
  :
  (X_{n+1} \oplus Y_{n+1}) \oplus X_{n}
  \to
  (X_{n} \oplus Y_{n}) \oplus X_{n-1}
  \,.
$$

=--

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

The [[colimits]] in a [[category of chain complexes]] $Ch_\bullet(\mathcal{A})$ are computed in the underlying
[[presheaf category]] of [[towers]] in $\mathcal{A}$.
There they are computed
degreewise in $\mathcal{A}$
(see at [limits in presheaf categories](limits+and+colimits+by+example#limitsinpresheafcat)).
Here the statement is evident:

the pushout identifies one [[direct sum|direct summand]] $X_n$
with $Y_n$ along $f_n$ and so where previously a $id_{X_n}$
appeared on the diagonl, there is now $f_n$.

=--

The last part of definition \ref{CylindersAndCones} now reads:

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

For $f_\bullet : X_\bullet \to Y_\bullet $ a [[chain map]],
the **mapping cone** $cone(f)$ is the [[pushout]]

$$
  \array{
    cone(f) &\leftarrow& cyl(f)
    \\
    \uparrow && \uparrow
    \\
    cone(X) &\leftarrow& X \otimes I
    \\
    \uparrow && \uparrow^{\mathrlap{i_1}}
    \\
    0 &\leftarrow& X
  }
$$

=--

In the literature this appears for instance as ([Schapira, def. 3.2.2](#Schapira)).

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

The components of the mapping cone $cone(f)$ are

$$
  cone(f)_n =   Y_n \oplus X_{n-1}
$$

with differential given by

$$
  \array{
    Y_{n+1} &\stackrel{\partial^Y}{\to}& Y_n
    \\
    \oplus &\nearrow_{f_n}& \oplus
    \\
    X_{n} &\underset{-\partial^X}{\to}& X_{n-1}
  }
  \,,
$$

and hence in [[matrix calculus]] by

$$
  \partial^{cone(f)}
   =
  \left(
    \array{
      \partial^Y_{n+1} & f_n
      \\
      0 & -\partial^X_n
    }
  \right)
  :
  Y_{n+1} \oplus X_{n}
  \to
  Y_{n} \oplus X_{n-1}
  \,.
$$

=--

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

As before the pushout is computed degreewise.
This identifies the remaining unshifted copy of $X$ with 0.

=--

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

For $f : X_\bullet \to Y_\bullet$ a [[chain map]],
the canonical inclusion $i : Y_\bullet \to cone(f)_\bullet$
of $Y_\bullet$ into the  mapping cone of $f$ is given in
components

$$
 i_n : Y_n \to cone(f)_n = Y_n \oplus X_{n-1}
$$

by the canonical inclusion of a summand into a [[direct sum]].

=--

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

This follows by starting with
remark \ref{InclusionOfChainComplexIntoItsCylinder} and then
following these inclusions through the formation of the two colimits
as discussed above.

=--

The construction above builds the mapping cone explicitly via the standard formula for [[homotopy pushouts]]. Often however other presentations are more convenient:

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

For $f_\bullet \colon X_\bullet \to Y_\bullet$ a [[chain map]],
consider the [[double complex]] $D_{\bullet,\bullet}$ concentrated
in degrees $D_{1,\bullet} \coloneqq X_\bullet$ and
$D_{0,\bullet} \coloneqq Y_\bullet$ with
$\partial_{0,\bullet} \coloneqq f_\bullet \colon D_{1,\bullet} \to D_{0,\bullet}$.

Then the [[total complex]] of $D_{\bullet, \bullet}$ is also
a model for the mapping cone of $f$:

$$
 Cone(f) \simeq tot(D_{\bullet,\bullet})
  \,.
$$

=--

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

One checks by inspection that
$tot(D_{\bullet,\bullet}) = Cone(\tilde f)$
for $\tilde f\colon X_\bullet \to Y_\bullet$
for which there is a [[chain homotopy]] $f \Rightarrow f'$
(given only by multiplication by signs).

=--

This appears for instance as ([Weibel, Exercise 1.2.8](#Weibel)).

### For cochain complexes
 {#InCochainComplexes}

We spell out the situation in more detail in a [[category of cochain complexes]].

Let $\mathcal{A}$ be some [[concrete category|concrete]] [[additive category]] and $Ch^\bullet(\mathcal{A})$ the [[category of chain complexes]] in $\mathcal{A}$. For

$$
  f : V^\bullet \to W^{\bullet}
$$

a morphism, the mapping cone is the complex

$$
  \begin{aligned}
   Cone(f)
    & \coloneqq
  (\cdots \to Cone(f)^{k-1} \stackrel{d_{Cone(f)}}{\to} Cone(f)^k) \to \cdots)
   \\
   & \coloneqq
  \left(
    \array{
      \cdots \to & V^k &\stackrel{- d_V}{\to}& V^{k+1} & \to \cdots
      \\
      & \oplus &\searrow^{f^k}& \oplus
      \\
      \cdots \to & W^{k-1} &\underset{d_W}{\to}& W^k & \to \cdots
    }
  \right)
  \end{aligned}
  \,.
$$

