homotopy pullback

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### $(\infty,1)$-Limits and colimits
+-- {: .hide}
[[!include infinity-limits - contents]]
=--
#### Homotopy theory
+-- {: .hide}
[[!include homotopy - contents]]
=--
#### Model category theory
+-- {: .hide}
[[!include model category theory - contents]]
=--
=--
=--

# Contents
* table of contents
{: toc}

## Idea

A **homotopy pullback** is a special kind of [[homotopy limit]]: the appropriate notion of [[pullback]] in the context of [[homotopy theory]]. Homotopy pullbacks model the [[limit in quasi-categories|quasi-category pullbacks]] in the [[(infinity,1)-category]] that is [[presentable (infinity,1)-category|presented]] by a given [[homotopical category]].  Since pullbacks are also called fiber products, homotopy pullbacks are also called **homotopy fiber products**.

The notion of __[[homotopy pushout]]__ is the [[duality|dual]] concept.

For more details see *[[homotopy limit]]*.

In the context of [[homotopy type theory]] a homotopy pullback is the [[categorical semantics|interpretation]] of the space of solutions to an [[equation]].

## Definition

### In category theory

As with all homotopy limits, there is both a _local_ and a _global_ notion of homotopy pullback.

The _global_ definition says that the homotopy pullback of a [[co-span|cospan]] $X \to Z \leftarrow Y$ in a [[category with weak equivalences]] $C$ is its image under the right [[derived functor]] of the [[base change]] functor $pb: C^{\to \leftarrow} \to C$.

The _local_ definition says that the homotopy pullback of $X \to Z \leftarrow Y$ in a category with a notion of [[homotopy]] consists of a square
$$
  \array{
   P & \to& Y
   \\
    \downarrow && \downarrow
   \\
   X &\to& Z
  }
$$
that [[commutative square|commutes]] up to [[homotopy]], and such that for any other square
$$
  \array{
   T & \to& Y
   \\
    \downarrow && \downarrow
   \\
   X &\to& Z
  }
$$
that commutes up to homotopy, there exists a morphism $T\to P$, unique up to homotopy, making the two triangles commute up to homotopy, and similarly for homotopies and higher homotopies.  In other words, there is an equivalence
$$Map(T,P) \simeq HoSq(T,X\to Z\leftarrow Y)$$
between the space of maps $T\to P$ and the space of homotopy commutative squares with vertex $T$.

In good situations, such as when $X,Y,Z$ are [[fibrant object|fibrant]] in a [[model category]], the two constructions agree up to weak equivalence.

Note that in both cases, there is a canonical map from the actual pullback $X\times_Z Y$ to the homotopy pullback $X\times_Z^h Y$.  In the global case this comes by the definition of a derived functor; in the local case it comes because a commutative square is, in particular, a homotopy commutative one.

### In homotopy type theory
 {#InHomotopyTypeTheory}

In [[homotopy type theory]] the homotopy pullback of a [[term]] of [[function type]]

$$
  f : A \to C
$$

along a term of function type

$$
  g : B \to C
$$

is given formally by precisely the same formula that would also define the ordinary [[fiber product]] of functions of sets, namely by

$$
  \left\{
     a : A, b : B \;\; | \;\; f(a) = g(b)
  \right\}
  \,.
$$

Spelled out, this is the [[dependent sum]]

$$
  \underset{
    \array{
      {a \colon A}
      \\
      {b\colon B}
    }
   }{\sum}
   \big(
     f(a) = g(b)
   \big)
$$

over the [[dependent type|dependent]] [[identity type]] over the [[evaluation]] of $f$ and $g$.

What in classical logic is interpreted as the set of pairs $(a,b)$ such that $f(a)$ and $g(b)$ are equal here becomes the restriction of a [[mapping cocylinder]].

Formal proof that this is the homotopy pullback in homotopy type theory is in ([Brunerie](#Brunerie)). Proof in the [[categorical semantics of homotopy type theory]] is [below](#ConstructionInHomotopyTypeTheory).

## Concrete constructions
  {#ConcreteConstructions}

We discuss various concrete constructions by ordinary [[pullbacks]] and ordinary [[limits]] such that under some sufficient conditions these compute homotopy pullbacks, up to weak equivalence.

### General
 {#ConstructionsGeneral}

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

Let $A \to C \leftarrow B$ be a [[diagram]] in some [[model category]]. Then sufficient conditions for the ordinary (1-categorical) [[pullback]] $A \times_C B$ to present the homotopy pullback of the diagram are

* one of the two morphisms is a [[fibration]] and all three objects are [[fibrant objects]];

* one of the two morphisms is a [[fibration]] and the model category is [[right proper model category|right proper]].

=--

Both statements are classical. They are reviewed for instance as [Lurie, prop. A.2.4.4](#Lurie). The proof of the second statement is spelled out [here](proper+model+category#HomotopyLimits).

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

Notice that a  [[fibrant resolution]] of the [[cospan]] [[diagram]] in the injective [[model structure on functors]] has _both_ morphisms be a fibration. So the first point in Prop. \ref{HomotopyPullbackByOrdinaryPullback} says that (in the special case of pullbacks) something weaker than fibrancy  is sufficient for [[derived functor|deriving]] the [[limit]] functor on cospan diagrams.

This can be explained in model-categorical terms by the fact that the category of [[cospans]] also has a [[Reedy model structure]] in which the fibrant objects are precisely those considered in the first point in Prop. \ref{HomotopyPullbackByOrdinaryPullback}, and that homotopy limits can equally well be computed using this model structure (specifically, the [[adjunction]] $Const \dashv Lim$ is [[Quillen adjunction|Quillen]] with respect to it).

In this spirit one may ask for the largest class of morphisms such that their ordinary pullbacks are already homotopy pullbacks. This is related to the concept of _[[sharp morphisms]]_.

=--

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

In the first point of Prop. \ref{HomotopyPullbackByOrdinaryPullback} it is indeed necessary that all three objects are fibrant. As a ([[counterexample|counter-]])[[example]] consider the
pullback of $\Lambda_1^2 \to \Delta^2$ along $d_1 \colon \Delta^1 \to \Delta^2$ in [[SimplicialSets]]: This is not a homotopy pullback with respect to the [[Joyal model structure]], even though the latter morphism is a fibration between fibrant objects.

=--

Due to prop. \ref{HomotopyPullbackByOrdinaryPullback} one typically computes homotopy pullbacks of a diagram by first forming a [[resolution]] of one of the two morphisms by a fibration and then forming an ordinary pullback.

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

If in $A \stackrel{f}{\to} C \stackrel{g}{\leftarrow} B$
all three objects are [[fibrant objects]], then
the homotopy pullback of this diagram is presented by the ordinary [[pullback]]

$$
  \array{
     A\times_C^h B & \to & C^I \times_C B
    \\
    \downarrow && \downarrow
    \\
     A & \stackrel{f}{\to} & C
  }
  \,.
$$

or, equivalently up to [[isomorphism]], as the ordinary [[pullback]]

$$
  \array{
     A\times_C^h B & \to & C^I
    \\
    \downarrow && \downarrow
    \\
     A\times B & \stackrel{(f,g)}{\to} & C\times C
  }
  \,.
$$

=--

See also at _[[Mayer-Vietoris sequence]]_ ([this proposition](Mayer-Vietoris+sequence#SequenceFromDiagonal)) and at *[[homotopy equalizer]]* ([this section](homotopy+equalizer#RelationToHomotopyPullbacks)).

+-- {: .proof}
###### Proof
Since the objects are already fibrant, prop. \ref{HomotopyPullbackByOrdinaryPullback} implies that it is sufficient to replace one of the morphisms by a fibrant resolution.
Such a resolution is provided by the [[factorization lemma]]: by [Lemma 3](factorization+lemma#FibrantResolution), $B \to C$ admits a canonical fibrant resolution
  $$ C^I \times_C B \twoheadrightarrow C $$
where $C \stackrel{\simeq}{\to} C^I \to C \times C$ is a [[path space object]] for $C$ (for instance, when $C$ is a [[closed monoidal homotopical category]] then this can be taken to be the [[internal hom]] with an [[interval object]] $I$).
=--

The homotopy pullback constructed in this way is an example of a _strict homotopy limit_, as mentioned at [[homotopy limit]].  In such a case, one can say that an arbitrary homotopy-commutative square

$$
  \array{
   W & \to& Y
   \\
    \downarrow && \downarrow
   \\
   X &\to& Z
  }
$$

is a homotopy pullback square if the induced morphism from $W$ to the strict homotopy pullback is a [[weak equivalence]].

A useful class of examples of this is implied by the following:

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

Let $\mathcal{C}$ be a category of [[simplicial presheaves]] over some [[site]]
equipped with a _local_ injective [[model structure on simplicial presheaves]] with respect to that site.

Then an ordinary pullback of $A \to C \leftarrow B$ in $\mathcal{C}$ is a homotopy pullback already when one of the two morphisms is an objectwise [[Kan fibration]].

=--

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

The _global_ projective [[model structure on simplicial presheaves]] is [[right proper model category|right proper]]. So by prop. \ref{HomotopyPullbackByOrdinaryPullback} the ordinary pullback in question presents the homotopy pullback in the global structure. By the discussion at [[homotopy limit]] and [[Bousfield localization of model categories]], this presents the [[(∞,1)-pullback]] of the diagram of [[(∞,1)-presheaves]], and the fibrant replacement of that pullback in the _local_ model structure presents the [[(∞,1)-sheafification]] of this [[(∞,1)-presheaf]]. This is (essentially by definition, see [[(∞,1)-topos]]) a [[left exact functor|left exact]] [[(∞,1)-functor]] and hence preserves finite [[(∞,1)-limits]].

=--

### In homotopy type theory
 {#ConstructionInHomotopyTypeTheory}

If we unwind the [[categorical semantics]] of the [above definition](#InHomotopyTypeTheory)

$$
  A \times_C^h B
  \;\simeq\;
  \Big\{
    a \colon A
    ,\,
    b \colon B
    \;\big\vert\;
    \big(
      f(a) = g(b)
    \big)
  \Big\}
$$

of the homotopy pullback in [[homotopy type theory]], we re-obtain the
[above prescription](#ConstructionsGeneral) for how to construct homotopy pullbacks.

So let the ambient category be a suitable [[type-theoretic model category]].

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

The type $ a : A, b : B \vdash (f(a) = g(b))$ is obtained by [[substitution]] from the [[identity type]] of $C$. By the discussion there, the [[categorical semantics]] of substitution is given by [[pullback]] of the fibrations that interpret the [[dependent types]], and so this is interpreted as the pullback $[a : A, b : B \vdash (f(a) = g(b))] \coloneqq (f,g)^* C^I$ of the [[path space object]] of $C$:

$$
  \array{
    [a : A, b : B \vdash (f(a) = g(b))]
    &\to&
    [Id C]
    \\
    \downarrow && \downarrow
    \\
    [a : A , b : B]
    &\stackrel{(f,g)}{\to}&
    [c_1 : C , c_2 : C]
  }
  \;\;
   =
  \;\;
  \array{
    (f,g)^* C^I &\to& C^I
    \\
    \downarrow && \downarrow
    \\
    A \times B
    &
    \stackrel{(f,g)}{\to}
    &
    C \times C
  }
  \,.
$$

Forming the [[dependent sum]] over $a : A, b : B$ is simply interpreted as regarding the resulting object $(f,g)^* C^I$ as an object in $\mathcal{C} \simeq \mathcal{C}_{/*}$ instead of as an object in the [[slice category]] $\mathcal{C}_{/ A \times B}$.

