combinatorial geometry

## Related concepts

[[!include geometries of physics -- table]]

[[matroid]]

Early authors such as Rota and Crapo used combinatorial pregeometry for matroid and combinatorial geometry for simple matroid, one with no 0-rank or loop, or 1-rank pair or parallel elements.

## References

See

* Wikipedia, _[Discrete geometry](http://en.wikipedia.org/wiki/Discrete_geometry)_

[[!redirects combinatorial geometries]]

[[!redirects discrete geometry]]
[[!redirects discrete geometries]]


combinatorial group theory

__Combinatorial group theory__ is an approach to the theory of discrete [[groups]] based on [[presentations of groups]], that is describing the groups by generators and relations. Despite the apparent simplicity of the idea at first look, some of the most basic problems of combinatorial group theory are difficult and unsolvable in general; for example S.P. Novikov and collaborators proved the existence of a group for which the __[[word problem]]__, that is the question of equality modulo the relations of elements given by generators, is algorithmically undecidable; in other words there is a presentation of this group such that no [[algorithm]] can take as an input two arbitrary words and decide in finite time if they represent the same element of the group.

While sometimes the presentation method is used in the theory of finite groups, most often not very deep facts are used; instead finite group theory is rather dominated with interaction with [[representation theory]], [[ring]] theory, [[group cohomology]] and so on. Typical groups for which combinatorial methods give deep results are infinite discrete groups which are close in some sense to [[free group]]s, e.g. groups with a presentation having only one relation. Such groups often make their appearance  as [[fundamental groups]] of interesting [[topological spaces]]. (Of course, all discrete groups can be realised as fundamental groups of spaces, so the exchange is two way.)

Nowadays combinatorial group theory uses fruitful connections to [[topology]] and [[graph]] theory: for example, the consequences if some group can be realized say as a fundamental group of a space in a nice way, or [[action|acts]] on a topological space are widely explored. They are apt  to invite topological /  geometric methods. In fact there is a hybrid field, [[geometric group theory]]  (see also [English Wikipedia](http://en.wikipedia.org/wiki/Geometric_group_theory)).

##Related entries

* [[group presentation]];

* [[Tietze transformation]];

* [[Nielsen-Schreier theorem]];

* [[Cayley graph]]

## References

Textbook accounts:

* [[Wilhelm Magnus]], A. Karras, D. Solitar, *Combinatorial group theory: Presentation of groups in terms of generators and relations*, Courier Corporation (1966, 2004) [[ISBN:0-486-43830-9](https://www.maa.org/press/maa-reviews/combinatorial-group-theory-presentations-of-groups-in-terms-of-generators-and-relations), [GoogleBooks](https://books.google.com/books?hl=en&lr=&id=1LW4s1RDRHQC&oi=fnd&pg=PP1&ots=WszNa-0hEz&sig=Hn_MTbyLcEreBBNZ8jJtQUVKlZk)]

* R. Lindon, P. Schupp, _Combinatorial group theory_, Springer 1977 (Russian transl. Mir, Moskva 1980)

* [[David Lawrence Johnson]], *Topics in Theory of Group Presentations*, London Mathematical Society Lecture notes series **42**, Cambridge Univ. Press (1980) [[doi:10.1017/CBO9780511629303](https://doi.org/10.1017/CBO9780511629303)]

* [[David Lawrence Johnson]], *Presentations of Groups*, London Mathematical Society Student Texts **15**, Cambridge Univ. Press (1990) [[doi:10.1017/CBO9781139168410](https://doi.org/10.1017/CBO9781139168410)]

category: group theory, algebra


combinatorial homotopy

As detailed in the entry [[algebraic homotopy]], in his ICM talk of 1950, [[Henry Whitehead]] said:

>The ultimate aim of [[algebraic homotopy]] is to construct a purely
  algebraic theory, which is equivalent to homotopy theory in the same sort of
  way that 'analytic' is equivalent to 'pure' projective geometry.

then later
> A statement of the aims of 'algebraic homotopy' might thus include the
following homotopy classification problem (from the same source,
J.H.C.Whitehead, ICM 1950).

>Classify the homotopy types of polyhedra, $X$, $Y$, ... , by
  algebraic data.

>Compute the set of homotopy classes of maps, $[X,Y]$, in terms of the
classifying data for $X$, $Y$.

He had already started on this program in two (famous) papers [[Combinatorial homotopy I]] and [[Combinatorial homotopy II]].  The term _combinatorial homotopy_ was, it seems, in part inspired by the similar usage in _[[combinatorial group theory]]_ and his hope would seem to have been that the ideas from the theory of group presentations would allow new insights on combinatorially defined 'complexes'.

## References

* [[J. H. C. Whitehead]], 1950, _Algebraic Homotopy Theory_ , in Proc. Int. Cong. of Mathematics, Harvard, volume 2, 354 – 357. 51

* J. H. C. Whitehead, _Combinatorial homotopy I_ , Bull. Amer. Math. Soc, 55, (1949), 213–245.