There is a canonical co[[chain homotopy]]

$$
  \array{
    Cone(f) &\leftarrow& 0
    \\
    {}^{\mathllap{i}}\uparrow &\swArrow_{\eta}& \uparrow
    \\
    W &\stackrel{f}{\leftarrow}& V
  }
$$

where $i : W \to Cone(f)$ is the canonical inclusion, componentwise given by

$$
  i^k : W^k \stackrel{(0,Id)}{\to} V_{k+1} \oplus W^k
$$

and where the cochain homotopy $\eta$ has components

$$
  \eta^k : V^k \stackrel{(Id,0)}{\to} Cone(f)^{k-1} = V^k \oplus W^{k-1}
$$

which we denote on $v \in V^k$ by

$$
  \eta : v \mapsto (f(v))[1]
  \,.
$$

The fact that this is a cochain homotopy means that

$$
  [d,\eta] = i \circ f - 0
  \,,
$$

which we check on any $v \in V^k$ by computing

$$
  \begin{aligned}
    [d, \eta](v) &=
    d_{Cone(f)} \circ \eta (v)
    + \eta(d_V v)
    \\
    & = d_{Cone(f)} (f(v)[1]) + (f(d_V v))[1]
    \\
    & =
    \left(
      f(v)
      -
      (d_{W}f(v))[1]
    \right)
    +
    (d_W (f(v)))[1]
    \\
    & = f(v)
  \end{aligned}
  \,,
$$

where we used the above definition of $d_{Cone(f)}$ and the fact that $f$ is a chain homomorphism and hence intertwines the differentials.

This cochain homotopy is universal in that
for any other cochain homotopy

$$
  \array{
    X &\leftarrow& 0
    \\
    {}^{\mathllap{j}}\uparrow &\swArrow_{\rho}& \uparrow
    \\
    W &\stackrel{f}{\leftarrow}& V
  }
$$

hence

$$
  j \circ f = [d,\rho]
$$

we have a morphism

$$
  (j,\rho) :  Cone(f) \to X
$$

given on $W$ by $j$ and on $V$ by $\rho$

$$
  (j,\rho)^k : V^{k+1} \oplus W^k \stackrel{(\rho, j)}{\to} X^k
$$

which is indeed a cochain homomorphism because for all $v + w \in Cone(f)$ we have

$$
  \begin{aligned}
    d_X (j,\rho)(v + w)
     & =
    d_X (\rho v) +  d_X (j(w))
     \\
     & = [d,\rho](v) - \rho d_V v + j (d_W w)
     \\
     & = j f (v) - \rho d_V v + j (d_{Cone(f)} w)
     \\
     & = (j,\rho) d_{Cone(f)}(v + w)
  \end{aligned}
$$

and which is unique with the property that [[whiskering]] of [[2-morphism]]s gives

$$
  \array{
    X &\leftarrow& 0
    \\
    {}^{\mathllap{j}}\uparrow &\swArrow_{\rho}& \uparrow
    \\
    W &\stackrel{f}{\leftarrow}& V
  }
  \;\;\;\;\;
   =
  \;\;\;\;\;
  \array{
    X
    \\
    & \nwarrow^{\mathrlap{(j,\rho)}}
    \\
    &&  Cone(f) &\leftarrow& 0
    \\
    && {}^{\mathllap{i}}\uparrow &\swArrow_{\eta}& \uparrow
    \\
    && W &\stackrel{f}{\leftarrow}& V
  }
$$

hence that

$$
  j = (j,\rho) \circ i
$$

and

$$
  \rho = (j,\rho) \circ \eta
  \,.
$$

### In additive categories with translation

Let $\mathcal{A}$ be an [[additive category]] [[category with translation|with translation]] $T=[1] : \mathcal{A} \to \mathcal{A}$. Let $X$ and $Y$ be two [[differential objects]] in $(\mathcal{A},T)$ and $f : X \to Y$ any [[morphism]] in $C$.

The **mapping cone** $Cone(f)$ of $f$ is the [[differential object]] whose underlying object is the [[direct sum]] $T X \oplus Y$ and whose differential $d_{cone f} : T X \oplus X \to T T X \oplus T X$ is given in [[matrix calculus]] notation by

$$
  d_{cone f} :=
  \left(
   \array{
    d_{T X} & 0
    \\
    T(f) & d_Y
   }
  \right)
  =
  \left(
   \array{
    - T(d_X) & 0
    \\
    T(f) & d_Y
   }
  \right)
  \,.
$$

Notice the minus sign here, coming from the definition of a [[differential object|shifted differential object]].

## Properties

### Homology exact sequences and fiber sequences
 {#HomologyExactSequencesAndFiberSequences}

We discuss the relation between mapping cones
in [[categories of chain complexes]], as [above](#InChainComplexes),
and  [[long exact sequences in homology]]. For an exposition of the following see there the section _[Relation to homotopy fiber sequences](long+exact+sequence+in+homology#RelationToHomotopyFiberSequences)_.

Let $f : X_\bullet \longrightarrow Y_\bullet$ be a [[chain map]]
and write $cone(f) \in Ch_\bullet(\mathcal{A})$
for its mapping cone
as explicitly given in prop. \ref{ComponentsOfMappingConeInChainComplexes}.

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

Write $X[1]_\bullet \in Ch_\bullet(\mathcal{A})$
for the [[suspension of a chain complex]] of $X$. Write

$$
  p : cone(f) \to X[1]_\bullet
$$

for the [[chain map]] which in components

$$
  p_n : cone(f)_n \to X[1]_n
$$

is given, via prop. \ref{ComponentsOfMappingConeInChainComplexes},
by the canonical projection out of a direct sum

$$
  p_n : Y_\n \oplus X_{n-1} \to X_{n-1}
  \,.
$$

=--

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

The chain map $p : cone(f)_\bullet \to X[1]_\bullet$
represents the [[homotopy cofiber]] of the
canonical map
$i : Y_\bullet \to cone(f)_\bullet$.