Since by assumption on the categorical interpretation of a type, all objects here are fibrant, this coincides with the expression of the homotopy pullback from corollary \ref{HomotopyPullbackByFactorizationLemma} above.

=--

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

Specifically, let $f \colon A \longrightarrow B$ be a [[function]], then the [[categorical semantics]] for the expression

$$
  \underset{b \colon B}{\sum} fib(f,b)
  =
  \underset{b \colon B}{\sum} \underset{a \colon A}{\sum} (f(a) = b)
$$

is the canonical [[fibration]] replacement of $f$ as it appears notably in the [[factorization lemma]]

$$
  \array{
    A \times_B B^I &\longrightarrow& B^I
    \\
    \downarrow && \downarrow
    \\
    A \times B
    &\stackrel{(f, Id)}{\longrightarrow}&
    B \times B
  }
  \,.
$$

=--

## Properties

### Fiber-wise characterization
 {#HomotopyFiberCharacterization}

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

In plain [[homotopy types]] (i.e. in [[∞-groupoids]], in the [[classical model structure on simplicial sets]] etc.) the following holds:

a [[diagram]]

$$
  \array{
    A &\stackrel{f}{\longrightarrow}& B
    \\
    \downarrow && \downarrow^{\mathrlap{p}}
    \\
    C &\stackrel{g}{\longrightarrow}& D
  }
$$

is a homotopy pullback diagram precisely if it induces a [[weak equivalence]] on all [[homotopy fibers]]

$$
  \array{
    hfib_b(f) &\longrightarrow& A &\stackrel{f}{\longrightarrow}& B
    \\
    \downarrow^{\mathrlap{\simeq}}  && \downarrow && \downarrow^{\mathrlap{p}}
    \\
    hfib_{p(b)}(g) &\longrightarrow& C &\stackrel{g}{\longrightarrow}& D
  }
$$

for all elements $b \in B$.

=--

 (e.g. [CPS 05, 5.2](#CPS05), [MO](http://mathoverflow.net/a/192303/381))

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

For the analog of prop. \ref{FiberwiseCharacterizationInInfinityGrpds} to hold in [[(∞,1)-categories]] more general than [[∞-groupoids]] one would need either "enough [[global elements]]" of the object $B$ to detect all homotopy fibers, or else one would need a suitable "[[internal logic|internal]]" formulation of the statement.

=--

On the other hand in [[stable homotopy theory]] the statement holds generally:

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

For $\mathcal{C}$ a [[stable model category]], then for a diagram

$$
  \array{
    A &\stackrel{f}{\longrightarrow}& B
    \\
    \downarrow && \downarrow^{\mathrlap{p}}
    \\
    C &\stackrel{g}{\longrightarrow}& D
  }
$$

the following are equivalent

1. the square is a homotopy pullback square;

1. the square is a [[homotopy pushout]] squre;

1. the induced morphism on the homotopy fiber over the [[zero object]] 0 is a weak equivalence

   $$
      hfib_0(f) \stackrel{\simeq}{\longrightarrow} hfib_0(g)
      \,.
   $$

=--

A proof in terms of [[stable model categories]] including the third item is spelled out for instance in ([Hovey 07, remark 7.1.12](#Hovey07)). (Beware: this is not in the original volume from 1999, but is in the version "reprinted with corrections" from 2007.) A proof in terms of [[stable (∞,1)-categories]] is in ([[Higher Algebra|Lurie HA, prop. 1.1.3.4 (2) & lemma 1.2.4.14]]).

### Pasting law

* [[pasting law]]

## Examples

### Fiber sequences

Of particular interest are consecutive homotopy pullbacks of point inclusions. These give rise to [[fiber sequence]]s and [[loop space object]]s.

## Related concepts

* [[equation]]

* [[homotopy limit]], [[(∞,1)-limit]]

* [[homotopy pushout type]]

[[!include notions of pullback -- contents]]

## References

### General

See also the references at _[[homotopy limit]]_ .

Homotopy pullbacks are at least mentioned in almost any textbook on [[homotopy theory]].

Dedicated textbook introductions:

* [[Jeffrey Strom]], *Homotopy Pullback Squares*, §7.3 in: *Modern classical homotopy theory*, Graduate Studies in Mathematics **127**, American Mathematical Society (2011) [[doi:10.1090/gsm/127](http://www.ams.org/books/gsm/127/)]

* [[Martin Arkowitz]], *Homotopy Pushouts and Pullbacks*, §6 in: *Introduction to Homotopy Theory*, Springer (2011) [[doi:10.1007/978-1-4419-7329-0](https://doi.org/10.1007/978-1-4419-7329-0)]

* [[Peter May]], [[Kate Ponto]], *Some basic homotopy limits*: §2.2 in: _[[More concise algebraic topology]]_,   University of Chicago Press (2012) [[ISBN:9780226511795](https://press.uchicago.edu/ucp/books/book/chicago/M/bo12322308.html), [pdf](https://www.math.uchicago.edu/~may/TEAK/KateBookFinal.pdf)]

Exposition in [[model category]]-theory:

* [[Urs Schreiber]], *[Homotopy Pullbacks](Introduction+to+Homotopy+Theory#HomotopyPullbacks)*, section in: *[[Introduction to Homotopy Theory]]*

Fairly comprehensive general resources:

* {#Hovey07} [[Mark Hovey]], _Model categories_, Mathematical Surveys and Monographs, vol 63, 1999,reprinted 2007

* {#Lurie} [[Jacob Lurie]], appendix of _[[Higher Topos Theory]]_

See also:

* {#CPS05} [[Wojciech Chacholski]], Wolfgang Pitsch, and [[Jerome Scherer]], _Homotopy pullback squares up to localization_, in [[Dominique Arlettaz]], [[Kathryn Hess]] (eds.) _An Alpine Anthology of Homotopy Theory_ ([arXiv:math/0501250](http://arxiv.org/abs/math/0501250), [pdf](http://sma.epfl.ch/~jscherer/articles/hopullbacks2.pdf), [doi:10.1090/conm/399](http://dx.doi.org/10.1090/conm/399))

* [[Alisa Govzmann]], [[Damjan Pištalo]], [[Norbert Poncin]], *Indeterminacies and models of homotopy limits* [[arXiv:2109.12395](https://arxiv.org/abs/2109.12395)]

### In terms of homotopy type theory
 {#ReferencesForHoTT}

Discussion of homotopy pullbacks in [[homotopy type theory]]:

* [[Univalent Foundations Project]], Exc. 2.11 in: *[[HoTT book|Homotopy Type Theory – Univalent Foundations of Mathematics]]* (2013)

* [[Egbert Rijke]], *Homotopy pullbacks* [[pdf](https://www.andrew.cmu.edu/user/erijke/hott/pullback.pdf)], Lecture 10 in: *[[Introduction to Homotopy Type Theory]]*, lecture notes, CMU (2018) [[pdf](http://www.andrew.cmu.edu/user/erijke/hott/hott_intro.pdf), [[Rijke-IntroductionHoTT-2018.pdf:file]], [webpage](https://www.andrew.cmu.edu/user/erijke/hott/)]

[[!redirects homotopy pullback]]
[[!redirects homotopy pullbacks]]

[[!redirects homotopy fiber product]]
[[!redirects homotopy fiber products]]

[[!redirects homotopy cofiber product]]
[[!redirects homotopy cofiber products]]

[[!redirects homotopy base change]]
[[!redirects homotopy base changes]]


homotopy pushout

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### $(\infty,1)$-Limits and colimits
+-- {: .hide}
[[!include infinity-limits - contents]]
=--
#### Homotopy theory
+-- {: .hide}
[[!include homotopy - contents]]
=--
#### Model category theory
+-- {: .hide}
[[!include model category theory - contents]]
=--
=--
=--

#Comtents#
* table of contents
{:toc}

## Idea

The notion of *homomtopy pushout* is

* the generalization of the notion of *[[pushout]]* to [[homotopy theory]], hence from [[1-category]]-[[category theory]] to [[(infinity,1)-category|$\infty$-category]] theory, and here often understood as a [[derived functor]] of the pushout-functor

* the [[formal duality|formal dual]] to the notion of [[homotopy pullback]]. For the moment, see there for more details.

## Related concepts

* [[homotopy pushout type]]

## References
 {#References}

Homotopy pushouts are at least mentioned in essentially any textbook on [[homotopy theory]].

Dedicated textbook introductions:

* [[Jeffrey Strom]], *Homotopy Pushout Squares*, §7.1 in: *Modern classical homotopy theory*, Graduate Studies in Mathematics **127**, American Mathematical Society (2011) [[doi:10.1090/gsm/127](http://www.ams.org/books/gsm/127/)]

* [[Martin Arkowitz]], *Homotopy Pushouts and Pullbacks*, §6 in: *Introduction to Homotopy Theory*, Springer (2011) [[doi:10.1007/978-1-4419-7329-0](https://doi.org/10.1007/978-1-4419-7329-0)]

* [[Peter May]], [[Kate Ponto]], *Some basic homotopy colimits*, §2.1 in: _[[More concise algebraic topology]]_,   University of Chicago Press (2012) [[ISBN:9780226511795](https://press.uchicago.edu/ucp/books/book/chicago/M/bo12322308.html), [pdf](https://www.math.uchicago.edu/~may/TEAK/KateBookFinal.pdf)]

For more references see at

* *[[homotopy colimit]]*

* *[[homotopy pullback]]*

[[!redirects homotopy pushout]]
[[!redirects homotopy pushouts]]

[[!redirects homotopy co-fiber product]]
[[!redirects homotopy co-fiber products]]

[[!redirects homotopy cobase change]]
[[!redirects homotopy cobase changes]]


homotopy quotient

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

#Contents#
* table of contents
{:toc}

## Idea

A _homotopy quotient_ is a [[quotient]] (say of a [[group]] [[action]]) in the context of _[[homotopy theory]]_.

Just as a [[quotient]] is a special case of [[colimit]], so a homotopy quotient is a special case of [[homotopy colimit]].

The homotopy quotient of a [[group]] [[action]] may be modeled by the corresponding  _[[action groupoid]]_, which in the context of [[higher geometry]] means the corresponding  _[[quotient stack]]_.

## Related concepts

* [[subquotient]]

* [[quotient space]], [[geometric invariant theory]]

* [[action groupoid]], [[quotient stack]], [[global quotient orbifold]]

* [[BRST complex]]

* In [[type theory]]/[[homotopy type theory]] the analogous concept is that of [[quotient types]].

* [[derived intersection]]

[[!redirects weak quotient]]

[[!redirects homotopy quotients]]


homotopy realization

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Limits and colimits
+--{: .hide}
[[!include infinity-limits - contents]]
=--
#### $(\infty,1)$-Category theory
+--{: .hide}
[[!include quasi-category theory contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

__Homotopy realizations__ are a special case of [[homotopy colimits]],
when the indexing diagram is $\Delta^{op}$,
the [[opposite category]] of the [[category of simplices]].