 * [[J. H. C. Whitehead]],  _Combinatorial Homotopy II_, Bull.
Amer. Math. Soc., 55 (1949), 453--496.


##Combinatorial Homotopy and 4-Dimensional Complexes.

This book by [[Baues]] handles the classification of 4-dimensional complexes. [[Henry Whitehead]] commented in his famous paper, _[[Combinatorial homotopy I]]_, (1949), 'What has been achieved so far is a purely algebraic description of the homotopy type of any 3-dimensional complex and of any finite, simply connected 4-dimensional complex.' (This case of simply connected 4-dimensional complexes had been treated by him in another paper in that same year of 1949.)
This book by Baues gives a solution in the non-simply connected case.

#Contents#

* automatic table of contents goes here
{:toc}

##  Preface by Ronnie Brown.

##  Introduction

## I  Homotopy, homology, and Whitehead's classification of simply connected 4-dimensional CW-complexes.

## II The CW-tower of categories

## III Crossed modules and homotopy systems of order 3

## IV Quadratic modules and homotopy systems of order 4

## V Cohomological invariants

## VI The cohomology of categories and the calculus of tracks

##Bibliography

##List of Symbols

##Index

#References

* H.-J. Baues, Combinatorial Homotopy and 4-Dimensional Complexes , de Gruyter Expositions in Mathematics 2, Walter de Gruyter, (1991).

category: reference


Combinatorial homotopy I

[This](http://www.ams.org/journals/bull/1949-55-03/S0002-9904-1949-09175-9/S0002-9904-1949-09175-9.pdf) is a link to the online copy of J. H. C. Whitehead's first paper on Combinatorial Homotopy.


Combinatorial homotopy II

[This](http://www.ams.org/journals/bull/1949-55-05/S0002-9904-1949-09213-3/S0002-9904-1949-09213-3.pdf) is a link to the online copy of J. H. C. WHitehead's second paper on Combinatorial Homotopy.


combinatorial Hopf algebra

Some [[Hopf algebra]]s encode relevant combinatorial information and are quite common and useful in [[algebraic combinatorics]] in general. More recently Hopf algebras appeared also in the study of [[renormalization]] of QFT, controlling its combinatorics. Combinatorial Hopf algebra may refer either to a  Hopf algebra coming from combinatorics (and, in particular, to the study of various rings of symmetric functions), but also to the now standard notion of a pair, $(H,\zeta)$, consisting of a Hopf algebra $H$ over $k$ together with a character $\zeta: H\to k$, the zeta function of a Hopf algebra.

## Literature

One of the original sources is:

* S. A. Joni and [[Gian-Carlo Rota]], _Coalgebras and bialgebras in combinatorics_, Umbral Calculus and Hopf Algebras (Norman, OK, 1978), Contemp. Math., vol. 6, AMS, Providence, RI, 1982.

There is also another early article from Rota:

* [[Gian-Carlo Rota]], _Hopf algebra methods in combinatorics_, Problèmes combinatoires et théorie des graphes (Colloq. Internat. CNRS, Univ. Orsay, Orsay, 1976), pp. 363–365, Colloq. Internat. CNRS __260__, CNRS, Paris 1978.

(Despite the title, though, this is just a short note that mainly discusses applications of Rota's theory to the so-called [[umbral calculus]], without making the Hopf algebra structure explicit.)

For the basic intuition for Hopf algebras in combinatorics see the chapter 5 of [[S. Majid]]'s _Foundation of quantum group theory_. Nowadays the major book on combinatorial Hopf algebras is

* [[Marcelo Aguiar]], Swapneel Mahajan, _[[Monoidal Functors, Species and Hopf Algebras]]_, With forewords by [[Kenneth Brown]], Stephen Chase and [[André Joyal]]. CRM Monograph Series __29__ Amer. Math. Soc. 2010. lii+784 pp. ([pdf draft](http://www.math.cornell.edu/~maguiar/a.pdf))

Other references include:

