Kan (mathematics)
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In mathematics, Kan complexes and Kan fibrations are part of the theory of
simplicial set In mathematics, a simplicial set is a sequence of sets with internal order structure ( abstract simplices) and maps between them. Simplicial sets are higher-dimensional generalizations of directed graphs. Every simplicial set gives rise to a "n ...
s. Kan fibrations are the fibrations of the standard
model category A model is an informative representation of an object, person, or system. The term originally denoted the plans of a building in late 16th-century English, and derived via French and Italian ultimately from Latin , . Models can be divided in ...
structure on simplicial sets and are therefore of fundamental importance. Kan complexes are the fibrant objects in this model category. The name is in honor of
Daniel Kan Daniel Marinus Kan (or simply Dan Kan) (August 4, 1927 – August 4, 2013) was a Dutch mathematician working in category theory and homotopy theory. He was a prolific contributor to both fields for six decades, having authored or coauthored sever ...
. For various kinds of fibrations for simplicial sets, see Fibration of simplicial sets.


Definitions


Definition of the standard n-simplex

For each ''n'' â‰¥ 0, recall that the standard n-simplex, \Delta^n, is the representable simplicial set :\Delta^n(i) = \mathrm_ ( Applying the geometric realization functor to this simplicial set gives a space homeomorphic to the topological standard n-simplex: the convex subspace of \mathbb^ consisting of all points (t_0,\dots,t_n) such that the coordinates are non-negative and sum to 1.


Definition of a horn

For each ''k'' â‰¤ ''n'', this has a subcomplex \Lambda^n_k, the ''k''-th horn inside \Delta^n, corresponding to the boundary of the ''n''-simplex, with the ''k''-th face removed. This may be formally defined in various ways, as for instance the union of the images of the ''n'' maps \Delta^ \rightarrow \Delta^n corresponding to all the other faces of \Delta^n. Horns of the form \Lambda_k^2 sitting inside \Delta^2 look like the black V at the top of the adjacent image. If X is a simplicial set, then maps :s: \Lambda_k^n \to X correspond to collections of n (n-1)-simplices satisfying a compatibility condition, one for each 0 \leq k \leq n-1. Explicitly, this condition can be written as follows. Write the (n-1)-simplices as a list (s_0,\dots,s_,s_,\dots,s_) and require that :d_i s_j = d_ s_i\, for all i < j with i,j \neq k. These conditions are satisfied for the (n-1)-simplices of \Lambda_k^n sitting inside \Delta^n.


Definition of a Kan fibration

A map of simplicial sets f: X\rightarrow Y is a Kan fibration if, for any n\ge 1 and 0\le k\le n, and for any maps s:\Lambda^n_k\rightarrow X and y:\Delta^n\rightarrow Y\, such that f \circ s=y \circ i (where i is the inclusion of \Lambda^n_k in \Delta^n), there exists a map x:\Delta^n \rightarrow X such that s=x \circ i and y=f \circ x. Stated this way, the definition is very similar to that of
fibration The notion of a fibration generalizes the notion of a fiber bundle and plays an important role in algebraic topology, a branch of mathematics. Fibrations are used, for example, in Postnikov systems or obstruction theory. In this article, all ma ...
s in
topology Topology (from the Greek language, Greek words , and ) is the branch of mathematics concerned with the properties of a Mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformat ...
(see also
homotopy lifting property In mathematics, in particular in homotopy theory within algebraic topology, the homotopy lifting property (also known as an instance of the right lifting property or the covering homotopy axiom) is a technical condition on a continuous function fr ...
), whence the name "fibration".


Technical remarks

Using the correspondence between n-simplices of a simplicial set X and morphisms \Delta^n \to X (a consequence of the
Yoneda lemma In mathematics, the Yoneda lemma is a fundamental result in category theory. It is an abstract result on functors of the type ''morphisms into a fixed object''. It is a vast generalisation of Cayley's theorem from group theory (viewing a group as a ...
), this definition can be written in terms of simplices. The image of the map fs: \Lambda_k^n \to Y can be thought of as a horn as described above. Asking that fs factors through yi corresponds to requiring that there is an n-simplex in Y whose faces make up the horn from fs (together with one other face). Then the required map x: \Delta^n\to X corresponds to a simplex in X whose faces include the horn from s. The diagram to the right is an example in two dimensions. Since the black V in the lower diagram is filled in by the blue 2-simplex, if the black V above maps down to it then the striped blue 2-simplex has to exist, along with the dotted blue 1-simplex, mapping down in the obvious way.