Homotopy realizations can be defined in any [[relative category]],
just like [[homotopy colimits]],
but practical computations are typically carried out
in presence of additional structures such as [[model structures]],
in fact, [[enriched model categories]] are the most common setup.

## Computation

In any $V$-[[enriched model category]], the homotopy realization
of a [[simplicial object]]
$$X\colon \Delta^{op}\to C$$
can be computed in three different ways,
all of which use the notion of [[tensor product of functors]] (i.e., [[weighted colimits]]).

Specifically, consider the [[functor]]
$$\otimes\colon V^\Delta \times C^{\Delta^{op}} \to C$$
that takes the [[tensor product of functors]], i.e., the [[weighted colimit]],
where $V$ is the [[monoidal model category]] over which $C$ is enriched.

If we set the first argument to the [[constant functor]]
with value $1$ (the [[monoidal unit]] of $V$),
then the resulting functor is the [[colimit functor]] $C^{\Delta^{op}}\to C$.

The functor $\otimes$ becomes a [[left Quillen bifunctor]]
if we equip $V^\Delta$ and $C^{\Delta^{op}}$ with one
of the three following pairs of model structures:

* injective and projective;
* projective and injective;
* Reedy and Reedy.

Accordingly, the homotopy realization of a [[simplicial object]] $X\colon \Delta^{op}\to C$ can be computed as follows.

* Cofibrantly resolve the constant weight $1$ in one of the three model structures listed above.
* Cofibrantly resolve $X$ in the other model structure in the same pair.
* Compute $Q 1 \otimes Q X$, which is the homotopy realization of $X$.

Cofibrant resolutions in the [[injective model structure]]
can be computed by applying some [[cofibrant resolution functor]] of $C$ objectwise.

Cofibrant resolutions in the [[Reedy model structure]]
can be computed inductively, by repeatedly factoring the [[latching map]] of $X$
as a [[cofibration]] followed by a [[weak equivalence]]
and adjusting $X$ accordingly.

Cofibrant resolutions in the [[projective model structure]]
can be computed explicitly in some practical examples.

### Reduction to semisimplicial objects

The inclusion of the
[[category of semisimplices]] (i.e., fininite inhabited totally ordered sets and injective order-preserving maps) into the [[category of simplices]]
(with the injectivity condition dropped) is a [[homotopy initial functor]],
i.e., restricting along the inclusion of opposite categories preserves [[homotopy colimits]].

Thus, homotopy realizations can be computed as
[[homotopy colimits]] over the opposite [[category of semisimplices]].
The latter category is a [[direct category]],
which makes cofibrancy conditions particularly easy.

## Examples

### Simplicial sets

A Reedy [[cofibrant replacement]] of the constant weight $1\colon \Delta\to sSet$
can be computed as the [[Yoneda embedding]] $Y_\Delta\colon\Delta\to sSet$.

The [[Reedy model structure]] on [[simplicial objects]] in [[simplicial sets]]
with [[simplicial weak equivalences]]
coincides with the [[injective model structure]],
as explained in the article [[elegant Reedy category]].
In particular, all objects are [[cofibrant]].

Thus, the homotopy realization of
$$X\colon \Delta^{op} \to sSet$$
can be computed as
$$Y_\Delta\otimes X,$$
which is isomorphic to the [[diagonal]] of $X$.

### Chain complexes of abelian groups

For [[chain complexes]] with [[quasi-isomorphisms]]
(which we equip with the [[injective model structure on chain complexes]]),
a computation analogous to the one for simplicial sets above
Reedy cofibrantly resolves the constant weight as
$$\mathrm{N} \mathbf{Z} [-]\colon \Delta \to Ch,$$
where $\mathrm{N}$ denotes the [[normalized chains functor]]
and $\mathbf{Z}[-]$ denotes the [[free simplicial abelian group functor]].

Once again, all [[simplicial objects]] in [[chain complexes]] are Reedy cofibrant.

Thus, the homotopy realization of $X\colon \Delta^{op} \to Ch$
can be computed as
$$\mathrm{N} \mathbf{Z} [-] \otimes X,$$
which is isomorphic to the [[direct sum]] [[total complex]]
of the [[double chain complex]]
obtained by applying the [[Dold–Kan correspondence]]
to the [[simplicial object]] $X\colon \Delta^{op}\to Ch$.

### Topological spaces

Consider [[topological spaces]] with [[weak homotopy equivalences]].
Below, we use the [[Serre model structure]].

The [[topological simplex]] $\mathbf{\Delta}\colon \Delta\to Top$ is Reedy cofibrant as
a [[cosimplicial topological space]].

Not all [[simplicial objects]] in [[topological spaces]] are Reedy [[cofibrant]],
since the [[latching map]] need not be a [[cofibration of topological spaces]],
i.e., a [[retract]] of a relative cellular map.

However, we can pass to the semisimplicial setting, as explained above.
In this, case Reedy cofibrancy boils down to the objectwise cofibrancy.

Thus, the homotopy realization of $X\colon \Delta^{op} \to Top$
can be computed as
$$\mathbf{\Delta}\otimes Q X,$$
where $Q$ denotes objectwise [[cofibrant replacement]].
This is precisely the classical [[fat geometric realization]]
of [[simplicial topological spaces]].

## Related notions

* [[realization]]

* [[homotopy colimit]]

* [[homotopy coend]]

* [[homotopy pushout]]

* [[homotopy coproduct]]

* [[homotopy coequalizer]]

* [[homotopy totalization]]

## References

For the case of [[chain complexes]], see

* [[Kensuke Arakawa]], _Homotopy Limits and Homotopy Colimits of Chain Complexes_, [arXiv:2310.00201](https://arxiv.org/abs/2310.00201).

[[!redirects homotopy realizations]]


homotopy relative boundary

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

## For paths

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

Let $X$ be a [[topological space]] and let

$$
  \gamma_1, \gamma_2 \;\colon\; [0,1] \longrightarrow X
$$

be two [[paths]] in $X$, i.e. two [[continuous functions]] from the [[closed interval]] to $X$,
such that their endpoints agree:

$$
  \gamma_1(0) = \gamma_2(0)
  \phantom{AAAA}
  \gamma_1(1) = \gamma_2(1)
  \,.
$$

Then a [[homotopy relative boundary]] from $\gamma_1$ to $\gamma_2$ is a [[homotopy]] (def. \ref{LeftHomotopy})

$$
  \eta \;\colon\; \gamma_1 \Rightarrow \gamma_2
$$

such that it does not move the endpoints:

$$
  \eta(0,-) = const_{\gamma_1(0)} = const_{\gamma_2(0)}
  \phantom{AAAAAA}
  \eta(1,-) = const_{\gamma_1(1)} = const_{\gamma_2(1)}
  \,.
$$

=--

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

Let $X$ be a [[topological space]] and let $x, y \in X$ be two points. Write

$$
  P_{x,y} X
$$

for the set of [[paths]] $\gamma$ in $X$ with $\gamma(0) = x$ and $\gamma(1) = y$.

Then [[homotopy relative boundary]] (def. \ref{PathHomotopyRelativeBoundary})
is an [[equivalence relation]] on $P_{x,y}X$.

The corresponding set of [[equivalence classes]] is denoted

$$
  Hom_{\Pi_1(X)}(x,y)
  \;\coloneqq\;
  (P_{x,y}X)/\sim
  \,.
$$

The operation of [[path concatenation]] descends to these equivalence
classes, so that for all $x,y, z \in X$ there is a function

$$
  Hom_{\Pi_1(X)}(x,y)
   \times
  Hom_{\Pi_1(X)}(y,z)
    \longrightarrow
  Hom_{\Pi_1(X)}(x,z)
  \,.
$$

Moreover, this composition operation is [[associativity|associative]] in the evident sense.

Set of points of $X$ together with the set $Hom_{\Pi_1(X)}(x,y)$ for all points of points
and equipped with this composition operation is called the _[[fundamental groupoid]]_ of $X$, denoted

$$
  \Pi_1(X)
  \,.
$$

If we pick a single point $x \in X$, then one writes

$$
  \pi_1(X,x)
  \;\coloneqq\;
  Hom_{\Pi_1(X)}(x,x)
  \,.
$$

Under the above composition this is a [[group]], and as such is called the _[[fundamental group]]_
of $X$ at $x$.

=--

[[!redirects homotopies relative boundary]]


homotopy spectral sequence

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

#Contents#
* table of contents
{:toc}

## Idea

The notion of _homotopy spectral sequence_ adapts the notion of _[[spectral sequence]]_ from [[homological algebra]]/[[stable homotopy theory]] to unstable/[[non-abelian cohomology|non-abelian]] [[homotopy theory]].

Where a [[spectral sequence]] consists of [[objects]] and [[morphisms]] in an [[abelian category]] -- which typically come from the ([[cohomology group|co-]])[[homology groups]] of a ([[cochain complex|co-]])[[chain complex]] or generally from the  [[homotopy groups]] of a [[stable homotopy type]] -- the notion of homotopy spectral sequence is modeled on the [[homotopy groups]] of ([[pointed object|pointed]]) [[topological spaces]]/[[homotopy types]] which in the lowest two degrees consist instead of [[nonabelian groups]] and ([[pointed sets|pointed]]) [[sets]].

## Examples

The classical and motivating examples of a homotopy spectral sequence is the [[Bousfield-Kan spectral sequence]] which computes [[homotopy groups]] of a [[topological space]]/[[simplicial set]]/[[homotopy type]] realized as the [[totalization]] of a [[cosimplicial homotopy type]]. It may be regarded as the unstable analog of the [[Adams spectral sequence]], which computes homotopy groups of certain [[spectra]]. The corresponding [[spectral sequence]] for [[homology groups]] of the totalization of a [[cosimplicial homotopy type]] is the [[Eilenberg-Moore spectral sequence]].

## References

The [[Bousfield-Kan spectral sequence]] was introduced and originally studied in

* [[Aldridge Bousfield]], [[Daniel Kan]], _The homotopy spectral sequence of a space with coefficients in a ring_. Topology, 11, pp. 79–106, 1972.

* [[Aldridge Bousfield]], [[Daniel Kan]], _A second quadrant homotopy spectral sequence_, Transactions of the American Mathematical Society Vol. 177, Mar., 1973

* [[Aldridge Bousfield]], [[Daniel Kan]], _Pairings and products in the homotopy
spectral sequence_. Transactions of the American Mathematical Society, 177,
pp. 319–343, 1973.

* [[Aldridge Bousfield]], _Homotopy Spectral Sequences and Obstructions_, Isr. J. Math. 66 (1989), 54-104.