* G-C. Rota, J. A. Stein, _Plethystic Hopf algebras_, Proc. Nat. Acad. Sci. U.S.A. __91__ (1994), no. 26, 13057–13061, [MR96e:16054](http://www.ams.org/mathscinet-getitem?mr=1306900), _Plethystic algebras and vector symmetric functions_, Proc. Nat. Acad. Sci. U.S.A. __91__ (1994), no. 26, 13062–13066, [MR96e:16055](http://www.ams.org/mathscinet-getitem?mr=1306901)
* Kurusch Ebrahimi-Fard, _Combinatorial Hopf algebras+, lectures at CIMAT summer school, slides [pdf](http://www-fourier.ujf-grenoble.fr/~peters/CIMAT/Abstracts/kurusch.pdf)
* F. Hivert, J.-C. Novelli, J.-Y. Thibon, _Trees, functional equations and combinatorial Hopf algebras_, Europ. J. Comb. __29__ (1) (2008), 1682–1695.
* Mercedes H. Rosas, Gian-Carlo Rota, Joel Stein, _A combinatorial overview of the Hopf algebra of MacMahon symmetric functions_, Ann. Comb. __6__ (2002), no. 2, 195–207.
* Bertfried Fauser, _On the Hopf algebraic origin of Wick normal ordering_, J. Phys. A 34 (2001), no. 1, 105–115
* [[Damien Calaque]], Kurusch Ebrahimi-Fard, Dominique Manchon, _Two interacting Hopf algebras of trees: a Hopf-algebraic approach to composition and substitution of B-series_, Adv. in Appl. Math. __47__ (2011), no. 2, 282–308
* K. Ebrahimi-Fard, Li Guo, _Mixable shuffles, quasi-shuffles and Hopf algebras_, J. Algebraic Combin. __24__ (2006), no. 1, 83–101, [MR2007d:05152](http://www.ams.org/mathscinet-getitem?mr=2245782),[doi](http://dx.doi.org/10.1007/s10801-006-9103-x)
* K. Ebrahimi-Fard, D. Kreimer, _The Hopf algebra approach to Feynman diagram calculations_, J. Phys. A 38 (2005), no. 50, R385–R407, [MR2006k:81266](http://www.ams.org/mathscinet-getitem?mr=2199729), [doi](http://dx.doi.org/10.1088/0305-4470/38/50/R01)
* [[Alain Connes]], [[Dirk Kreimer]], _Renormalization in quantum field theory and the Riemann-Hilbert problem. I. The Hopf algebra structure of graphs and the main theorem_, Comm. Math. Phys. __210__ (2000), no. 1, 249–273, [hep-th/9912092](http://arxiv.org/abs/hep-th/9912092), [MR2002f:81070](http://www.ams.org/mathscinet-getitem?mr=1748177), [doi](http://dx.doi.org/10.1007/s002200050779), _II. The $\beta$-function, diffeomorphisms and the renormalization group_, Comm. Math. Phys. __216__ (2001), no. 1, 215--241; [hep-th/0003188](http://arxiv.org/abs/hep-th/0003188), [MR2002f:81071](http://www.ams.org/mathscinet-getitem?mr=1748177), [doi](http://dx.doi.org/10.1007/PL00005547)
* [[Jean-Louis Loday]] and María O. Ronco. Combinatorial Hopf algebras. _Quanta of maths_, Clay Math. Proc. 11, 347–383, Amer. Math. Soc., Providence, RI, 2010. [pdf](http://www-irma.u-strasbg.fr/~loday/PAPERS/2011LodayRonco%28CHA%29.pdf)
* Bill Schmitt, Hopf algebras of combinatorial structures, Canadian Journal of Mathematics 45 (1993), 412-428. [pdf](http://home.gwu.edu/~wschmitt/papers/hacs.pdf)
* Bill Schmitt, Hopf algebra methods in graph theory, Journal of Pure and Applied Algebra 101 (1995), 77-90.  [pdf](http://home.gwu.edu/~wschmitt/papers/hamgt.pdf)
* Bill Schmitt, Incidence Hopf algebras, Journal of Pure and Applied Algebra 96 (1994), 299-330. [pdf](http://home.gwu.edu/~wschmitt/papers/iha.pdf)

See also this nice lecture series on YouTube:

* Federico Ardila, _Hopf algebras and combinatorics_, 30 lectures, [youtube list](http://www.youtube.com/playlist?list=PL-XzhVrXIVeRLeezwY9h4M68k6yB3yOo-), with [index and overview](http://math.sfsu.edu/federico/Clase/hopfindex.pdf) by Sara Billey.

[[!redirects combinatorial Hopf algebras]]


combinatorial map

#Contents#
* table of contents
{:toc}

## Idea

A **combinatorial map** is a representation of a graph embedded in a surface (i.e., a [[topological map]]) as a set and a list of permutations acting on that set, with the property that two combinatorial maps are equivalent up to [[conjugation]] if and only if the corresponding graphs are isomorphic by a [[homeomorphism]] of the underlying surfaces. The precise formulation depends upon assumptions made about the underlying surface (e.g., whether it is [[connected space|connected]] and/or [[oriented]]) and about the underlying graph (e.g., whether all vertices have a fixed degree), but typically different classes of topological maps can be represented by lists of permutations satisfying different sets of constraints.