Kan complexes defined from Kan fibrations

A simplicial set X is called a Kan complex if the map from X \to \, the one-point simplicial set, is a Kan fibration. In the
model category A model is an informative representation of an object, person, or system. The term originally denoted the plans of a building in late 16th-century English, and derived via French and Italian ultimately from Latin , . Models can be divided in ...
for simplicial sets, \ is the terminal object and so a Kan complex is exactly the same as a fibrant object. Equivalently, this could be stated as: if every map \alpha: \Lambda^n_k \to X from a horn has an extension to \Delta^n, meaning there is a lift \tilde: \Delta^n \to X such that
\alpha = \tilde\circ \iota
for the inclusion map \iota: \Lambda^n_k \hookrightarrow \Delta^n, then X is a Kan complex. Conversely, every Kan complex has this property, hence it gives a simple technical condition for a Kan complex.


Examples


Simplicial sets from singular homology

An important example comes from the construction of singular simplices used to define
singular homology In algebraic topology, singular homology refers to the study of a certain set of algebraic invariants of a topological space X, the so-called homology groups H_n(X). Intuitively, singular homology counts, for each dimension n, the n-dimensional ...
, called the singular functorpg 7
S: \text \to s\text.
Given a space X, define a singular n-simplex of X to be a continuous map from the standard topological n-simplex (as described above) to X, :f: \Delta_n \to X Taking the set of these maps for all non-negative n gives a graded set, :S(X) = \coprod_n S_n(X). To make this into a simplicial set, define face maps d_i: S_n(X)\to S_(X) by :(d_i f)(t_0,\dots,t_) = f(t_0,\dots,t_,0,t_i,\dots,t_)\, and degeneracy maps s_i: S_n(X)\to S_(X) by :(s_i f)(t_0,\dots,t_) = f(t_0,\dots,t_,t_i + t_,t_,\dots,t_)\,. Since the union of any n+1 faces of \Delta_ is a strong
deformation retract In topology, a retraction is a continuous mapping from a topological space into a subspace that preserves the position of all points in that subspace. The subspace is then called a retract of the original space. A deformation retraction is a mappi ...
of \Delta_, any continuous function defined on these faces can be extended to \Delta_, which shows that S(X) is a Kan complex.


Relation with geometric realization

It is worth noting the singular functor is
right adjoint In mathematics, specifically category theory, adjunction is a relationship that two functors may exhibit, intuitively corresponding to a weak form of equivalence between two related categories. Two functors that stand in this relationship are k ...
to the
geometric realization functor In mathematics, a simplicial set is a sequence of sets with internal order structure ( abstract simplices) and maps between them. Simplicial sets are higher-dimensional generalizations of directed graphs. Every simplicial set gives rise to a "n ...
, \cdot, :s\text \to \text
giving the isomorphism
\text_(, X, ,Y) \cong \text_(X,S(Y))
See also:
Milnor's theorem on Kan complexes In mathematics, especially algebraic topology, a theorem of Milnor says that the geometric realization functor from the homotopy category of the category Kan of Kan complexes to the homotopy category of the category Top of (reasonable) topological s ...
.


Simplicial sets underlying simplicial groups

It can be shown that the simplicial set underlying a
simplicial group In mathematics, more precisely, in the theory of simplicial sets, a simplicial group is a simplicial object in the category of groups. Similarly, a simplicial abelian group is a simplicial object in the category of abelian groups. A simplicial grou ...
is always fibrantpg 12. In particular, for a
simplicial abelian group In mathematics, more precisely, in the theory of simplicial sets, a simplicial group is a simplicial object in the category of groups. Similarly, a simplicial abelian group is a simplicial object in the category of abelian groups. A simplicial grou ...
, its geometric realization is homotopy equivalent to a product of Eilenberg-Maclane spaces
\prod_ K(A_i,n_i)
In particular, this includes
classifying space In mathematics, specifically in homotopy theory, a classifying space ''BG'' of a topological group ''G'' is the quotient of a weakly contractible space ''EG'' (i.e., a topological space all of whose homotopy groups are trivial) by a proper free ...
s. So the spaces S^1 \simeq K(\mathbb,1), \mathbb^\infty \simeq K(\mathbb,2), and the infinite lens spaces L^\infty_q \simeq K(\mathbb/q, 2) are correspond to Kan complexes of some simplicial set. In fact, this set can be constructed explicitly using the
Dold–Kan correspondence In mathematics, more precisely, in the theory of simplicial sets, the Dold–Kan correspondence (named after Albrecht Dold and Daniel Kan) states that there is an equivalence between the category of (nonnegatively graded) chain complexes and the ...
of a chain complex and taking the underlying simplicial set of the simplicial abelian group.