An early textbook account of this work is in

* [[Aldridge Bousfield]], [[Daniel Kan]], _Homotopy limits, completions and localization_, Springer 1972

A general abstract category-theoretic discussion is in

* [[Marco Grandis]], _Homotopy spectral sequences_ ([arXiv:1007.0632](http://arxiv.org/abs/1007.0632))

Lecture notes on the [[Bousfield-Kan spectral sequence]] include

* Bertrand Guillo, _The Bousfield-Kan spectral sequence_ [pdf](http://www.ms.uky.edu/~guillou/BKss.pdf)

Discussion of this using [[effective homology]] includes

* Romero Ibanez, _Effective homology and spectral sequences_, 2007 ([pdf](http://www.unirioja.es/cu/anromero/zgz07.pdf))

Discussion in [[homotopy type theory]] is in

* {#Shulman13} [[Mike Shulman]], _Spectral sequences_ 2013 ([part I](https://golem.ph.utexas.edu/category/2013/08/what_is_a_spectral_sequence.html), [part II](http://homotopytypetheory.org/2013/08/08/spectral-sequences/))

[[!redirects homotopy spectral sequences]]


homotopy sphere

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

#Contents#
* table of contents
{:toc}

## Idea

A _homotopy sphere_ is a [[topological space]] which need not be [[homeomorphism|homeomorphic]] to an [[n-sphere]], but which has the same [[homotopy type]] as an $n$-sphere.

## Properties

* Every [[smooth manifold]] homotopy-[[4-sphere]] is [[homeomorphism|homeomorphic]] to the actual [[4-sphere]].

  (due to [[Michael Freedman]], see [Siebenmann](#Siebenmann))

## Related concepts

* [[homology sphere]]

## Redirects

* Wikipedia, _[Homotopy sphere](https://en.wikipedia.org/wiki/Homotopy_sphere)_

## References

* {#Siebenmann} Laurent Siebenmann, _Topological Poincaré conjecture in dimension 4 (the work of M. H. Freedman)_ (pdf)

[[!redirects homotopy spheres]]

[[!redirects homotopy n-sphere]]
[[!redirects homotopy n-spheres]]


homotopy T-algebra

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

#Contents#
* table of contents
{:toc}

## Idea

A _homotopy $T$-algebra_ over a [[Lawvere theory]] $T$ is a model for an $\infty$-algebra over $T$, when the latter is regarded as an [[(∞,1)-algebraic theory]].

As a model, homotopy $T$-algebras are equivalent to strict [[simplicial algebra]]s.

## Definition

For $T$ (the syntactic category of) a [[Lawvere theory]] with generating object $x$ an ordinary [[algebra over a Lawvere theory]]  [[functor]] $T \to Set$ that preserves products, in that for all $n \in \mathbb{N}$ the canonical morphism

$$
  \prod_{i = 1}^n A(p_i) :  A(x^n) \to (A(x))^n
$$

is an [[isomorphism]].

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

A **homotopy $T$-algebra** is a functor $A : T \to $ [[sSet]] with values in [[Kan complex]]es such that for all $n \in \mathbb{N}$ this canonical morphism is a [[weak homotopy equivalence]].

=--

For $n \in \mathbb{N}$ write $F_T(n)$ for the free simplicial $T$-algebra on $n$-generators, which is the image of $x^n$ under the [[Yoneda embedding]] $j : T^{op} \to [T,sSet]$. (See [[Lawvere theory]] for more on this.)

+-- {: .un_prop}
###### Proposition
A homotopy $T$-algebra is precisely

* a fibrant object in the projective [[model structure on simplicial presheaves]];

* which is a [[local object]] with respect to the canonical morphisms

  $$
    \coprod F_T(1) \to F_T(n)
  $$

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

=--

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

The fibrant objects in $[T,sSet]_{proj}$ are precisely the [[Kan complex]]-valued co-presheaves. Because $F_T(n)$ is [[representable functor|representable]], it is cofibrant in $[T,sSet]_{proj}$ (as one easily checks). Therefore the [[derived hom-space]]s between $F_T(\cdots)$ and a degreewise Kan complex-valued $A$ may be computed simply as the [[sSet]]-[[hom-object]]s of the [[simplicial model category]] $[T,sSet]$ and so the degreewise fibrant $A$ being a [[local object]] means that all morphisms of [[sSet]]-[[hom-object]]s

$$
  [T,sSet](F_T(n),A) \to [T,sSet](\coprod_n F_T(1), A)
  \,.
$$

Due to the respect of the [[hom-functor]] for [[limit]]s the expression on the right is

$$
  \cdots = \prod_n [T,sSet](F_T(1), A)
  \,.
$$

Using the [[Yoneda lemma]] the morphism in question is indeed isomorphic to

$$
  A(x^n) \to A(x)^n
  \,.
$$

=--

This observation motivated the following definition.

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

The **[[model category]] structure for homotopy $T$-algebras** is the left [[Bousfield localization of model categories|Bousfield localization]] $[T,sSet]_{proj,loc}$ of the projective [[model structure on simplicial presheaves]] $[T,sSet]_{proj}$ at the set of morphisms $\{\coprod_n F_T(1) \to F_T(b)\}_{n \in \mathbb{N}}$.

=--

## Properties

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

The model structure for homotopy $T$-algebra $[T,sSet]_{proj,loc}$ is a [[left proper model category|left proper]] [[simplicial model category]].

=--

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

Because the [[model structure on simplicial presheaves]] is and left [[Bousfield localization of model categories]] preserves these properties.

=--

+-- {: .un_lemma}
###### Lemma

The inclusion

$$
  i : T Alg^{\Delta^{op}} \hookrightarrow [T,sSet]
$$

has a [[left adjoint]]

$$
  F : [T,sSet] \to T Alg^{\Delta^{op}}
$$

=--

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

The [[limit]]s in $T Alg$ are easily seen to be limits in the underlying sets. Hence $i$ preserves all limits. The statement then follows by observing that the assumptions of the special [[adjoint functor theorem]] are met:

* $T Alg$ is complete;

* it is a [[well powered category]] since $[T,Set]$ is and the subobject in $T Alg$ are special subobjects in $[T,Set]$;

* it has a small [[cogenerating set]] given by the representables.

=--

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

An explicit description of $F$ is around [[Higher Topos Theory|HTT, lemma 5.5.9.5]].

=--

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

Let $T Alg^{\Delta^{op}}_{proj}$ be the category of [[simplicial object|simplicial]] [[T-algebra]]s equipped with the standard [[model structure on simplicial algebras]] (with weak equivalences and fibrations the degreewise weak equivalences and fibrations in simplicial sets).

The adjunction from the previous lemma

$$
  T Alg^{\Delta^{op}} \stackrel{\overset{F}{\leftarrow}}{\hookrightarrow}
  [T,sSet] = [T,Set]^{\Delta^{op}}
$$

is a [[Quillen adjunction]] which is a [[Quillen equivalence]]

$$
  T Alg^{\Delta^{op}}_{proj} \simeq
  [T,sSet]_{proj,loc}
  \,.
$$

=--

This is theorem 1.3 in ([Badzioch](#Badzioch))

## Examples

The model structure on homotopy $T$-algebras for $T = $ [[CartSp]] the Lawvere theory of [[smooth algebra]]s is considered in ([Spivak](#Spivak)) in the study of [[derived smooth manifold]]. (There is also a bit of disucssion of the relation to the model structure on simplicial algebras there.)

## Related concepts

* [[algebra over a monad]]

  [[∞-algebra over an (∞,1)-monad]]

* [[algebra over an algebraic theory]]

  [[∞-algebra over an (∞,1)-algebraic theory]]

  * **homotopy T-algebra** / [[model structure on simplicial T-algebras]]

* [[algebra over an operad]]

   [[∞-algebra over an (∞,1)-operad]]

   * [[model structure on algebras over an operad]]

## References

In

* [[Bernard Badzioch]], _Algebraic theories in homotopy theory_ Annals of Mathematics, 155 (2002), 895-913 ([JSTOR](http://www.jstor.org/stable/3062135))
{#Badzioch}

the model structure on homotopy $T$-algebras is discussed and its Quillen equivalence to simplicial $T$-algebras is proven.

A related discussion showing that simplicial $T$ algebras model all $\infty$-$T$-algebras is in

* [[Julie Bergner]], _Rigidification of algebras over multi-sorted theories_, Algebraic and Geometric Topology 7, 2007.
{#Bergner}

The model structure on homotopy $T$-algebras for $T = $ [[CartSp]] the Lawvere theory of [[smooth algebra]]s is considered in

* [[David Spivak]], _Derived smooth manifolds_ ([arXiv:0810.5174](http://arxiv.org/abs/0810.5174))
{#Spivak}

[[!redirects homotopy T-algebras]]

[[!redirects model structure on homotopy T-algebras]]


Homotopy theories and model categories

This page collects material related to

* [[William Dwyer]], [[Jan Spalinski]],

  _Homotopy theories and model categories_

  [pdf](https://math.jhu.edu/~eriehl/616-s16/DwyerSpalinski.pdf), [[DwyerSpalinski_HomotopyTheories.pdf:file]]

  in:

  [[Ioan Mackenzie James]],

  _[[Handbook of Algebraic Topology]]_

  North Holland 1995

   [ISBN:9780080532981](https://www.elsevier.com/books/handbook-of-algebraic-topology/james/978-0-444-81779-2)

  [doi:10.1016/B978-0-444-81779-2.X5000-7](https://doi.org/10.1016/B978-0-444-81779-2.X5000-7)

on [[homotopy theory]] and [[model category]] theory,

category: reference


homotopy theory

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

#Contents#
* table of contents
{:toc}

## Idea
 {#Idea}

### General idea
 {#GeneralIdea}

In generality, _homotopy theory_ is the study of mathematical contexts in which [[function|functions]] or rather ([[homomorphism|homo]]-)[[morphism|morphisms]] are equipped with a concept of _[[homotopy]]_ between them, hence with a concept of "equivalent [[deformations]]" of morphisms,

\begin{imagefromfile}
    "file_name": "2Cell.jpg",
    "width": 200,
    "unit": "px",
    "margin": {
        "top": -30,
        "bottom": 10,
        "right": 0,
        "left": 10
    }
\end{imagefromfile}

and then iteratively with [[homotopies of homotopies]] between those:

\begin{imagefromfile}
    "file_name": "3Cell.jpg",
    "width": 215,
    "unit": "px",
    "margin": {
        "top": -30,
        "bottom": 0,
        "right": 0,
        "left": 10
    }
\end{imagefromfile}

and so forth.

A key aspect here is that in such homotopy theoretic contexts the concept of _[[isomorphism]]_ is relaxed to that of _[[homotopy equivalence]]_: Where a [[morphism]] is regarded as [[inverse morphism|invertible]] if there is a reverse function such that both [[composition|composites]] are _[[equality|equal]]_ to the [[identity morphism]], for a [[homotopy equivalence]] one only requires the composites to be [[homotopy|homotopic]] to the identity. Regarding objects in a homotopical context up to [[homotopy equivalence]] this way is to regard them as _[[homotopy types]]_.

It is not wrong to summarize this by saying that:

> *Homotopy theory embodies the [[gauge principle]] in mathematics*.

Namely, in [[physics]], the [[gauge principle]] says that it is wrong to ask for any two [[field history|things]] to be [[equal]], instead one always has to ask whether there is a [[gauge transformation]] relating them, and then a [[gauge-of-gauge transformation]] relating these, etc. Similarly in homotopy theory it is wrong to ask whether any two objects are equal, instead one has to ask whether there is a [[homotopy equivalence]] between them and then [[higher homotopies]] between these, etc.