=--

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

By prop. \ref{InclusionIntoChainComplexCone} and
def. \ref{ProjectionOutOfChainComplexMappingCone} the sequence

$$
  Y_\bullet
    \stackrel{i}{\to}
  cone(f)_\bullet
    \stackrel{p}{\to}
  X[1]_\bullet
$$

is a [[short exact sequence]] of chain complexes (since it is so degreewise, in fact degreewise it is even a [[split exact sequence]]). In particular we have a [[cofiber]] [[pushout]] diagram

$$
  \array{
     Y_\bullet &\stackrel{i}{\hookrightarrow}& cone(f)_\bullet
     \\
     \downarrow && \downarrow
     \\
     0 &\to& X[1]_\bullet
  }
  \,.
$$

Now, in the [[injective model structure on chain complexes]] all chain complxes are [[cofibrant objects]] and an inclusion such as $i : Y_\bullet \hookrightarrow cone(f)_\bullet$ is a [[cofibration]]. By the detailed discussion at [[homotopy limit]] this means that the ordinary colimit here is in fact a [[homotopy colimit]], hence exhibits $p$ as the [[homotopy cofiber]] of $i$.

=--

+-- {: .num_cor #HomotopyCofiberSequenceOfChainMap}
###### Corollary

For $f_\bullet : X_\bullet \to Y_\bullet$ a [[chain map]],
there is a **[[homotopy cofiber sequence]]** of the form

$$
  X_\bullet
    \stackrel{f_\bullet}{\to}
  Y_\bullet
    \stackrel{i_\bullet}{\to}
  cone(f)_\bullet
    \stackrel{p_\bullet}{\to}
  X[1]_\bullet
    \stackrel{f[1]_\bullet}{\to}
  Y_\bullet
    \stackrel{i[1]_\bullet}{\to}
  cone(f)_\bullet
    \stackrel{p[1]_\bullet}{\to}
  X[2]_\bullet
    \to
  \cdots
$$

=--

In order to compare this to the discussion of [[nLab:connecting homomorphisms]], we now turn attention to the case that $f_\bullet$ happens to be a [[monomorphism]]. Notice that this we can always assume, up to [[quasi-isomorphism]], for instance by prolonging $f$ by the map into its [[mapping cylinder]]

$$
  X_\bullet \to Y_\bullet \stackrel{\simeq}{\to} cyl(f)
  \,.
$$

By the axioms on an [[abelian category]] in this case we have a [[short exact sequence]]

$$
  0 \to X_\bullet \stackrel{f_\bullet}{\to} Y_\bullet \stackrel{p_\bullet}{\to} Z_\bullet \to 0
$$

of chain complexes. The following discussion revolves around the fact that now $cone(f)_\bullet$ as well as $Z_\bullet$ are both models for the homotopy cofiber of $f$.

+-- {: .num_lemma #ConeInjectionEquivalentToZigzag}
###### Lemma

Let
$$
  X_\bullet \stackrel{f_\bullet}{\to} Y_\bullet \stackrel{p_\bullet}{\to} Z_\bullet
$$

be a [[short exact sequence]] of [[chain complexes]].

The collection of linear maps

$$
  h_n : Y_n \oplus X_{n-1} \to Y_n \stackrel{}{\to} Z_n
$$

constitutes a [[chain map]]

$$
  h_\bullet : cone(f)_\bullet \to Z_\bullet
  \,.
$$

This is a [[quasi-isomorphism]]. The inverse of $H_n(h_\bullet)$ is given by sending a representing [[cycle]] $z \in Z_n$ to

$$
  (\hat z_n, \partial^Y \hat z_n) \in Y_n \oplus X_{n+1}
  \,,
$$

where $\hat z_n$ is any choice of lift through $p_n$ and where $\partial^Y \hat z_n$ is the formula expressing the [[connecting homomorphism]] in terms of elements, as discussed at [Connecting homomorphism -- In terms of elements](connecting%20homomorphism#OnHomologyInTermsOfElements).

Finally, the morphism $i_\bullet : Y_\bullet \to cone(f)_\bullet$ is eqivalent in the [[homotopy category]] (the [[derived category]]) to the [[zigzag]]

$$
  \array{
    && cone(f)_\bullet
    \\
    && \downarrow^{\mathrlap{h}}_{\mathrlap{\simeq}}
    \\
    Y_\bullet &\to& Z_\bullet
  }
  \,.
$$

=--

In the literature this appears for instance as ([Schapira, cor. 7.2.2](#Schapira)).

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

To see that $h_\bullet$ defines a chain map recall the differential $\partial^{cone(f)}$ from prop. \ref{ComponentsOfMappingConeInChainComplexes}, which acts by

$$
  \partial^{cone(f)} (x_{n-1}, \hat z_n)
  =
  (  -\partial^X x_{n-1} , \partial^Y \hat z_n + x_{n-1} )
$$

and use that $x_{n-1}$ is in the [[kernel]] of $p_n$ by exactness, hence

$$
  \begin{aligned}
    h_{n-1}\partial^{cone(f)}(x_{n-1}, \hat z_n)
    &=
    h_{n-1}( -\partial^X x_{n-1}, \partial^Y \hat z_n + x_{n-1}  )
    \\
    & = p_{n-1}( \partial^Y \hat z_n + x_{n-1})
    \\
    & = p_{n-1}( \partial^Y \hat z_n )
    \\
    & = \partial^Z p_n \hat z_n
    \\
    & = \partial^Z h_n(x_{n-1}, \hat z_n)
  \end{aligned}
  \,.
$$

It is immediate to see that we have a [[commuting diagram]] of the form

$$
  \array{
    && cone(f)_\bullet
    \\
    & {}^{\mathllap{i_\bullet}}\nearrow& \downarrow^{\mathrlap{h}}_{\mathrlap{\simeq}}
    \\
    Y_\bullet &\to& Z_\bullet
  }
$$

since the composite morphism is the inclusion of $Y$ followed by the bottom morphism on $Y$.

Abstractly, this already implies that $cone(f)_\bullet \to Z_\bullet$ is a [[quasi-isomorphism]],
for this diagram gives a morphism of [[cocones]] under the diagram defining $cone(f)$ in prop. \ref{HomotopyCofiberByFactorizationLemma} and by the above both of these cocones are [[homotopy colimit|homotopy-colimiting]].