## Oriented maps and hypermaps

### Classical combinatorial maps

The following definition corresponds to the case which is perhaps most well-studied:

+-- {: .num_definition #def_cmap}
###### Definition
A **combinatorial map** $M$ is a set $D$ (whose elements are called _darts_) equipped with a triple of [[permutations]] $(\sigma,\alpha,\phi)$ on $D$ such that:

1. the [[subgroup]] $\langle \sigma,\alpha,\phi\rangle$ of $Aut(D)$ generated by the permutations acts [[transitive action|transitively]] on $D$;

1. the composition of the permutations is the identity: $\phi\alpha\sigma = id$;

1. $\alpha$ is a [[fixed point]]-free [[involution]].
=--

A [[homomorphism]] of combinatorial maps

$$M \to M'$$

where $M = (D,(\sigma,\alpha,\phi)), M' = (D',(\sigma',\alpha',\phi'))$ is a function $h : D \to D'$ which commutes with each of the respective permutations,

$$h\sigma = \sigma'h \qquad h\alpha = \alpha'h \qquad h\phi = \phi' h.$$

Hence, $M$ and $M'$ are [[isomorphic]] as maps if and only there is a bijection $h : D \cong D'$ between their underlying sets of darts, such that the triple of permutations $(\sigma,\alpha,\phi)$ is mapped by [[conjugation]] along $h$ to the triple of permutations $(\sigma',\alpha',\phi')$.

In the literature, many authors equivalently define a combinatorial map by a pair $(\sigma,\alpha)$ of permutations, since by condition (2) above, the third permutation $\phi$ can always be recovered uniquely as $\phi = \sigma^{-1}\alpha^{-1}$.

### Representing an embedded graph as a combinatorial map

Let $\theta$ be an embedding of a [[graph]] $G$ with $n$ edges into a [[surface]] $X$ which is [[compact]], [[connected]], [[oriented]], and without [[boundary]] (below we will just call these "oriented surfaces" for short).  We can associate a combinatorial map to this topological embedding, which is unique up to isomorphism:

* $D$ is defined as a set containing $2n$ elements, with a pair of distinct darts $e^+,e^- \in D$ for each edge $e \in G$.  These darts may be visualized as oriented edges, or alternatively as half-edges (splitting each of the original edges in the middle).

* the collection of edges induces a fixed point-free involution $\alpha$ on $D$, by swapping $e^+$ with $e^-$ for each edge $e \in G$.

* suppose darts are visualized as oriented edges: we will say that a dart $d$ is _incident_ to a vertex $x$ if $x$ is the unique vertex located at the source of $d$.  Then every vertex $x\in G$ determines a cyclic ordering of the darts incident to $x$, by considering (say) a counterclockwise-oriented [[loop]] around $\theta(x)$.  In this way (which relies on the assumption that $X$ is oriented), each vertex determines a cycle of the permutation $\sigma$.

* similarly, we say that a dart $d$ is incident to a face $f$ (i.e., a connected component of $X \setminus \theta(G)$) if $f$ is the unique face appearing to the left of $d$.  Then each face $f$ determines a cyclic ordering of darts incident to $f$ (corresponding to a counterclockwise traversal along the perimeter of $f$), and we can combine these disjoint cycles into a single permutation $\phi$.

* if we start at any dart $d \in D$, then apply the permutation $\sigma$ followed by $\alpha$ followed followed by $\phi$, we always end up back at $d$.

### Hypermaps and constellations

There are several directions for generalizing the classical definition. A small but useful generalization is to drop the requirement that $\alpha$ be [[fixed point]] free, while keeping the requirement that it be an [[involution]] (i.e., that $\alpha^2 = id$).  This can be seen as allowing the possibility for the graph to include "dangling edges".  In particular, every $\alpha$-[[orbit]] has length 2 or 1, with orbits of length 2 representing complete edges and orbits of length 1 representing dangling edges.

As a bigger step, it is possible to drop condition (3) altogether:

+-- {: .num_definition #def_hmap}
###### Definition
A **combinatorial hypermap** is a set $D$ equipped with a triple of [[permutations]] $(\sigma,\alpha,\phi)$ on $D$ such that:

1. the [[subgroup]] $\langle \sigma,\alpha,\phi\rangle$ of $Aut(D)$ generated by the permutations acts [[transitive action|transitively]] on $D$;

1. the composition of the permutations is the identity: $\phi\alpha\sigma = id$.
=--

A combinatorial hypermap gives a representation of a topological embedding of a [[hypergraph]] in a oriented surface, but (as noted by Walsh) this is equivalent to an embedding of an ordinary graph equipped with a [[vertex coloring|2-coloring]] of its vertices. (Such a graph is necessarily a [[bipartite graph]].) Then, the permutations $\sigma$ and $\alpha$ represent the cycles around the black and white vertices, respectively, while $\phi$ as before represents the faces.  Any ordinary map with all black vertices can be seen as a hypermap with a white vertex dividing each edge (thinking of darts as half-edges), and from that perspective, the definition of hypermap simply corresponds to lifting the restriction that white vertices have degree 2.