Geometric realizations of small groupoids

Another important source of examples are the simplicial sets associated to a small groupoid \mathcal. This is defined as the geometric realization of the simplicial set Delta^,\mathcal/math> and is typically denoted B\mathcal. We could have also replaced \mathcal with an infinity groupoid. It is conjectured that the homotopy category of geometric realizations of infinity groupoids is equivalent to the homotopy category of homotopy types. This is called the
homotopy hypothesis In category theory, a branch of mathematics, Grothendieck's homotopy hypothesis states, homotopy theory speaking, that the ∞-groupoids are space (mathematics), spaces. One version of the hypothesis was claimed to be proved in the 1991 paper by M ...
.


Mapping space

Let X be an ∞-category. Then for each object x, y, let \operatorname_X(x, y) be the fiber of (s, t) : \underline(\Delta^1, X) \to X^2 over the point (x, y). Then \operatorname_X(x, y) is a Kan complex.


Postnikov section

Let ''X'' be a Kan complex. Then the n''-th Postnikov section'' P_n X is a simplicial set such that P_n X_m is the coequalizer of X_m \rightrightarrows \operatorname_n X_m. Then the following can be verified directly: *P_n X is a Kan complex and X \to P_nX is a Kan fibration. *The induced map \pi_i X \to \pi_i P_n X is an isomorphism for i = 0 and for each 0 < i \le n and choice of a base point on ''X''. *\pi_i X = 0 for each i > n and choice of a base point on ''X''. For a simplicial set ''X'', we then let P_n X = P_n \operatorname^ X where \operatorname^ is an
Ex∞ functor In higher category theory in mathematics, the extension of simplicial sets (extension functor or Ex functor) is an endofunctor on the category of simplicial sets. Due to many remarkable properties, the extension functor has plenty and strong appli ...
.


Non-example: standard n-simplex

It turns out the standard n-simplex \Delta^n is not a Kan complexpg 38. The construction of a counter example in general can be found by looking at a low dimensional example, say \Delta^1. Taking the map \Lambda_0^2 \to \Delta^1 sending
\begin ,2\mapsto ,0& ,1\mapsto ,1\end
gives a counter example since it cannot be extended to a map \Delta^2 \to \Delta^1 because the maps have to be order preserving. If there was a map, it would have to send
\begin 0 \mapsto 0 \\ 1 \mapsto 1 \\ 2 \mapsto 0 \end
but this isn't a map of simplicial sets.


Categorical properties


Simplicial enrichment and function complexes

For simplicial sets X,Y there is an associated simplicial set called the function complex \textbf(X,Y), where the simplices are defined as
\textbf_n(X,Y) = \text_(X\times\Delta^n, Y)
and for an ordinal map \theta : \to /math> there is an induced map
\theta^*: \textbf(X,Y)_n \to \textbf(X,Y)_m
(since the first factor of Hom is contravariant) defined by sending a map f:X\times\Delta^n \to Y to the composition
X\times\Delta^m \xrightarrowX\times\Delta^n \xrightarrow Y


Exponential law

This complex has the following exponential law of simplicial sets
\text_*:\text_(K, \textbf(X,Y)) \to \text_(X\times K, Y)
which sends a map f: K \to \textbf(X,Y) to the composite map
X\times K \xrightarrowX\times\textbf(X,Y) \xrightarrow Y
where ev(x,f) = f(x,\iota_n) for \iota_n \in \text_\Delta( lifted to the n-simplex \Delta^n.


Kan fibrations and pull-backs

Given a (Kan) fibration p:X \to Y and an inclusion of simplicial sets i: K \hookrightarrow L, there is a fibration pg 21
\textbf(L,X) \xrightarrow\textbf(K,X)\times_\textbf(L, Y)
(where \textbf is in the function complex in the category of simplicial sets) induced from the commutative diagram
\begin \textbf(L,X) & \xrightarrow & \textbf(L,Y) \\ i^* \downarrow & & \downarrow i^* \\ \textbf(K,X) & \xrightarrow & \textbf(K,Y) \end
where i^* is the pull-back map given by pre-composition and p_* is the pushforward map given by post-composition. In particular, the previous fibration implies p_*:\textbf(L,X) \to \textbf(L,Y) and i^*:\textbf(L,Y) \to \textbf(K,Y) are fibrations. The above is a consequence of a
theorem of Gabriel and Zisman In mathematics, especially in homotopy theory, a left fibration of simplicial sets is a map that has the right lifting property with respect to the horn inclusions \Lambda^n_i \subset \Delta^n, 0 \le i < n. A right fibration is defined simila ...
.