But in checking the claimed inverse of the induced map on homology groups, we verify this also explicity:

We first determine those cycles $(x_{n-1}, y_n) \in cone(f)_n$ which lift a cycle $z_n$. By lemma \ref{HomotopyCofiberByFactorizationLemma}
a lift of chains is any pair of the form $(x_{n-1}, \hat z_n)$ where $\hat z_n$ is a lift of $z_n$ through $Y_n \to X_n$. So $x_{n-1}$ has to be found such that this pair is a cycle.
By prop. \ref{ComponentsOfMappingConeInChainComplexes} the differential acts on it by

$$
  \partial^{cone(f)} (x_{n-1}, \hat z_n)
  =
  (  -\partial^X x_{n-1} , \partial^Y \hat z_n + x_{n-1} )
$$

and so the condition is that

$x_{n-1} \coloneqq -\partial^Y \hat z_n$ (which implies $\partial^X x_{n-1} = -\partial^X \partial^Y \hat z_n = -\partial^Y \partial^Y \hat z_n = 0$ due to the fact that $f_n$ is assumed to be an inclusion, hence that $\partial^X$ is the restriction of $\partial^Y$ to elements in $X_n$).

This condition clearly has a unique solution for every lift $\hat z_n$ and a lift $\hat z_n$ always exists since $p_n : Y_n \to Z_n$ is surjective, by assumption that we have a [[short exact sequence]] of chain complexes. This shows that $H_n(h_\bullet)$ is surjective.

To see that it is also injective we need to show that
if a [[cycle]] $(-\partial^Y \hat z_n, \hat z_n) \in cone(f)_n$ maps to a cycle $z_n = p_n(\hat z_n)$ that is trivial in $H_n(Z)$ in that there is $c_{n+1}$ with $\partial^Z c_{n+1} = z_n$, then also the original cycle was trivial in homology, in that there is $(x_n, y_{n+1})$ with

$$
  \partial^{cone(f)}(x_n, y_{n+1})
  \coloneqq
  (-\partial^X x_n, \partial^Y y_{n+1} + x_n)
  =
  (-\partial^Y \hat z_n, \hat z_n)
  \,.
$$

For that let $\hat c_{n+1} \in Y_{n+1}$ be a lift of $c_{n+1}$ through $p_n$, which exists again by surjectivity of $p_{n+1}$. Observe that

$$
  p_{n}( \hat z_n -  \partial^Y \hat c_{n+1})
  =
  z_n -\partial^Z ( p_n \hat c_{n+1} )
  =
  z_n - \partial^Z ( c_{n+1} )
  =
  0
$$

by assumption on $z_n$ and $c_{n+1}$,
and hence that $\hat z_n - \partial^Y \hat c_{n+1}$ is in $X_n$ by exactness.

Hence
$(z_n - \partial^Y \hat c_{n+1}, \hat c_{n+1}) \in cone(f)_n$
trivializes the given cocycle:

$$
  \begin{aligned}
    \partial^{cone(f)}( \hat z_n - \partial^Y \hat c_{n+1}  , \hat c_{n+1})
    & =
   (-\partial^X(\hat z_n - \partial^Y \hat c_{n+1} ), \partial^Y \hat c_{n+1} + (\hat z_n - \partial^Y \hat c_{n+1} ) )
    \\
    & = (-\partial^Y(\hat z_n - \partial^Y \hat c_{n+1}),  \hat z_n )
    \\
    & =
    ( -\partial^Y \hat z_n, \hat z_n )
  \end{aligned}
  \,.
$$

=--

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

Let
$$
  X_\bullet \stackrel{f_\bullet}{\to} Y_\bullet \to Z_\bullet
$$

be a [[short exact sequence]] of [[chain complexes]].

Then the  [[chain homology]] functor

$$
  H_n(-) : Ch_\bullet(\mathcal{A}) \to \mathcal{A}
$$

sends the homotopy cofiber sequence of $f$, cor. \ref{HomotopyCofiberSequenceOfChainMap}, to the
[[long exact sequence in homology]] induced by the given short
exact sequence, hence to

$$
  H_n(X_\bullet)
  \to
  H_n(Y_\bullet)
   \to
  H_n(Z_\bullet)
   \stackrel{\delta}{\to}
  H_{n-1}(X_\bullet)
   \to
  H_{n-1}(Y_\bullet)
   \to
  H_{n-1}(Z_\bullet)
   \stackrel{\delta}{\to}
  H_{n-2}(X_\bullet)
   \to \cdots
  \,,
$$

where $\delta_n$ is the $n$th [[connecting homomorphism]].

=--

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

By lemma \ref{ConeInjectionEquivalentToZigzag} the
homotopy cofiber sequence is equivalen to the [[zigzag]]

$$
  \array{
     && && && && && cone(f)[1]_\bullet &\to& \cdots
     \\
     && && && && && \downarrow^{\mathrlap{h[1]_\bullet}}_{\mathrlap{\simeq}}
     \\
     && && cone(f)_\bullet &\to& X[1]_\bullet &\stackrel{f[1]_\bullet}{\to}& Y[1]_\bullet &\to& Z[1]_\bullet
     \\
     && && \downarrow^{\mathrlap{h_\bullet}}_{\mathrlap{\simeq}}
     \\
     X_\bullet
       &\stackrel{f_\bullet}{\to}&
     Y_\bullet
       &\stackrel{}{\to}&
     Z_\bullet
  }
  \,.
$$

Observe that

$$
  H_n( X[k]_\bullet) \simeq H_{n-k}(X_\bullet)
  \,.
$$

It is therefore sufficient to check that

$$
  H_n
  \left(
    \array{
      cone(f)_\bullet &\to& X[1]_\bullet
      \\
      \downarrow^{\mathrlap{\simeq}}
      \\
      Z_\bullet
    }
  \right)
  \;\;
   :
  \;\;
  H_n(Z_\bullet) \to H_n(cone(f)_\bullet) \to H_{n-1}(X_\bullet)
$$

equals the [[connecting homomorphism]] $\delta_n$ induced by the short exact sequence.