Both of these generalizations of the standard notion of combinatorial map are studied in the literature on so-called [[dessins d'enfants]], sometimes using the following terminology to distinguish the different cases:

* $\alpha$ is a fixed point free involution : "clean dessin"

* $\alpha$ is an involution : "pre-clean dessin"

* no conditions on $\alpha$ (besides (1) and (2) above): "dessin"

Another line of generalization is to consider an arbitrary number of permutations (rather than only three).  This more general notion of hypermap is called a "constellation" in [Lando & Zvonkin](#LandoZvonkin04).

+-- {: .num_definition #def_constellation}
###### Definition
A **k-constellation** is a set $D$ equipped with a $k$-[[tuple]] of [[permutations]] $\alpha_1,\dots,\alpha_k$ on $D$ such that:

1. the [[subgroup]] $\langle \alpha_1,\dots,\alpha_k\rangle$ of $Aut(D)$ generated by the permutations acts [[transitive action|transitively]] on $D$;

1. the composition of the permutations is the identity: $\alpha_k\dots\alpha_1 = id$.
=--

### Fixed vertex/edge/face degrees

If one adds, say, the condition that

* $\sigma^3 = id$ and $\sigma$ has no fixed points

to the definition of combinatorial map, then the resulting class of permutations represent embeddings of _trivalent_ graphs, i.e., where every vertex has degree 3.  Dually, placing the condition that

* $\phi^3 = id$ and $\phi$ has no fixed points

gives a way of representing arbitrary [[triangulations]] of oriented surfaces.  Analogous to what we saw above in the case of the edge involution $\alpha$, dropping the condition that $\sigma$ or $\phi$ be fixed point free also allows for the possibility of degenerate vertices/faces of degree 1.

More generally, we can say that a map has type $(m,n)$ if the permutations $\sigma$ and $\phi$ have [[order#in_the_sense_of_group_theory|order]]s $m$ and $n$ respectively (allowing for the possibility that $m = \infty$ or $n = \infty$ in the case where the set of darts is infinite).  This algebraic condition translates to the geometric condition that $m$ is the [[least common multiple]] of the degrees of the vertices of the map, and $n$ is the least common multiple of the degrees of the faces.

### Realizing a combinatorial map as a graph embedded in a Riemann surface

Conversely, any combinatorial map $M = (D,(\sigma,\alpha,\phi))$ can be realized as a graph embedded in a surface, in other words as a [[topological map]].  The first explicit algorithm for computing such an embedding was given by [Edmonds 1960](#Edmonds60).

The underlying graph $G(M)$ of $M$ is easy to describe using the "Serre" definition of a graph (as explained [[graph#definition_in_terms_of_action_on_a_set_of_halfedges|here]]) $G(M) = (H,V,i,s)$.  The set of half-edges $H$ and the involution $i$ are just $D$ and $\alpha$, respectively, while the set of vertices $V$ is the set of $\sigma$-orbits, and the function $s : H \to V$ sends any dart to its orbit under $\sigma$.

Some more work is required in order to define a surface $X$ and an embedding of $G(M)$ into $X$.  [Jones and Singerman 1978](#JonesSingerman78) actually proved a stronger result than Edmonds' algorithm, showing that any combinatorial map (of finite type $(m,n)$) can be realized as a graph embedded in a [[Riemann surface]].  As a corollary, this implies that any topological map (on a compact oriented surface without boundary) is isomorphic to some canonical map on a Riemann surface.

### Genus of a combinatorial map

The **genus** $g$ of a combinatorial map $M = (\sigma,\alpha,\phi)$ can be defined directly using the [[Euler characteristic|Euler-Poincaré formula]],

$$\chi(M) = c(\sigma) - c(\alpha) + c(\phi) = 2-2g$$

where $c(\pi)$ counts the number of cycles in the [[permutation#via_cycle_decompositions|cyclic decomposition]] of $\pi$ (i.e., the number of $\pi$-[[orbits]]).  This definition of genus agrees with the genus of the underlying surface of the embedded graph associated to $M$.

### Dual map

Any embedded graph has a dual graph embedded into the same surface, constructed by placing a vertex in the middle of each face, and connecting two vertices whenever the corresponding faces share an edge.  This construction is particularly easy to express on combinatorial maps: for any $M = (D,(\sigma,\alpha,\phi))$, the corresponding **dual map** is defined by $M^* = (D,(\phi^{-1},\alpha^{-1},\sigma^{-1}))$.  This operation is clearly an involution (with fixed points, since some maps are self-dual).  Moreover, it extends to an operation on hypermaps.