\linebreak

For more exposition on the general idea of homotopy theory see:

* *[[schreiber:Higher Structures]]*

\linebreak

### Topological homotopy theory
 {#TopologicalHomotopyTheoryInIdeaSection}

\begin{imagefromfile}
    "file_name": "AHomotopy.jpg",
    "float": "right",
    "width": 380,
    "unit": "px",
    "margin": {
        "top": -40,
        "bottom": 20,
        "right": 0,
        "left": 10
    }
\end{imagefromfile}

The archetypical example of a homotopy theory is the [[classical model structure on topological spaces|classical homotopy theory of topological spaces]], where one considers [[topological spaces]] with [[continuous functions]] between them, and with the original concept of [topological homotopies](Introduction+to+Topology+--+2#LeftHomotopy) between these continuous functions.

The [[category]] whose [[objects]] are [[topological spaces]] and whose [[morphisms]] are [[homotopy equivalence]]-[[equivalence class|classes]] of [[continuous functions]] is also called the _[[classical homotopy category]]_.

Classical constructions in topological homotopy theory are *[[homotopy groups]]*, *[[Toda brackets]]*, *[[long exact sequences of homotopy groups]]*, ...

This [[classical model structure on topological spaces|classical homotopy theory of topological spaces]] has many applications, for example to _[[covering space]] theory_, to _[[classifying space]] theory_, to _[[Whitehead-generalized cohomology theory]]_ and many more. (See also at _[[shape theory]]_.) Accordingly, homotopy theory has a large overlap with _[[algebraic topology]]_.

\linebreak

For more exposition of topological homotopy theory see:

* _[[Introduction to Topology -- 2|Introduction to Basic Homotopy Theory]]_

\linebreak

### Abstract homotopy theory

The basic structures seen in  [[classical model structure on topological spaces|classical homotopy theory of topological spaces]] may be abstracted to yield an "abstract homotopy theory" that applies to a large variety of contexts. There are several more or less equivalent formalizations of the concept of "abstract homotopy theory", including

* _[[model categories]]_

* _[[(∞,1)-categories]]_

* _[[homotopy type theory|homotopy type theories]]_.

* _[[higher observational type theories]]_.

The terminology _[[model category]]_ is short for "category of models of homotopy types". The idea here is to consider [[categories]] equipped with suitable [[stuff, structure, property|extra structure and properties]] that encodes the existence of [[homotopies]] between all [[morphisms]] and convenient means to handle and control these, in particular a means to construct the corresponding [[homotopy category]].

\linebreak

For a detailed introduction to abstract homotopy theory via model cateories see:

* _[[Introduction to Homotopy Theory]]_,

* _[[geometry of physics -- homotopy types]]_.

\linebreak

The approach of [[(∞,1)-categories]] to homotopy theory is meant to be more truthful to the intrinsic nature of homotopy theory. Instead of equipping an ordinary category with a extra concept of homotopy between its morphisms, here one regards the resulting structure as a [[higher category]] where the [[homotopies]] themselves appear as a kind of higher order morphisms, called _[[2-morphisms]]_ and where higher [[homotopies of homotopies]] are regarded as _[[k-morphisms]]_ for all $k$.

The terminology "[[(∞,1)-category]]" signifies that homotopy theory is but one special case of general [[higher category theory]], namely [[(∞,1)-categories]] (hence homotopy theories) are those [[infinity-categories]] in which all [[k-morphisms]] for $k \gt 1$ are [[invertible morphisms|invertible]] up to [[homotopy]]. If one drops this constraint, so that homotopies become "directed" then one might still speak of "[[directed homotopy theory]]".

The archetypical example of an [[(∞,1)-category]] is the $(\infty,1)$-category _[[∞Grpd]]_ of [[∞-groupoids]], just as [[Set]] is the archetypical 1-[[category]].

This turns out to be equivalent, as homotopy theories, to the [[classical model structure on topological spaces|classical homotopy theory of topological spaces]] if restricted to those that admit the structure of [[CW-complexes]].

\linebreak

### Relation between topological and abstract homotopy theory: Cohesion.
{#RelationBetweenTopologicalAndAbstractHomotopyTheoryInIdeaSection}

The case of the homotopy theory of topological spaces ([above](#TopologicalHomotopyTheoryInIdeaSection))
is so classical that often the distinction between [[topology]] and homotopy theory has been (certainly so in the early literature) and still is often blurred (for instance in textbook titles or lecture notes). A transparent conceptual way to understand how [[topology]] and homotopy theory are closely related and yet different is provided by the notion of [[cohesion]]:

A [[topological space]] $X$ *has* (or *presents*) a [[homotopy type]] through the data that is retained in its [[path ∞-groupoid]], conveniently modeled by its [[singular simplicial set]] $Sing(X)$ (which is a [[Kan complex]] and as such an [[∞-groupoid]]).
But the topological space contains more information than that retained in its [[path ∞-groupoid]] -- the latter is just a shadow of it, called the *[[shape modality|shape]]* of the topological space.

The extra information in a topological space is the [[cohesion]] on its [[underlying]] [[set]] of points: The [[geometry|geometric]] information of how these [[points]] stick together within [[open subsets]]. This information is lost when one regards topological spaces as stand-ins for their [[shape modality|shapes]] ([[path ∞-groupoids]]) in plain homotopy theory.

More formally:

The [[Top|category of]] [[D-topological spaces]] is a [[full sub-(∞,1)-category|full subcategory]] of the [[cohesive (infinity,1)-topos|cohesive $(\infty,1)$-topos]] of [[D-topological infinity-groupoid|D-topological $\infty$-groupoids]] (also of [[smooth infinity-groupoids|smooth $\infty$-groupoids]]). These carry a [[modal homotopy type theory|qualitative aspect]] called their **[[shape modality|shape]]**, which is a [[shape via cohesive path ∞-groupoid|generalization]] of the [[fundamental ∞-groupoid]]-construction. Now, regarding a [[topological space]] as an object in (plain) [[homotopy theory]], as traditionally done, really means to first regard it as a [[topological groupoid]] in [[cohesive (∞,1)-topos|cohesive homotopy theory]] and then, as such, to retain only its [[shape modality|shape]]. The result is the *[[homotopy type]]* which is "presented" by the topological space:

\begin{imagefromfile}
    "file_name": "RelationTopologyHomotopyTheory_20210921.jpg",
    "width": 700,
    "unit": "px",
    "margin": {
        "top": -20,
        "bottom": 40,
        "right": 0,
        "left": 10
    }
\end{imagefromfile}

\linebreak

### Simplicial homotopy theory

The most immediate way to model an [[∞-groupoid]] is as a [[simplicial set]] which is a [[Kan complex]].  Accordingly, another homotopy theory equivalent to archetypical homotopy theory of [[∞-groupoids]] is the [[classical model structure on simplicial sets|classical homotopy theory of simplicial sets]], typically referred to as _[[simplicial homotopy theory]]_.

\linebreak

### Flavors of homotopy theory

But there are many other homotopy theories besides (the various incarnations of) this classical one. Important sub-classes of homotopy theories include:

* _[[stable homotopy theory]]_ modeled by [[stable model categories]] and [[stable (∞,1)-categories]], where the operations of [[looping and delooping]] are an [[equivalence of (∞,1)-categories|equivalence]]. This includes the [[stable (∞,1)-category of spectra]] as well as those of [[sheaves of spectra]].

* _[[geometric homotopy theory]]_ modeled by [[model toposes]] and [[(∞,1)-toposes]], which are the homotopy theoretic analogs of ordinary [[toposes]]. This includes the  homotopy theoretic analog of [[categories of sheaves]], called _[[(∞,1)-categories of (∞,1)-sheaves]]_ or _of [[∞-stacks]]_, but it potentially also contains "[[elementary (∞,1)-toposes]]".

The [[geometric homotopy theory]] of [[(∞,1)-toposes]] in particular serves as the foundation for [[higher geometry]]/[[derived geometry]]. This is relevant notably in the [[physics]] of [[gauge theory]], where [[gauge transformations]] are identified with [[homotopies]] in [[geometric homotopy theory]]. For more on this see at _[[geometry of physics -- homotopy types]]_.

On the other hand, the incarnation of homotopy theory as _[[homotopy type theory]]_ exhibits the remarkably [[foundation|foundational]] nature of homotopy theory. Contrary to its original appearance as a fairly complicated-looking theory built on top of classical [[set theory]] and classical [[topology]], homotopy theory turns out to be intrinsically simple: it arises from plain [[dependent type theory]] just by adopting a fully [[constructive mathematics|constructive]] attitude towards the concept of [[identity]]/[[equality]], see at _[[identity type]]_ for more on this. For exposition of this perspective see ([Shulman 17](#Shulman17)).

\linebreak

## Presentations ##

A convenient, powerful, and traditional way to deal with [[(∞,1)-categories]] is to "present" them by 1-categories with specified classes of morphisms called _[[weak equivalence|weak equivalences]]_ : a [[category with weak equivalences]] or [[homotopical category]].  The idea is as follows.  Given a category $C$ with a class $W$ of weak equivalences, we can form its **[[homotopy category]]** or **[[calculus of fractions|category of fractions]]** $C[W^{-1}]$ by adjoining formal inverses to all the morphisms in $W$.  The [[simplicial localization|(∞,1)-category presented by]] $(C,W)$" can be thought of as the result of regarding $C$ as an $\infty$-category with only identity $k$-cells for $k\gt 1$, then adjoining formal inverses to morphisms in $W$ in the $\infty$-categorical sense; that is, making them into [[equivalence|equivalences]] rather than isomorphisms.  It is remarkable that most $(\infty,1)$-categories that arise in mathematics can be presented in this way.

As with presentations of [[group]]s and other algebraic structures, very different presentations can give rise to equivalent $(\infty,1)$-categories.  For example, several different presentations of the $(\infty,1)$-category of $\infty$-groupoids are:

 * $C=$ [[CW-complex]]es, $W=$ [[homotopy equivalence]]s - see _[[Quillen model structure on topological spaces]]_
 * $C=$ [[topological space]]s, $W=$ [[weak homotopy equivalences]]
 * $C=$ [[Kan complex]]es, $W=$ simplicial homotopy equivalences
 * $C=$ [[simplicial set]]s, $W=$ weak homotopy equivalences - see _[[Quillen model structure on simplicial sets]]_
 * $C=$ [[small category|small categories]], $W=$ functors whose nerves are weak homotopy equivalences -- see _[[Thomason model structure]]_.

The latter three can hence be regarded as providing "combinatorial models" for the homotopy theory of topological spaces.

### Model Categories

The value of working with presentations of $(\infty,1)$-categories rather than the $(\infty,1)$-categories themselves is that the presentations are ordinary 1-categories, and thus much simpler to work with.  For instance, ordinary limits and colimits are easy to construct in the category of topological spaces, or of simplicial sets, and we can then use these to get a handle on $(\infty,1)$-categorical limits and colimits in the $(\infty,1)$-category of $\infty$-groupoids.  However, we always have to make sure that we use only 1-categorical constructions that are _homotopically meaningful_, which essentially means that they induce $(\infty,1)$-categorical meaningful constructions in the presented $(\infty,1)$-category.  In particular, they must be invariant under weak equivalence.