Homotopy groups of Kan complexes

The
homotopy group In mathematics, homotopy groups are used in algebraic topology to classify topological spaces. The first and simplest homotopy group is the fundamental group, denoted \pi_1(X), which records information about loops in a space. Intuitively, homo ...
s of a fibrant simplicial set may be defined combinatorially, using horns, in a way that agrees with the homotopy groups of the topological space which realizes it. For a Kan complex X and a vertex x:\Delta^0 \to X, as a set \pi_n(X,x) is defined as the set of maps \alpha:\Delta^n \to X of simplicial sets fitting into a certain commutative diagram:
\pi_n(X,x) = \left\
Notice the fact \partial\Delta^n is mapped to a point is equivalent to the definition of the sphere S^n as the quotient B^n / \partial B^n for the standard unit ball
B^n = \
Defining the group structure requires a little more work. Essentially, given two maps \alpha,\beta:\Delta^n \to X there is an associated (n+1)-simplice \omega:\Delta^ \to X such that d_n\omega:\Delta^n \to X gives their addition. This map is well-defined up to simplicial homotopy classes of maps, giving the group structure. Moreover, the groups \pi_n(X,x) are Abelian for n \geq 2. For \pi_0(X), it is defined as the homotopy classes /math> of vertex maps x:\Delta^0 \to X.


Homotopy groups of simplicial sets

Using model categories, any simplicial set X has a fibrant replacement \hat which is homotopy equivalent to X in the homotopy category of simplicial sets. Then, the homotopy groups of X can be defined as
\pi_n(X,x) := \pi_n(\hat,\hat)
where \hat is a lift of x:\Delta^0 \to X to \hat. These fibrant replacements can be thought of a topological analogue of resolutions of a chain complex (such as a
projective resolution In mathematics, and more specifically in homological algebra, a resolution (or left resolution; dually a coresolution or right resolution) is an exact sequence of modules (or, more generally, of objects of an abelian category) that is used to defi ...
or a
flat resolution In mathematics, and more specifically in homological algebra, a resolution (or left resolution; dually a coresolution or right resolution) is an exact sequence of modules (or, more generally, of objects of an abelian category) that is used to defi ...
).


Kan

Kan complexes themselves form a
weak Kan complex In mathematics, more specifically category theory, a quasi-category (also called quasicategory, weak Kan complex, inner Kan complex, infinity category, ∞-category, Boardman complex, quategory) is a generalization of the notion of a category. Th ...
called Kan. Namely, first consider the category ''K'' where objects are Kan complexes and morphisms maps of simplicial sets. As a category of presheaves has internal Hom, each hom-set in ''K'' has a structure of a simplicial set; in short, ''K'' is a simplicial category. The
homotopy coherent nerve In category theory, a discipline within mathematics, the nerve ''N''(''C'') of a small category ''C'' is a simplicial set constructed from the objects and morphisms of ''C''. The geometric realization of this simplicial set is a topological space, ...
of it :\textbf := N^(K) is then a weak Kan complex (∞-category).https://kerodon.net/tag/00TZ In view of
homotopy hypothesis In category theory, a branch of mathematics, Grothendieck's homotopy hypothesis states, homotopy theory speaking, that the ∞-groupoids are space (mathematics), spaces. One version of the hypothesis was claimed to be proved in the 1991 paper by M ...
, it is often taken as the ∞-category of spaces = ∞-groupoids and is also denoted as \mathsf or some other variants. See also: universal left fibration.


See also

* Simplicial homotopy theory *
Simplicially enriched category In mathematics, a simplicially enriched category, is a category enriched over the category of simplicial sets. Simplicially enriched categories are often also called, more ambiguously, simplicial categories; the latter term however also applies t ...
*
Weak Kan complex In mathematics, more specifically category theory, a quasi-category (also called quasicategory, weak Kan complex, inner Kan complex, infinity category, ∞-category, Boardman complex, quategory) is a generalization of the notion of a category. Th ...
(also called quasi-category, ∞-category) *
∞-groupoid In category theory, a branch of mathematics, an ∞-groupoid is an abstract homotopical model for topological spaces. One model uses Kan complexes which are fibrant objects in the category (mathematics), category of simplicial sets (with the standa ...
* Minimal fibration


Footnotes


References

* * * * Pierre Gabriel, Michel Zisman, Calculus of fractions and homotopy theory, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 35, Springer (1967

* *{{cite journal , doi=10.1016/j.indag.2010.12.004 , title=Algebraic models for higher categories , date=2011 , last1=Nikolaus , first1=Thomas , journal=Indagationes Mathematicae , volume=21 , issue=1–2 , pages=52–75 , arxiv=1003.1342 Simplicial sets Homotopy theory