## Equivalence with other families of objects

### Tetravalent maps with bicolored faces

To any graph $G$ embedded in an oriented surface $X$, one can associate a tetravalent graph $G^m$ embedded in the same surface $X$ by the following construction:

* one vertex $v_e \in G^m$ for every edge $e \in G$;

* one edge $v_{e_1} \sim v_{e_2} \in G^m$ for every ordered pair of edges $e_1,e_2 \in G$ which occur consecutively along the same face in $\theta(G)$.

In graph theory, this is known as the **medial graph** construction.  As with the dual graph construction, though, note that this is only well-defined as an operation on embedded graphs (i.e., topological maps) rather than on abstract graphs: the same graph can have different medial graphs, depending on how it is embedded into a surface.  To avoid confusion, we refer to this as the **medial map** construction.

Two topological maps $M_1$ and $M_2$ have isomorphic medial maps $M_1^m \cong M_2^m$ if and only if they are either isomorphic $M_1 \cong M_2$ or dual $M_1 \cong M_2^*$.  To distinguish between duals, one can define a 2-coloring of the _faces_ of the medial map as follows: a face of $M^m$ is colored in black just in case it contains a vertex of $M$, and white otherwise.  The construction of the 2-colored medial map thus defines a one-to-one correspondence between general maps and face 2-colored tetravalent maps on oriented surfaces.

### Alternating virtual links

Any _alternating_ virtual link (in the sense of [[virtual knot theory]]) is naturally represented by a face 2-colored tetravalent map: each ordinary crossing becomes a vertex of degree 4, with faces colored according to the pattern of under/over crossings (virtual crossings appearing in the link diagram disappear when the knot is embedded into the appropriate surface of higher genus).  Thus, any alternating virtual link can be encoded by a unique combinatorial map, and vice versa: see Section 4 of [Zinn-Justin and Zuber 2004](#ZJZ2004) where they discuss this representation.

## The category of oriented maps

In this section we will adopt the more liberal notion of combinatorial map described above, for which the involution $\alpha$ is allowed to have fixed points corresponding to "dangling" edges.  We will also sometimes write _map_ as shorthand for combinatorial map in this more liberal sense.

+-- {: .num_definition #def_cm}
###### Definition
The **category of combinatorial maps** $CM$ is defined as follows:

* objects are pairs of a set $D$ together with a triple of permutations $(\sigma,\alpha,\phi)$ on $D$ such that $\langle\sigma,\alpha,\rho\rangle$ acts transitively on $D$ and $\alpha^2 = \phi\alpha\sigma = id$.

* morphisms $(D,(\sigma,\alpha,\phi)) \to (D',(\sigma',\alpha',\phi'))$ are functions $h : D \to D'$ such that $h\sigma = \sigma' h$, $h\alpha = \alpha' h$, and $h\phi = \phi' h$.
=--

The category $CM$ was defined by [Jones and Singerman 1978](#JonesSingerman78) essentially as above (they call it $AM$ for "algebraic maps").  Specifically, Jones and Singerman take this category to include combinatorial maps in which the involution $\alpha$ may contain fix points.

+-- {: .num_prop}
###### Proposition
$CM$ is a [[concrete category]], with a faithful functor $U : CM \to Set$ which sends any combinatorial map $(D,(\sigma,\alpha,\phi))$ to its underlying set of darts $D$, and any homomorphism of maps $h : M \to M'$ to a function $h : D \to D'$ between the underlying sets of darts.
=--

### Terminal object

We note that the following property of $CM$ relies on allowing maps with dangling edges.

+-- {: .num_prop}
###### Proposition
The set $1 = \{ * \}$ equipped with the triple of identity permutations $\sigma = \alpha = \phi = id_1$ is a map, and is a [[terminal object]] in $CM$.
=--

### The oriented cartographic group

(See also article: [[cartographic group]].)

+-- {: .num_definition #def_cartgp}
###### Definition
The **oriented cartographic group** $\mathcal{C}_2^+$ is the group generated by three elements $\rho_0,\rho_1,\rho_2$ satisfying the relations $\rho_1^2 = \rho_0\rho_1\rho_2 = 1$.
=--

+-- {: .num_prop}
###### Proposition
The oriented cartographic group is isomorphic to a [[free product of groups|free product]] of the integers with the integers modulo 2, $\mathcal{C}_2^+ \cong \mathbb{Z} \star \mathbb{Z}_2$.
=--

Every combinatorial map is a $\mathcal{C}_2^+$-set, i.e., a set equipped with an action of the oriented cartographic group.  In fact, $CM$ is just the [[full subcategory]] of $\mathcal{C}_2^+Set$ consisting of sets equipped with a [[transitive action]] of $\mathcal{C}_2^+$.