Most presentations of $(\infty,1)$-categories come with additional classes of morphisms, called _fibrations_ and _cofibrations_, that are very useful in performing constructions in a homotopically meaningful way.  Quillen defined a [[model category]] to be a 1-category together with classes of morphisms called weak equivalences, cofibrations, and fibrations that fit together in a very precise way (the term is meant to suggest "a category of models for a homotopy theory").  Many, perhaps most, presentations of $(\infty,1)$-categories are model categories.  Moreover, even when we do not have a model category, we often have classes of cofibrations and fibrations with many of the properties possessed by cofibrations and fibrations in a model category, and even when we do have a model category, there may be classes of cofibrations and fibrations, different from those in the model structure, that are useful for some purposes.

Unlike the weak equivalences, which determine the "homotopy theory" and the $(\infty,1)$-category that it presents, fibrations and cofibrations should be regarded as _technical tools_ which make working directly with the presentation easier (or possible).  Whether a morphism is a fibration or cofibration has no meaning after we pass to the presented $(\infty,1)$-category.  In fact, _every_ morphism is weakly equivalent to a fibration and to a cofibration.  In particular, despite the common use of double-headed arrows for fibrations and hooked arrows for cofibrations, they do not correspond to $(\infty,1)$-categorical epimorphisms and monomorphisms.

In a model category, a morphism which is both a fibration and a weak equivalence is called an **acyclic fibration** or a **trivial fibration**.  Dually we have **acyclic** or **trivial cofibrations**. An object $X$ is called **cofibrant** if the map $0\to X$ from the initial object to $X$ is a cofibration, and **fibrant** if the map $X\to 1$ to the terminal# object is a fibration.  The axioms of a model category ensure that for every object $X$ there is an acyclic fibration $Q X \to X$ where $Q X$ is cofibrant and an acyclic cofibration $X\to R X$ where $R X$ is fibrant.

### Examples

* [[topological homotopy theory]], [[classical model structure on topological spaces]],

* [[simplicial homotopy theory]], [[classical model structure on simplicial sets]]

* [[localic homotopy theory]]

For a (higher) category theorist, the following examples of model categories are perhaps the most useful to keep in mind:

 * $C=$ sets, $W=$ isomorphisms.  _All_ morphisms are both fibrations and cofibrations.  The $(\infty,1)$-category presented is again the 1-category $Set$.
 * $C=$ categories, $W=$ equivalences of categories.  The cofibrations are the functors which are injective on objects, and the fibrations are the [[isofibration|isofibrations]].  The acyclic fibrations are the equivalences of categories which are literally surjective on objects.  Every object is both fibrant and cofibrant.  The $(\infty,1)$-category presented is the 2-category $Cat$.  This is often called the _folk model structure_.
 * $C=$ (strict) 2-categories and (strict) 2-functors, $W=$ 2-functors which are [[equivalence of categories|equivalences of bicategories]].  The fibrations are the 2-functors which are isofibrations on hom-categories and have an equivalence-lifting property.  Every object is fibrant; the cofibrant 2-categories are those whose underlying 1-category is freely generated by some directed graph.  The $(\infty,1)$-category presented is the (weak) 3-category $2Cat$.  This model structure is due to Steve Lack.

### Generalized Morphisms

The morphisms from $A$ to $B$ in the $(\infty,1)$-category presented by $(C,W)$ are zigzags
$
  \stackrel{\simeq}{\leftarrow}
  \to
  \stackrel{\simeq}{\leftarrow}
  \to
  \cdots
$; these are sometimes called **generalized morphisms**.  Many presentations (including every model category) have the property that any such morphism is equivalent to one with a single zag, as in $\stackrel{\simeq}{\leftarrow} \to \stackrel{\simeq}{\leftarrow}$.  In a model category,  a canonical form for such a zigzag is
$X \stackrel{\simeq}{\leftarrow} Q X \to  R Y \stackrel{\simeq}{\leftarrow} Y$
where $Q X$ is cofibrant and $R Y$ is fibrant.  In this case we can moreover take $Q X\to X$ to be an acyclic fibration and $Y\to R Y$ to be an acyclic cofibration.

Often it suffices to consider even shorter zigzags of the form $\stackrel{\simeq}{\leftarrow} \to$ or $\to \stackrel{\simeq}{\leftarrow}$.  In particular, this is the case if every object is fibrant or every object is cofibrant.  For example:

 * If $X$ and $Y$ are strict 2-categories, then pseudofunctors $X\to Y$ are equivalent to strict 2-functors $Q X \to Y$, where $Q X$ is a cofibrant replacement for $X$.
 * [[anafunctor|anafunctors]] are zigzags $\stackrel{\simeq}{\leftarrow} \to$ in the [[folk model structure]] on 1-categories whose first factor is an acyclic (i.e. surjective) fibration.
 * [[Morita equivalence|Morita morphisms]] in the theory of [[Lie groupoid|Lie groupoids]] are generalized morphisms of length one where both maps are acyclic fibrations.

If $X$ is cofibrant and $Y$ is fibrant, then every generalized morphism from $X$ to $Y$ is equivalent to an ordinary morphism.  For example, if $X$ is a cofibrant 2-category, then every pseudofunctor $X\to Y$ is equivalent to a strict 2-functor $X\to Y$

### Quillen Equivalences

Quillen also introduced a highly structured notion of equivalence between model categories, now called a [[Quillen equivalence]], which among other things ensures that they present the same $(\infty,1)$-category.  Quillen equivalences are now being used to compare different definitions of higher categories.

## Related entries
 {#RelatedConcepts}

### General

* [[homotopy theory FAQ]]

* [[higher structures]]

* [[algebraic homotopy]]

* [[homotopy type theory]]

### Flavors of homotopy theory

* [[synthetic homotopy theory]]

* [[simplicial homotopy theory]]

* [[stable homotopy theory]]

* [[proper homotopy theory]]

* [[parameterized homotopy theory]]

* [[equivariant homotopy theory]], [[global equivariant homotopy theory]]

* [[equivariant stable homotopy theory]]

* [[rational homotopy theory]]

* [[rational parameterized stable homotopy theory]]

* [[p-adic homotopy theory]]

* [[chromatic homotopy theory]]

* [[persistent homotopy theory]]

### Basic concepts in homotopy theory

* [[interval object]],

  [[cylinder object]], [[path space object]]

* [[homotopy]], [[homotopy group]]

* [[deformation retract]], [[neighborhood retract]]

### Categorical homotopy theory

* [[category with weak equivalences]]

  [[homotopical category]], [[relative category]]

* [[category of fibrant objects]], [[cofibration category]]

* [[model category]]

(...)

* [[closed monoidal deformation retract]]

* [[closed monoidal homotopical category]]

* [[cylinder functor]], [[cocylinder]]

* [[derived functor]]

* [[deformation retract of a homotopical category]]

* [[Dold fibration]]

* [[equivalence]]

* [[folk model structure]]

* [[fundamental groupoid]]

* [[homological resolution]]

* [[homotopical category]]

* [[homotopical cohomology theory]]

* [[homotopy category]]

* [[homotopy limit]], [[homotopy colimit]]

* [[homotopy coherent category theory]]

* [[Hurewicz fibration]], [[Hurewicz connection]], [[Hurewicz cofibration]]

* [[model structure on crossed complexes]]

* [[model structure on simplicial sets]], [[model structure on dendroidal
sets]]

* [[path object]], [[path groupoid]], [[path n-groupoid]]

* [[test category]]

* [[weak equivalence]]

## References
 {#References}

[[!include homotopy theory and algebraic topology -- references]]

[[!redirects Homotopy Theory]]
[[!redirects homotopy theories]]


Homotopy Theory > history

< [[Homotopy Theory]]
[[!redirects Homotopy Theory -- history]]


homotopy theory and algebraic topology -- references

The following lists basic references on *[[homotopy theory]]*, *[[algebraic topology]]* and some *[[(infinity,1)-category theory|$(\infty,1)$-category theory]]* and *[[homotopy type theory]]*, but see these entries for more pointers.

### Pre-history

Historical article at the origin of all these subjects:

* [[Henri Poincaré]], *[[Analysis Situs]]*, Journal de l'École Polytechnique. (2). 1: 1–123 (1895) ([gallica:12148/bpt6k4337198/f7](https://gallica.bnf.fr/ark:/12148/bpt6k4337198/f7), Engl: [pdf](https://www.maths.ed.ac.uk/~v1ranick/papers/poincare2009.pdf), [[Stillwell_AnalysisSitus.pdf:file]])

On early developments from there, such as the eventual understanding of the notion of higher [[homotopy groups]]:

* [[Peter Hilton]], *Subjective History of Homology and Homotopy Theory*, Mathematics Magazine **61** 5 (1988) 282-291 $[$[doi:10.2307/2689545](https://doi.org/10.2307/2689545)$]$

### Topological homotopy theory
 {#ReferencesTopologicalHomotopyTheory}

Textbook accounts of [[homotopy theory]] of [[topological spaces]] (i.e. via "[[point-set topology]]"):

* {#Hilton53} [[Peter J. Hilton]], *An introduction to homotopy theory*, Cambridge University Press 1953 ([doi:10.1017/CBO9780511666278](https://doi.org/10.1017/CBO9780511666278))

* {#SzeTsen59} [[Sze-Tsen Hu]], *Homotopy Theory*, Academic Press 1959 ([pdf](https://www.maths.ed.ac.uk/~v1ranick/papers/hu2.pdf))

* [[Robert E. Mosher]], [[Martin C. Tangora]], *Cohomology operations and applications in homotopy theory*, Harper & Row, 1968, reprinted by [Dover 2008](https://store.doverpublications.com/0486466647.html) [GoogleBooks](https://www.google.com/books/edition/Cohomology_Operations_and_Applications_i/wu79f-7V_6AC)

* {#Homotopietheorie} [[Tammo tom Dieck]], [[Klaus Heiner Kamps]], [[Dieter Puppe]], *Homotopietheorie*, Lecture Notes in Mathematics **157** Springer 1970 ([doi:10.1007/BFb0059721](https://link.springer.com/book/10.1007/BFb0059721))

* [[Brayton Gray]], *Homotopy Theory: An Introduction to Algebraic Topology*, Academic Press (1975) $[$[978-0-12-296050-5](https://www.sciencedirect.com/bookseries/pure-and-applied-mathematics/vol/64/suppl/C), [pdf](https://www.maths.ed.ac.uk/~v1ranick/papers/gray.pdf)$]$

* [[George W. Whitehead]], *Elements of Homotopy Theory*, Springer 1978 ([doi:10.1007/978-1-4612-6318-0](https://link.springer.com/book/10.1007/978-1-4612-6318-0))

* [[Ioan Mackenzie James]], *General Topology and Homotopy Theory*, Springer 1984 ([doi:10.1007/978-1-4613-8283-6](https://doi.org/10.1007/978-1-4613-8283-6))

* [[Renzo A. Piccinini]], *Lectures on Homotopy Theory*, Mathematics Studies **171**, North Holland 1992 ([ISBN:978-0-444-89238-6](https://www.sciencedirect.com/bookseries/north-holland-mathematics-studies/vol/171/suppl/C))

* {#Bredon93} [[Glen Bredon]], Chapter VII of: _Topology and Geometry_, Graduate texts in mathematics **139**, Springer 1993 ([doi:10.1007/978-1-4757-6848-0](https://link.springer.com/book/10.1007/978-1-4757-6848-0),  [pdf](http://virtualmath1.stanford.edu/~ralph/math215b/Bredon.pdf))