By prop. \ref{ConeInjectionEquivalentToZigzag} the inverse of the vertical map is given by choosing lifts and forming the corresponding element given by the connecting homomorphism.
By prop. \ref{ProjectionOutOfChainComplexMappingConeIsHoCofiber} the horizontal map is just the projection, and hence the assignment is of the form

$$
  [z_n] \mapsto [x_{n-1}, y_n] \mapsto [x_{n-1}]
  \,.
$$

So in total the image of the zig-zag under homology sends

$$
  [z_n]_Z \mapsto -[\partial^Y \hat z_n]_X
  \,.
$$

By the discussion [there](connecting%20homomorphism#OnHomologyInTermsOfElements), this is indeed the action of the [[connecting homomorphism]].

=--

### Distinguished triangles from mapping cones
 {#DistinguishedTriangles}

In summary, the [above](#HomologyExactSequencesAndFiberSequences) says that for every [[chain map]] $f_\bullet : X_\bullet \to Y_\bullet$ we obtain maps

$$
  X_\bullet \stackrel{f}{\to}
  Y_\bullet
  \stackrel{
    \left(
      \array{
          0
          \\
          id_{Y_\bullet}
      }
    \right)
  }{\to}
   cone(f)_\bullet
  \stackrel{
    \left(
      \array{
         id_{X[1]_\bullet} & 0
      }
    \right)
  }{\to}
  X[1]_\bullet
$$

which form a [[homotopy fiber sequence]] and such that this sequence continues by forming [[suspension of a chain complex|suspensions]], hence for all $n \in \mathbb{Z}$ we have

$$
  X[n]_\bullet \stackrel{f}{\to}
  Y[n]_\bullet
  \stackrel{
    \left(
      \array{
          0
          \\
          id_{Y[n]_\bullet}
      }
    \right)
  }{\to}
   cone(f)[n]_\bullet
  \stackrel{
    \left(
      \array{
         id_{X[n+11]_\bullet} & 0
      }
    \right)
  }{\to}
  X[n+1]_\bullet
$$

To amplify this quasi-cyclic behaviour one sometimes depicts the situation as follows:

$$
  \array{
     X_\bullet &&\stackrel{f}{\to}&& Y_\bullet
     \\
     & {}_{\mathllap{[1]}}\nwarrow && \swarrow
     \\
     && cone(f)_\bullet
  }
$$

and hence speaks of a "triangle", or **distinguished triangle** or **mapping cone triangle** of $f$.

* distinguished triangle = period of [[homotopy fiber sequence]] .

Due to these "triangles" one calls the [[homotopy category]] of chain complexes [[localization|localized]] at the [[quasi-isomorphisms]], hence the  [[derived category]], a **[[triangulated category]]**.

Notice that equivalently we can express the triangles via the
[[mapping cylinder]].
For every map of [[chain complex]]es $f:A\to B$, the cylinder $Cyl(f)$ is quasi-isomorphic to $B$, and moreover in the [[homotopy category]] of chain complexes, every distinguished triangle is quasi-isomorphic to a distinguished triangle of the form

$$ A\to Cyl(u)\to Cone(u)\to A[1]$$

for some $u:A\to B$ where all the morphisms in the triangle are appropriatedly induced by $u$.

## Related concepts

[[!include universal constructions of topological spaces -- table]]

## References

In the context of [[chain complexes]] the construction is discussed for instance in

* {#Schapira} [[Pierre Schapira]], section 3.2 and section 7 of _Categories and homological algebra_ (2011) ([pdf](http://people.math.jussieu.fr/~schapira/lectnotes/HomAl.pdf))

* {#Weibel} [[Charles Weibel]], section 1.5 of  _[[An Introduction to Homological Algebra]]_ .

In the context of [[spectra]] discussion includes

* {#Switzer75} [[Robert Switzer]], around 8. 17 of _Algebraic Topology - Homotopy and Homology_, Die  Grundlehren der Mathematischen Wissenschaften in Einzeldarstellungen, Vol. 212, Springer-Verlag, New York, N. Y., 1975.

Graphics taken from

* {#Muro10} [[Fernando Muro]], _Representability of Cohomology Theories_, Joint Mathematical Conference CSASC 2010, 22–27 January 2010, Prague, Czech Republic ([slides](https://personal.us.es/fmuro/files/slides/praha.pdf))

[[!redirects mapping cones]]


mapping cylinder

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

#Contents#
* table of contents
{:toc}

## In Top

### Definition

Given a [[continuous map]] $f:X\to Y$ of [[topological space]]s, one can define its **mapping cylinder** as a [[pushout]]

$$
\array{
  X &\stackrel{f}\to& Y
  \\
  {}^{\mathllap{\sigma_0}}\downarrow && \downarrow^{\mathrlap{ f_*(\sigma_0)}}
  \\
  X\times I &\stackrel{(\sigma_0)_* (f)}\to & Cyl(f)
}
$$

in [[Top]], where $I = [0,1]$ (the [[unit interval]]) and $\sigma_0:X\to X\times I$ is given by $x\mapsto (x,0)$. By tradition, [[homotopy theory|homotopy theorists]] sometimes use the inverted (upside-down) mapping cylinder where $\sigma_0$ is replaced by $\sigma_1:x\mapsto (x,1)$. The two mapping cylinders are [[homeomorphism|homeomorphic]] so it is a matter of convention which one to use, of course, compatibly with other constructions depending on the [[orientation]] of $I$.