+-- {: .num_remark}
###### Remark
The [[moebius transformation#abstract_structure_of_modular_group|modular group]] $PSL_2(\mathbb{Z}) \cong \mathbb{Z}_3 \star \mathbb{Z}_2$ is a quotient of the oriented cartographic group.  Sets equipped with a transitive action of $PSL_2(\mathbb{Z})$ are the same thing as trivalent combinatorial maps, which represent trivalent graphs (possibly including degeneracies) embedded in oriented surfaces (or dually, triangulations of oriented surfaces).
=--

For any given map $M$ with underlying set of darts $D$ we will denote the action of the oriented cartographic group by

$$*_M : D \times \mathcal{C}_2^+ \to D,$$

omitting subscripts when clear from context.  The fact that $\mathcal{C}_2^+$ acts transitively on $D$ corresponds to the topological condition that the graph represented by $M$ is connected.  As we will see, this places strong restrictions on possible morphisms between maps:

+-- {: .num_prop}
###### Proposition
A morphism $h : M_1 \to M_2$ of combinatorial maps is entirely determined given the image of a single dart $h(d_1) \in D_2$.
=--

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

Let $d_1' \in D_1$.  By assumption of transitivity, there exists a word $w \in \mathcal{C}_2^+$ such that $d_1' = d_1 *_1 w$.  Hence $h(d_1') = h(d_1 *_1 w) = h(d_1) *_2 w$.
=--

+-- {: .num_prop}
###### Proposition
If $h : M_1 \to M_2$ is a morphism of combinatorial maps and $D_1$ is [[inhabited]], then the underlying function $h : D_1 \to D_2$ is a [[surjection]].
=--

+-- {: .proof}
###### Proof
Let $d_2 \in D_2$.  By assumption that $D_1$ is inhabited there exists a $d_1 \in D_1$, and by assumption of transitivity there exists $w \in \mathcal{C}_2^+$ such that $d_2 = h(d_1) *_2 w$.  Since $h(d_1) *_2 w = h(d_1 *_1 w)$, the dart $d_1 *_1 w \in D_1$ is in the inverse image of $d_2 \in D_2$.
=--

### Rooted maps

Enumeration of maps is an active branch of [[combinatorics]], going back to the work of [[W. T. Tutte]] in the 1960s.  One of the difficulties with trying to count (combinatorial or topological) maps directly, though, is that they can have non-trivial symmetries, which would need to be accounted for to avoid double-counting.  This is why combinatorists typically begin with the easier problem of counting _rooted maps_, as suggested in this autobiographical account by Tutte:

> Having made no progress with the enumeration of these diagrams [(strict triangulations of the plane)] I bethought myself of Cayley's work on the enumeration of trees. His first successes had been with the rooted trees, in which one vertex is distinguished as the "root". Perhaps I should root the strict triangulations in some way and try to enumerate the rooted ones. Eventually I decided that the rooting should consist of the choice of a face, edge and vertex, mutually incident....

> I am not sure what I would have replied at this stage if I had been asked why I preferred these rooted triangulations to the unrooted ones.... Later I realized that the distinguishing feature and the great advantage of rooted maps is that they have no symmetry. Automorphisms seem to complicate enumerative problems.

> (From Chapter 10 of W. T. Tutte, _Graph Theory As I Have Known It_, Oxford, 1998.)

For a combinatorial map $(D,(\sigma,\alpha,\phi))$, choosing "a face, edge and vertex, mutually incident" is equivalent to choosing a dart $r \in D$: the root vertex, edge, and face can then be computed as the orbits of $r$ under the three permutations.  Rooted combinatorial maps may be organized into another category:

+-- {: .num_definition #def_rcm}
###### Definition
The **category of rooted combinatorial maps** $CM_\bullet$ is defined as follows:

* objects are pairs of a map $M = (D,(\sigma,\alpha,\phi))$ together with a distinguished dart $r \in D$ called the _root_.

* morphisms $(M,r) \to (M',r')$ are map morphisms $h : M \to M'$ in $CM$ which preserve the root $h(r) = r'$.
=--

Note that $CM_\bullet$ is equivalent to the [[comma category]] $1 \downarrow U$ of the singleton set 1 with the forgetful functor $U : CM \to Set$.  Now we have the following basic observations, in line with the above remarks by Tutte:

+-- {: .num_prop}
###### Proposition
For $M = (D,(\sigma,\alpha,\phi))$ a combinatorial map, the automorphism group $Aut(M)$ is by definition the subgroup of permutations on $D$ which commute with $\sigma$ and $\alpha$ and $\phi$, i.e., the [[centralizer]] of $\langle\sigma,\alpha,\phi\rangle$.
=--

+-- {: .num_prop}
###### Proposition
For $(M,r)$ a rooted combinatorial map, the automorphism group $Aut(M,r)$ is always trivial.  In particular, if $h : (M,r) \to (M,r)$ is an endomorphism of $(M,r)$ in the category of rooted maps $CM_\bullet$, then $h$ must be the identity morphism.
=--

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

Let $d \in D$ be any dart of $M$.  Since the action of $\mathcal{C}_2^+$ is transitive, there is some word $w \in \mathcal{C}_2^+$ such that $d = r * w$, and therefore $h(d) = h(r * w) = h(r) * w = r * w = d$.
=--

## Related concepts

* [[cartographic group]]

* [[topological map]]

* [[child's drawing]]

* [[ribbon graph]]

## References

* {#Edmonds60} J. Edmonds (1960), A combinatorial representation for polyhedral surfaces, _Notices Amer. Math. Soc._ 7, 646.