* [[Hans-Joachim Baues]], _Homotopy types_, in [[Ioan Mackenzie James]] (ed.) _[[Handbook of Algebraic Topology]]_, North Holland, 1995 ([ISBN:9780080532981](https://www.elsevier.com/books/handbook-of-algebraic-topology/james/978-0-444-81779-2), [doi:10.1016/B978-0-444-81779-2.X5000-7](https://doi.org/10.1016/B978-0-444-81779-2.X5000-7))

* [[Nicolas Bourbaki]], _Topologie Algébrique_, Chapitres 1 à 4, Springer (1998, 2016)  \[ISBN 978-3-662-49361-8, [doi:10.1007/978-3-662-49361-8](https://doi.org/10.1007/978-3-662-49361-8)\]

* [[Marcelo Aguilar]], [[Samuel Gitler]], [[Carlos Prieto]], *Algebraic topology from a homotopical viewpoint*, Springer (2008) ([doi:10.1007/b97586](https://link.springer.com/book/10.1007/b97586))

* [[Jeffrey Strom]], *Modern classical homotopy theory*, Graduate Studies in Mathematics **127**, American Mathematical Society (2011) $[$[doi:10.1090/gsm/127](http://www.ams.org/books/gsm/127/)$]$

* [[Martin Arkowitz]], *Introduction to Homotopy Theory*, Springer (2011) $[$[doi:10.1007/978-1-4419-7329-0](https://doi.org/10.1007/978-1-4419-7329-0)$]$

* [[Dai Tamaki]], *Fiber Bundles and Homotopy*, World Scientific (2021) $[$[doi:10.1142/12308](https://doi.org/10.1142/12308)$]$

  > (motivated from [[classifying spaces]] for [[principal bundles]]/[[fiber bundles]])

### Algebraic topology
 {#ReferencesAlegbraicTopology}

On [[algebraic topology]]:

Textbooks:

* [[Samuel Eilenberg]], [[Norman Steenrod]], _Foundations of Algebraic Topology_, Princeton University Press 1952 ([pdf](https://www.maths.ed.ac.uk/~v1ranick/papers/eilestee.pdf), [ISBN:9780691653297](https://press.princeton.edu/books/hardcover/9780691653297/foundations-of-algebraic-topology))

* {#Godement58} [[Roger Godement]], *Topologie algébrique et theorie des faisceaux*, Actualités Sci. Ind. **1252**, Hermann, Paris (1958) $[$[webpage](https://www.editions-hermann.fr/livre/topologie-algebrique-et-theorie-des-faisceaux-roger-godement), [[Godement-TopologieAlgebrique.pdf:file]]$]$

* {#Spanier66} [[Edwin Spanier]], _Algebraic topology_, McGraw Hill (1966), Springer (1982) ([doi:10.1007/978-1-4684-9322-1](https://link.springer.com/book/10.1007/978-1-4684-9322-1))

* [[William S. Massey]], *Algebraic Topology: An Introduction*, Harcourt Brace & World 1967, reprinted in: Graduate Texts in Mathematics, Springer 1977 ([ISBN:978-0-387-90271-5](https://link.springer.com/book/9780387902715))

* {#Maunder70} [[C. R. F. Maunder]], *Algebraic Topology*, Cambridge University Press, Cambridge (1970, 1980) $[$[pdf](https://www.maths.ed.ac.uk/~v1ranick/papers/maunder.pdf)$]$

* {#Switzer75} [[Robert Switzer]], _Algebraic Topology - Homotopy and Homology_, Die  Grundlehren der Mathematischen Wissenschaften in Einzeldarstellungen, Vol. 212, Springer-Verlag, New York, N. Y., 1975 ([doi:10.1007/978-3-642-61923-6](https://link.springer.com/book/10.1007/978-3-642-61923-6))

* [[Raoul Bott]], [[Loring Tu]], _[[Differential Forms in Algebraic Topology]]_, Graduate Texts in Mathematics 82, Springer (1982) $[$[doi:10.1007/978-1-4757-3951-0](https://link.springer.com/book/10.1007/978-1-4757-3951-0)\]

  > (with focus on [[differential forms]], [[differential topology]])

* [[James Munkres]], *Elements of Algebraic Topology*, Addison-Wesley (1984) $[$[pdf](https://people.dm.unipi.it/benedett/MUNKRES-ETA.pdf)\]

* [[Joseph J. Rotman]], *An Introduction to Algebraic Topology*, Graduate Texts in Mathematics **119** (1988) $[$[doi:10.1007/978-1-4612-4576-6](https://doi.org/10.1007/978-1-4612-4576-6)\]

* {#Bredon93} [[Glen Bredon]], _Topology and Geometry_, Graduate texts in mathematics **139**, Springer 1993 ([doi:10.1007/978-1-4757-6848-0](https://link.springer.com/book/10.1007/978-1-4757-6848-0),  [pdf](http://virtualmath1.stanford.edu/~ralph/math215b/Bredon.pdf))

* {#Dold95} [[Albrecht Dold]], *Lectures on Algebraic Topology*, Springer 1995 ([doi:10.1007/978-3-642-67821-9](https://www.springer.com/gp/book/9783540586609), [pdf](https://link.springer.com/content/pdf/bfm%3A978-3-642-67821-9%2F1.pdf))

* [[William Fulton]], *Algebraic Topology -- A First Course*, Graduate Texts in Mathematics **153**, Springer (1995) \[doi:10.1007/978-1-4612-4180-5\]

* [[Peter May]], _[[A concise course in algebraic topology]]_, University of Chicago Press 1999 ([ISBN: 9780226511832](https://www.press.uchicago.edu/ucp/books/book/chicago/C/bo3777031.html), [pdf](http://www.math.uchicago.edu/~may/CONCISE/ConciseRevised.pdf))

* [[Tammo tom Dieck]], _Topologie_, De Gruyter 2000 ([doi:10.1515/9783110802542](https://doi.org/10.1515/9783110802542))

* [[Peter May]], [[Kate Ponto]], _[[More concise algebraic topology]]_,   University of Chicago Press (2012) ([ISBN:9780226511795](https://press.uchicago.edu/ucp/books/book/chicago/M/bo12322308.html), [pdf](https://www.math.uchicago.edu/~may/TEAK/KateBookFinal.pdf))

* [[Dai Tamaki]], [[Akira Kono]], _Generalized Cohomology_, Translations of Mathematical Monographs, American Mathematical Society, 2006 ([ISBN: 978-0-8218-3514-2](https://bookstore.ams.org/mmono-230))

* {#tomDieck2008} [[Tammo tom Dieck]],  _Algebraic topology_, European Mathematical Society, Zürich (2008) ([doi:10.4171/048](https://www.ems-ph.org/books/book.php?proj_nr=86), [pdf](https://www.maths.ed.ac.uk/~v1ranick/papers/diecktop.pdf))

* {#Hatcher02} [[Allen Hatcher]], *Algebraic Topology*, Cambridge University Press (2002) $[$[ISBN:9780521795401](https://www.cambridge.org/gb/academic/subjects/mathematics/geometry-and-topology/algebraic-topology-1?format=PB&isbn=9780521795401), [webpage](https://pi.math.cornell.edu/~hatcher/AT/ATpage.html)$]$

On [[constructive mathematics|constructive]] methods ([[constructive algebraic topology]]):

* [[Julio Rubio]], [[Francis Sergeraert]], *Constructive Algebraic Topology*, Bulletin des Sciences Mathématiques
**126** 5 (2002) 389-412 \[doi:10.1016/S0007-4497(02)01119-3, [arXiv:math/0111243](https://arxiv.org/abs/math/0111243)\]

Lecture notes:

* {#HopkinsMathew} [[Michael Hopkins]] (notes by [[Akhil Mathew]]), _algebraic topology -- Lectures_ ([pdf](http://people.fas.harvard.edu/~amathew/ATnotes.pdf))

* [[Friedhelm Waldhausen]], _Algebraische Topologie_ I ([pdf](https://www.math.uni-bielefeld.de/~fw/at.pdf)) , II ([pdf](https://www.math.uni-bielefeld.de/~fw/at_II.pdf)), III ([pdf](https://www.math.uni-bielefeld.de/~fw/at_III.pdf)) ([web](https://www.math.uni-bielefeld.de/~fw/))

* [[James F. Davis]] and [[Paul Kirk]], _Lecture notes in algebraic topology_ ([pdf](http://www.indiana.edu/~jfdavis/teaching/m623/book.pdf))

* [[Gereon Quick]], _[Advanced algebraic topology](https://folk.ntnu.no/gereonq/Math231br.html)_, 2014

Survey of various subjects in algebraic topology:

* {#James95} [[Ioan Mackenzie James]], _[[Handbook of Algebraic Topology]]_ 1995

Survey with relation to [[differential topology]]:

* [[Sergei Novikov]], _Topology I -- General survey_, in: Encyclopedia of Mathematical Sciences Vol. 12, Springer 1986 ([doi:10.1007/978-3-662-10579-5](https://link.springer.com/book/10.1007/978-3-662-10579-5), [pdf](https://web.math.rochester.edu/people/faculty/doug/otherpapers/novikovsurv.pdf))

* [[Jean Dieudonné]], _A History of Algebraic and Differential Topology, 1900 - 1960_, Modern Birkhäuser Classics 2009 ([ISBN:978-0-8176-4907-4](https://www.springer.com/de/book/9780817649067))

With focus on [[ordinary homology]], [[ordinary cohomology]] and [[abelian sheaf cohomology]]:

* [[Jean Gallier]], [[Jocelyn Quaintance]], *Homology, Cohomology, and Sheaf Cohomology for Algebraic Topology, Algebraic Geometry, and Differential Geometry*, World Scientific (2022) \[doi:10.1142/12495, [webpage](https://www.cis.upenn.edu/~jean/gbooks/sheaf-coho.html)\]

Some interactive 3D demos:

* [[Neil Strickland]], *Interactive pages for Algebraic Topology*, [web site](http://neil-strickland.staff.shef.ac.uk/courses/MAS435/demos/)

Further pointers:

*  [a thread on AlgTop literature at MathOverflow](http://mathoverflow.net/questions/18041/algebraic-topology-beyond-the-basicsany-texts-bridging-the-gap)

### Abstract homotopy theory
 {#ReferencesAbstractHomotopyTheory}

On [[localization]] at [[weak equivalences]] to [[homotopy categories]]:

* {#Brown65} [[Edgar Brown]], _Abstract homotopy theory_, Trans. AMS 119 no. 1 (1965) ([doi:10.1090/S0002-9947-1965-0182970-6](https://doi.org/10.1090/S0002-9947-1965-0182970-6))

On [[localization]] via [[calculus of fractions]]:

* [[Pierre Gabriel]], [[Michel Zisman]], _[[Calculus of fractions and homotopy theory]]_, _Ergebnisse der Mathematik und ihrer Grenzgebiete_, Band 35. Springer, New York (1967)

On [[homotopy category of a model category|localization via]] [[model category]]-theory:

* {#Quillen67} [[Daniel Quillen]], _Homotopical algebra_, Lecture Notes in Mathematics 43, Berlin, New York, 1967