Set-theoretically, the mapping cylinder is usually represented as the [[quotient space]] $(X\times I \coprod Y)/{\sim}$ where $\sim$ is the smallest [[equivalence relation]] identifying $(x,0)\sim f(x)$ for all $x\in X$.

### Properties

As any other pushout, the mapping cylinder has a [[universal property]]: for any space $Z$ and mapping $g_1:X\times I\to Z$, $g_2:Y\to Z$ such that $g_1(x,0)=g_2(f(x))$ for all $x\in X$, there is a unique $k:Cyl(f)\to Z$, such that the composition $X\times I\to Cyl(f)\stackrel{k}\to Z$ equals $g_1$ and the composition $Y\to Cyl(f)\stackrel{k}\to Z$ equals $g_2$.

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

For  $f \colon X\to Y$ a [[continuous function]],
the canonical map $j \coloneqq f_*(\sigma_0) \colon Y\to Cyl(f)$ is a [[homotopy equivalence]]. In fact its homotopy inverse can be chosen a [[deformation retraction]]. In particular every continuous function factors as a map into its mapping cylinder followed by a deformation retraction.

=--

+-- {: .proof}
###### Proof
We exhibit $j$ as a homotopy equivalence by constructing its homotopy inverse $\tilde{f}$ given by $\tilde{f}:[x,t]\mapsto f(x)$, where $[x,t]$ is a class of  $(x,t)\in X\times I$ and $\tilde{f}([y])=[y]$ for $y\in Y$.
Clearly this map is well-defined and
$\tilde{f}\circ j = \id_Y$. On the other hand,
$(j\circ\tilde{f})[x,t] = [f(x)]$. Homotopy
$H:\mathrm{Cyl}(f)\times I\to Y$ is given by
$$
H([x,t],\tau) = [x,t(1-\tau)], \,\,\,H([y],\tau)=[y].
$$
It is easy to see that $H(-,0) = \id_{Cyl(f)}$,
$H(-,1)=[-,0]=[f(-)]$ hence $j\circ\tilde{f}\sim id_{Cyl(f)}$.
=--

+-- {: .num_theorem}
###### Theorem
A continuous map $i:A\to X$ is a [[Hurewicz cofibration]] iff there is a [[retraction]] $r:X\times I\to Cyl(f)$ for the canonical map $X\times I \to Cyl(f)$.
=--

+-- {: .num_theorem}
###### Theorem
A continuous map $f:X\to Y$ is a [[homotopy equivalence]] iff $X = X\times\{0\}$ is a [[deformation retract]] of the cylinder $Cyl(f)$.
=--

+-- {: .num_theorem}
###### Theorem
For any $f:X\to Y$, the composition

$$X\stackrel{\sigma_1}\to X\times I\stackrel{(\sigma_0)_* (f)}\to Cyl(f)$$

is a [[Hurewicz cofibration]]. Furthermore, the map $r:Cyl(f)\to Y$ determined by $r([x,t])= f(x)$ (for all $x\in X$ and $t\in I$) and $r([y])=y$ (for $y\in Y$) is well defined and a homotopy equivalence.

The composition $r\circ (\sigma_0)_* (f)\circ \sigma_1 = f$, hence this is a decomposition of a continuous map into a cofibration followed by a homotopy equivalence.
=--

## Related entries

* [[mapping cone]]

* [[double mapping cylinder]]

* In [[homotopy type theory]] mapping cyclinders can be constructed as [[higher inductive types]]. See [there](higher+inductive+type#MappingCylinders).

[[!include universal constructions of topological spaces -- table]]

## References

### General

(...)

### In topology

Discussion of mapping cylinders of [[topological spaces]] in [[general topology]]:

* Alex Aguado, *A Short Note on Mapping Cylinders* ([arXiv:1206.1277](https://arxiv.org/abs/1206.1277))

See also:

* Wikipedia, *[Mapping cylinder](https://en.wikipedia.org/wiki/Mapping_cylinder)*

### In homological algebra

Discussion of mapping cylinders of [[chain complexes]] in [[homological algebra]]:

Textbook accounts:

* [[Charles Weibel]], Sec. 1.5.5 in _[[An introduction to homological algebra]]_,  Cambridge Studies in Adv. Math. 38, CUP 1994 ([pdf](https://web.math.rochester.edu/people/faculty/doug//otherpapers/weibel-hom.pdf), [doi:10.1017/CBO9781139644136](https://doi.org/10.1017/CBO9781139644136))

Lecture notes:

* Keller Vandebogert, Def. 3.1 in: *Homological Algebra Notes* (pdf)

[[!redirects mapping cylinders]]


mapping simplex

In the theory of [[nLab:cartesian fibration]]s of simplicial sets cartesian fibrations $X\to \Delta^n$ over a simplex play an important role since an arbitrary  morphism $X\to S$ is a cartesian fibration iff for all $n$, $X\times_S \Delta^n\to \Delta^n$ is a cartesian fibration.

A cartesian fibration $X\to \Delta^n$ is by the $(\infty,1)$-Grothendieck construction equivalently a functor $\Delta^n\to (\infty,1)Cat$; i.e. a composable sequence of $(\infty,1)$-categories and functors $\phi:A^0\leftarrow\dots\leftarrow A^n$.

+-- {: .un_defn}
###### Definition
The *mapping simplex* $M(\phi)$ of $\phi$ is defined by:

* For a nonempty finite finite linear order $L$ with greatest element $j$, a map $\Delta^L\to M(\phi)$ consists of a order preserving map $f:L\to [n]$ and a morphism $\sigma:\Delta^L\to A^{f(j)}$.