* {#Hovey99} [[Mark Hovey]], _[[Model Categories]]_, Mathematical Surveys and Monographs, Volume 63, AMS (1999) ([ISBN:978-0-8218-4361-1](https://bookstore.ams.org/surv-63-s), [doi:10.1090/surv/063](https://doi.org/http://dx.doi.org/10.1090/surv/063), [pdf](https://people.math.rochester.edu/faculty/doug/otherpapers/hovey-model-cats.pdf), [Google books](http://books.google.co.uk/books?id=Kfs4uuiTXN0C&printsec=frontcover))

* {#Hirschhorn02} [[Philip Hirschhorn]], _Model Categories and Their Localizations_, AMS Math. Survey and Monographs Vol 99 (2002) ([ISBN:978-0-8218-4917-0](https://bookstore.ams.org/surv-99-s/), [pdf toc](http://www.gbv.de/dms/goettingen/360115845.pdf), [pdf](http://www.maths.ed.ac.uk/~aar/papers/hirschhornloc.pdf))

* [[William G. Dwyer]], [[Philip S. Hirschhorn]], [[Daniel M. Kan]], [[Jeffrey H. Smith]], _[[Homotopy Limit Functors on Model Categories and Homotopical Categories]]_, Mathematical Surveys and Monographs 113 (2004) $[$[ISBN: 978-1-4704-1340-8](https://bookstore.ams.org/surv-113-s), [pdf](http://dodo.pdmi.ras.ru/~topology/books/dhks.pdf)$]$

On [[localization]] (especially of categories of [[simplicial sheaves]]/[[simplicial presheaves]]) via [[categories of fibrant objects]]:

* [[Kenneth S. Brown]], *[[BrownAbstractHomotopyTheory.pdf:file]]*, Transactions of the American Mathematical Society **186** (1973) 419-458 $[$[jstor:1996573](http://www.jstor.org/stable/1996573)$]$.

See also:

* [[Klaus Heiner Kamps]], [[Tim Porter]], _Abstract Homotopy and Simple Homotopy Theory_, World Scientific 1997 ([doi:10.1142/2215](https://doi.org/10.1142/2215), [GoogleBooks](http://books.google.de/books?id=7JYKxInRMdAC&dq=Porter+Kamps&printsec=frontcover&source=bl&ots=uuyl_tIjs4&sig=Lt8I92xQBZ4DNKVXD0x76WkcxCE&hl=de&sa=X&oi=book_result&resnum=3&ct=result#PPP1,M1))

* [[Haynes Miller]] (ed.), _[[Handbook of Homotopy Theory]]_, 2019

Lecture notes:

* [[William Dwyer]], _Homotopy theory and classifying spaces_, Copenhagen, June 2008 ([pdf](http://www.math.ku.dk/~jg/homotopical2008/Dwyer.CopenhagenNotes.pdf), [[Dwyer_HomotopyTheoryOfClassifyingSpaces.pdf:file]])

* [[Jesper Michael Møller]], _Homotopy theory for beginners_, 2015 ([pdf](http://www.math.ku.dk/~moller/e01/algtopI/comments.pdf), [[Moller_HomotopyTheory.pdf:file]])

* [[Urs Schreiber]], _[[Introduction to Homotopy Theory]]_ (2016)

* {#Martins20} [[Yuri Ximenes Martins]], _Introduction to Abstract Homotopy Theory_ ([arXiv:2008.05302](https://arxiv.org/abs/2008.05302))

Introduction, from [[category theory]] to (mostly [[(infinity,1)-category theory|abstract]], [[simplicial homotopy theory|simplicial]]) homotopy theory:

* [[Emily Riehl]], _[[Categorical Homotopy Theory]]_, Cambridge University Press, 2014 ([pdf](http://www.math.jhu.edu/~eriehl/cathtpy.pdf), [doi:10.1017/CBO9781107261457](https://doi.org/10.1017/CBO9781107261457))

* {#Richter19} [[Birgit Richter]], _From categories to homotopy theory_, Cambridge Studies in Advanced Mathematics 188, Cambridge University Press 2020 ([doi:10.1017/9781108855891](https://doi.org/10.1017/9781108855891), [book webpage](https://www.math.uni-hamburg.de/home/richter/catbook.html), [pdf](https://www.math.uni-hamburg.de/home/richter/bookdraft.pdf))

* [[Urs Schreiber]], _[[geometry of physics -- categories and toposes]]_

See also:

* [[William Dwyer]], [[Philip Hirschhorn]], [[Daniel Kan]], [[Jeff Smith]], _Homotopy Limit Functors on Model Categories and Homotopical Categories_, volume 113 of _Mathematical Surveys and Monographs_, American Mathematical Society (2004) (there exists [this](http://dodo.pdmi.ras.ru/~topology/books/dhks.pdf) pdf copy of what seems to be a preliminary version of this book)

* [[Zhen Lin Low]], _[[Notes on homotopical algebra]]_, 2015

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

  > (with emphasis on [[cubical objects]] such as in [[n-excisive functors]] and [[Goodwillie calculus]])

### Simplicial homotopy theory
 {#ReferencesSimplicialHomotopyTheory}

On [[simplicial homotopy theory]]:

* [[Peter May]], _Simplicial objects in algebraic topology_, University of Chicago Press 1967 ([ISBN:9780226511818](https://press.uchicago.edu/ucp/books/book/chicago/S/bo5956688.html), [djvu](http://www.math.uchicago.edu/~may/BOOKS/Simp.djvu), [[May_SimplicialObjectsInAlgebraicTopology.pdf:file]])

*  [[Edward B. Curtis]], _Simplicial homotopy theory_, Advances in Mathematics 6 (1971) 107–209 (doi:10.1016/0001-8708(71)90015-6, [MR279808](http://www.ams.org/mathscinet-getitem?mr=279808))

* {#GoerssJardine09} [[Paul Goerss]], [[J. F. Jardine]], Section V.4 of: _[[Simplicial homotopy theory]]_, Progress in Mathematics, Birkhäuser (1999) Modern Birkhäuser Classics (2009) ([doi:10.1007/978-3-0346-0189-4](https://link.springer.com/book/10.1007/978-3-0346-0189-4), [webpage](http://web.archive.org/web/19990208220238/http://www.math.uwo.ca/~jardine/papers/simp-sets/))

* {#JoyalTierney05} [[André Joyal]], [[Myles Tierney]] _Notes on simplicial homotopy theory_, Lecture at _[Advanced Course on Simplicial Methods in Higher Categories](https://lists.lehigh.edu/pipermail/algtop-l/2007q4/000017.html)_, CRM 2008  ([[JoyalTierneyNotesOnSimplicialHomotopyTheory.pdf:file]])

* {#JoyalTierney05} [[André Joyal]], [[Myles Tierney]], _An introduction to simplicial homotopy theory_, 2009  ([web](http://hopf.math.purdue.edu/cgi-bin/generate?/Joyal-Tierney/JT-chap-01), [[JoyalTierneySimplicialHomotopyTheory.pdf:file]])

* [[Paul Goerss]], [[Kirsten Schemmerhorn]], _Model categories and simplicial methods_, Notes from lectures given at the University of Chicago, August 2004, in: _Interactions between Homotopy Theory and Algebra_, Contemporary Mathematics 436, AMS 2007([arXiv:math.AT/0609537](http://arxiv.org/abs/math.AT/0609537), [doi:10.1090/conm/436](http://dx.doi.org/10.1090/conm/436))

* [[Francis Sergeraert]], _Introduction to Combinatorial Homotopy Theory_, 2008 ([pdf](https://www-fourier.ujf-grenoble.fr/~%20sergerar/Papers/Trieste-Lecture-Notes.pdf), [[SergeraertCombinatorialHomotopyTheory.pdf:file]])

### Basic $(\infty,1)$-category theory
 {#ReferencesBasicInfinityCategoryTheory}

On [[(∞,1)-category theory]] and [[(∞,1)-topos theory]]:

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

* {#Joyal08} [[André Joyal]], _Notes on Logoi_, 2008 ([pdf](http://www.math.uchicago.edu/~may/IMA/JOYAL/Joyal.pdf), [[JoyalOnLogoi2008.pdf:file]])

* [[Jacob Lurie]], _[[Higher Topos Theory]]_

* [[Denis-Charles Cisinski]], _Higher category theory and homotopical algebra_ ([pdf](http://www.mathematik.uni-regensburg.de/cisinski/CatLR.pdf))

### Basic homotopy type theory
  {#ReferencesBasicHomotopyTypeTheory}

On [[synthetic homotopy theory]] in [[homotopy type theory]]:

Exposition:

* [[Dan Licata]]: *Homotopy theory in type theory* (2013) $[$[pdf slides](http://dlicata.web.wesleyan.edu/pubs/bll13homotopy/bll13homotopy.pdf), [[Licata-HomotopyInTypeTheory.pdf:file]], [blog entry 1](https://homotopytypetheory.org/2013/03/08/homotopy-theory-in-homotopy-type-theory-introduction), [blog entry 2](https://homotopytypetheory.org/2013/05/20/homotopy-theory-in-type-theory-progress-report/)$]$

* {#Shulman17} [[Mike Shulman]], *The logic of space*, in: [[Gabriel Catren]], [[Mathieu Anel]] (eds.), *[[New Spaces for Mathematics and Physics]]*, Cambridge University Press (2021) 322-404  $[$[arXiv:1703.03007](https://arxiv.org/abs/1703.03007), [doi:10.1017/9781108854429.009](https://doi.org/10.1017/9781108854429.009)$]$

Textbook accounts:

* [[Univalent Foundations Project]]: *[[Homotopy Type Theory -- Univalent Foundations of Mathematics]]* (2013)  ([webpage](http://homotopytypetheory.org/book/), [pdf](http://hottheory.files.wordpress.com/2013/03/hott-online-323-g28e4374.pdf))

* {#Rijke19} [[Egbert Rijke]], _[[Introduction to Homotopy Type Theory]]_ (2019)  ([web](http://www.andrew.cmu.edu/user/erijke/hott/), [pdf](http://www.andrew.cmu.edu/user/erijke/hott/hott_intro.pdf), [GitHub](https://github.com/EgbertRijke/HoTT-Intro))

For more see also at *[homotopy theory formalized in homotopy type theory](homotopy+type+theory#HomotopyTheoryInHomotopyTyepTheoryReferences)*.

### Outlook
 {#ReferencesOutlook}

Indications of open questions and possible future directions in [[algebraic topology]] and ([[stable homotopy theory|stable]]) [[homotopy theory]]:

* [[Mark Hovey]], *[Algebraic Topology Problem List](https://www-users.cse.umn.edu/~tlawson/hovey/)*

* [[Tyler Lawson]], _The future_, Talbot lectures 2013 ([pdf](http://math.mit.edu/conferences/talbot/2013/19-Lawson-thefuture.pdf))

* {#ProblemsInHomotopyTheoryWiki} _Problems in homotopy theory_ ([wiki](http://topology-octopus.herokuapp.com/problemsinhomotopytheory/show/HomePage))

More regarding the sociology of the field (such as its [[folklore]] results):

* {#Barwick17} [[Clark Barwick]], _The future of homotopy theory_, 2017 ([pdf](http://www.maths.ed.ac.uk/~cbarwick/papers/future.pdf), [[BarwickFutureOfHomotopyTheory.pdf:file]])