* Given two such linear orders $L$ and $L^\prime$ with greatest elements $j$ resp. $j^\prime$ there is a natural map $M(\phi)(\Delta^{L^\prime})\to M(\phi)(\Delta^{L})$ sending $(f,\sigma)$  to $(f\circ p, e\circ \sigma)$, where $e:A^{f(j^\prime)}\to A^{f(p(j))}$ is obtained by $\phi$.

There is a natural map $h:M(\phi)\to \Delta^n$ (take $J=m$, then the Yoneda lemma gives a map $\Delta^m\to \Delta^n$).

An edge $e$ of $M(\phi)$ is defined by a pair of integers $0\le i\le j\le n$ and an edge $e^\prime\in A^j$. $M(\phi)$ becomes a [[nLab:marked simplicial set]] $(M(\phi), E)$ by marking those edges for which $e^\prime$ is degenerated.
=--

+-- {: .un_defn}
###### Definition
Let $p:X\to \Delta^n$ be a cartesian fibration, let $\phi:A^0\leftarrow\dots\leftarrow A^n$ be  a composable sequence of $(\infty,1)$-categories and functors. Then A map $q:M(\phi)\to X$ is called a *quasi-equivalence* if it satisfies:

(1) The map $h$ commutes with $p$ and $q$.

(2) $q$ sends marked edges of $M(\phi)$ to $p$-cartesian ones.

(3) For every $0\le i\le n$, the induced map $A^i\to p^{-1}\{ i \}$ is a [[nLab:categorical equivalence]].
=--

+-- {: .un_prop}
######  Proposition
Let $p:X\to \Delta^n$ be a cartesian fibration.

(1) There exists  a composable sequence of $(\infty,1)$-categories and functors $\phi:A^0\leftarrow\dots\leftarrow A^n$ and a quasi-equivalence $q:M(\phi)\to X$.

(2) If $\phi:A^0\leftarrow\dots\leftarrow A^n$ is a composable sequence and $q:M(\phi)\to X$ a quasi-equivalence. Then for any map $T\to \Delta^n$, the induced map

$$M(\phi)\times_{\Delta^n}T\to X\times_{\Delta^n}T$$

is a [[nLab:categorical equivalence]].
=--

## Related topics

* [[nLab:relative nerve]]

* [[nLab:(infinity,1)-Grothendieck construction]]

## Reference

* [[nLab:Higher Topos Theory]] section 3.2.2.


mapping space - contents

**[[internal hom]]/[[mapping space]]**

### General abstract

* [[hom-set]], [[hom-object]], [[internal hom]], [[exponential object]], [[derived hom-space]]

* [[loop space object]], [[free loop space object]], [[derived loop space]]

### Topology

* [[mapping space]] ([[compact-open topology]])

* [[topology of mapping spaces]]

* [[evaluation fibration of mapping spaces]]

* [[loop space]], [[free loop space]]

  * [[free loop space of a classifying space]]

### Simplicial homotopy theory

* [[simplicial mapping complex]]

* [[inertia groupoid]]

### Differential topology

* [[differential topology of mapping spaces]]

  * [[C-k topology]]

* [[manifold structure of mapping spaces]]

  * [[tangent spaces of mapping spaces]]

* [[smooth loop space]]

### Stable homotopy theory

* [[mapping spectrum]]

### Geometric homotopy theory

* [[mapping stack]]

* [[inertia stack]]

* [[free loop stack]]

[Edit this sidebar](/nlab/edit/mapping+space+-+contents)


mapping stack

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

#Contents#
* table of contents
{:toc}

## Definition

An [[internal hom]] in a [[(2,1)-topos]] of [[stacks]] or generally in an [[(∞,1)-topos]] of [[∞-stacks]] is often called a **mapping stack** or **[[mapping ∞-stack]]**, in analogy with _[[mapping space]]_.

## Properties

By the general formula for [[internal homs]] in [[toposes]], for $X, A \in \mathbf{H}$ two stacks or $\infty$-stacks, their mapping stack assigns to an object $U \in C$ in a [[site]] or [[(∞,1)-site]] of definition the [[groupoid]] or [[∞-groupoid]] given by

$$
 [X, A] \;\colon\; U \mapsto \mathbf{H}(X \times U, A)
  \,.
$$

## Related concepts

* [[mapping ∞-stack]]

## References

Mapping stacks of [[orbifolds]] have been discussed in

* [[Weimin Chen]], _On a notion of maps between orbifolds, I. Function spaces_, Commun. Contemp. Math. 8 (2006), no. 5, 569–620 ([doi:10.1142/S0219199706002246](https://doi.org/10.1142/S0219199706002246))

* [[André Haefliger]], _On the space of morphisms between étale groupoids_, In:  P. Robert Kotiuga (ed.) _A Celebration of the Mathematical Legacy of Raoul Bott_ ([arXiv:0707.4673](https://arxiv.org/abs/0707.4673), [ams:crmp-50](https://bookstore.ams.org/crmp-50))

More general results in the context of [[differentiable stacks]] ([[orbifolds]]) are in

* {#RobertsVozzo18} [[David Roberts]], [[Raymond Vozzo]], _The Smooth Hom-Stack of an Orbifold_, In : Wood D., de Gier J., Praeger C., Tao T. (eds) 2016 MATRIX Annals. MATRIX Book Series, vol 1. Springer, Cham (2018) ([arXiv:1610.05904](https://arxiv.org/abs/1610.05904), [doi:10.1007/978-3-319-72299-3_3](https://doi.org/10.1007/978-3-319-72299-3_3))

generalizing the discussion at _[[manifold structure of mapping spaces]]_ to stacks.

That the mapping stack out of the circle into a topological stack is again a topological stack is often attributed to

* [[Ernesto Lupercio]], [[Bernardo Uribe]], _Loop groupoids, gerbes, and twisted sectors on orbifolds_ In _Orbifolds in mathematics and physics_ (Madison, WI, 2001), volume 310 of Contemp. Math., pages 163184. Amer. Math. Soc., Providence, RI, 2002.

General mapping stacks of [[topological stacks]] are discussed in

* {#Noohi08} [[Behrang Noohi]], _Mapping stacks of topological stacks_, Journal für die reine und angewandte Mathematik, Volume 2010, Issue 646 ([arXiv:0809.2373](http://arxiv.org/abs/0809.2373), [doi:10.1515/crelle.2010.067](https://doi.org/10.1515/crelle.2010.067))

[[!redirects mapping stacks]]

[[!redirects Hom stack]]
[[!redirects Hom stacks]]
[[!redirects hom stack]]
[[!redirects hom stacks]]
[[!redirects Hom-stack]]
[[!redirects Hom-stacks]]
[[!redirects hom-stack]]
[[!redirects hom-stacks]]


mapping telescope

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

#Contents#
* table of contents
{:toc}

## Idea

Given a sequence

$$
  X_\bullet
  =
  \left(
  X_0
   \overset{f_0}{\longrightarrow}
  X_1
   \overset{f_1}{\longrightarrow}
  X_2
   \overset{f_2}{\longrightarrow}
  \cdots
  \right)
$$

of ([[pointed topological space|pointed]]) [[topological spaces]], then its _mapping telescope_ is the result of forming the (reduced) [[mapping cylinder]] $Cyl(f_n)$ for each $n$ and then attaching all these cylinders to each other in the canonical way.

At least if all the $f_n$ are inclusions, this is the sequential attachment of ever "larger" cylinders, whence the name "telescope".

The mapping telescope is a representation for the [[homotopy colimit]] over $X_\bullet$. It is used for instance for discussion of [[lim^1 and Milnor sequences]] (and that's maybe the origin of the concept?).

## Definition

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

For

$$
  X_\bullet
  =
  \left(
  X_0
   \overset{f_0}{\longrightarrow}
  X_1
   \overset{f_1}{\longrightarrow}
  X_2
   \overset{f_2}{\longrightarrow}
  \cdots
  \right)
$$

a sequence  in [[Top]], its **mapping telescope** is the [[quotient topological space]] of the [[disjoint union]] of [[product topological spaces]]

$$
  Tel(X_\bullet)
  \coloneqq
  \left(
  \underset{n \in \mathbb{N}}{\sqcup}
  \left(
    X_n \times [n,n+1]
  \right)
  \right)/_\sim
$$

where the [[equivalence relation]] quotiented out is

$$
  (x_n, n) \sim (f(x_n), n+1)
$$

for all $n\in \mathbb{N}$ and $x_n \in X_n$.

Analogously for $X_\bullet$ a sequence of [[pointed topological spaces]] then use [[reduced cylinders]] to set

$$
  Tel(X_\bullet)
  \coloneqq
  \left(
  \underset{n \in \mathbb{N}}{\sqcup}
  \left(
    X_n \wedge [n,n+1]_+
  \right)
  \right)/_\sim
  \,.
$$

=--

## Properties

### For CW-complexes

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

For $X_\bullet$ the sequence of stages of a ([[pointed topological space|pointed]]) [[CW-complex]] $X = \underset{\longleftarrow}{\lim}_n X_n$, then the canonical map

$$
  Tel(X_\bullet)
  \longrightarrow
  X
$$

from the [[mapping telescope]], def. \ref{MappingTelescope}, is a [[weak homotopy equivalence]].

=--

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

Write in the following $Tel(X)$ for $Tel(X_\bullet)$ and write $Tel(X_n)$ for the mapping telescop of the substages of the finite stage $X_n$ of $X$. It is intuitively clear that each of the projections at finite stage

$$
  Tel(X_n) \longrightarrow X_n
$$

is a [[homotopy equivalence]], hence a [[weak homotopy equivalence]]. A concrete construction of a homotopy inverse is given for instance in ([Switzer 75, proof of prop. 7.53](#Switzer75)).

Moreover, since spheres are [[compact object|compact]], so that elements of [[homotopy groups]] $\pi_q(Tel(X))$ are represented at some finite stage $\pi_q(Tel(X_n))$ it follows that

$$
  \underset{\longrightarrow}{\lim}_n \pi_q(Tel(X_n))
    \overset{\simeq}{\longrightarrow}
  \pi_q(Tel(X))
$$

are [[isomorphisms]] for all $q\in \mathbb{N}$ and all choices of basepoints (not shown).

Together these two facts imply that in the following commuting square, three morphisms are isomorphisms, as shown.

$$
  \array{
    \underset{\longleftarrow}{\lim}_n \pi_q(Tel(X_n))
    &\overset{\simeq}{\longrightarrow}&
    \pi_q(Tel(X))
    \\
    {}^{\mathllap{\simeq}}\downarrow && \downarrow
    \\
    \underset{\longleftarrow}{\lim}_n \pi_q(X_n)
    &\underset{\simeq}{\longrightarrow}&
    \pi_q(X)
  }
  \,.
$$

Therefore also the remaining morphism is an isomorphism ([[two-out-of-three]]). Since this holds for all $q$ and all basepoints, it is a weak homotopy equivalence.

=--

## Related concepts

* [[homotopy colimit]]

[[!include universal constructions of topological spaces -- table]]

## References

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


mapping ∞-stack

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

An [[internal hom]] in an [[(∞,1)-topos]]. The generalization of [[mapping stacks]] to [[higher topos theory]].

[[!redirects mapping ∞-stacks]]

[[!redirects mapping infininity-stack]]
[[!redirects mapping infinity-stacks]]