* {#Jacques70} A. Jacques (1970), Constellations et graphes topologiques, _Colloque Math. Soc. Janos Bolyai_, 657-672.

* {#Tutte73} W. T. Tutte. What is a map? In _New Directions in Graph Theory_ (ed. F. Harary), Academic Press, New York, 309-325, 1973.

* {#Tutte75} W. T. Tutte. Duality and Trinity. In _Infinite and Finite Sets (III)_, Colloq. Math. Soc. Janos Bolyai, vol. 10, 1459-1472, North Holland, Amsterdam,  1975.

* {#JonesSingerman78} [[Gareth A. Jones]] and [[David Singerman]], Theory of Maps on Orientable Surfaces. Proceedings of the London Mathematical Society, 37:273-307, 1978.

* {#JonesSingerman94} [[Gareth Jones]] and [[David Singerman]]. Maps, hypermaps, and triangle groups.  In _The Grothendieck Theory of Dessins d'Enfants_, [[L. Schneps]] (ed.), London Mathematical Society Lecture Note Series 200, Cambridge University Press, 1994.

* {#LandoZvonkin04} [[Sergei K. Lando]] and [[Alexander K. Zvonkin]], _Graphs on Surfaces and Their Applications_, Springer, 2004.

* {#VidalPhD} Samuel Vidal, _Groupe Modulaire et Cartes Combinatoires. Génération et Comptage._ PhD Thesis, Université Lille I, France, July 2010. [pdf](https://mathblog.uk/articles/thesis.pdf)

* [[Alexander K. Zvonkin]], Functional Composition is a Generalized Symmetry, _Symmetry: Culture and Science_ Vol. 21, Nos.1-4, 333-368, 2010. ([pdf](https://www.labri.fr/perso/zvonkin/Research/tesselations.pdf))

* {#ZJZ2004} P. Zinn-Justin and J.-B. Zuber, Matrix Integrals and the Generation and Counting of Virtual Tangles and Links, _Journal of Knot Theory and Its Ramifications_, 13:03, May 2004. ([arXiv](http://arxiv.org/abs/math-ph/0303049v2))

* Wikipedia page: [Combinatorial map](https://en.wikipedia.org/wiki/Combinatorial_map)

[[!redirects algebraic map]]
[[!redirects algebraic maps]]
[[!redirects combinatorial maps]]
[[!redirects combinatorial hypermap]]
[[!redirects combinatorial hypermaps]]
[[!redirects hypermap]]
[[!redirects hypermaps]]


This entry is about the article

* [[Daniel Dugger]]:

  \linebreak

  **Combinatorial model categories have presentations**

  \linebreak

   Adv. Math. **164** 1 (2001) 177-201

  [arXiv:math/0007068](http://arxiv.org/abs/math/0007068)

  [doi:10.1006/aima.2001.2015](https://doi.org/10.1006/aima.2001.2015)

on [[combinatorial model categories]] and their identifications with [[Bousfield localization of model categories|Bousfield localizations]] of [[model categories of simplicial presheaves]].

> **Abstract.** _We show that every combinatorial model category can be obtained, up to Quillen equivalence, by localizing a model category of diagrams of simplicial sets. This says that any combinatorial model category can be built up from a category of "generators" and a set of "relations" --- that is, any combinatorial model category has a presentation._

This builds on the companion paper

* [[Dan Dugger]], _[[Universal homotopy theories]]_

This shows that, up to [[Quillen equivalence]], every [[combinatorial model category]] $A$ arises as the [[Bousfield localization of model categories|left Bousfield localization]] of the global projective [[model structure on simplicial presheaves]] $[C^{op}, sSet_{Quillen}]_{proj}$, for some [[small category]] $C$

$$
  A \simeq
  [C^{op}, sSet_{Quillen}]_{proj, loc}
  \stackrel{\overset{}{\leftarrow}}{\underset{}{\to}}
  [C^{op}, sSet_{Quillen}]_{proj}
  \,.
$$

Notice (as discussed at the relevant entries) that

* every combinatorial model category can be enhanced to a [[combinatorial simplicial model category]];

* that the full [[enriched category|enriched]] subcategories $A^\circ$ of these on fibrant-cofibrant objects are the [[locally presentable (∞,1)-categories]];

* that under this correspondence the global [[model structure on simplicial presheaves]] models the corresponding [[(∞,1)-category of (∞,1)-presheaves]]

  $$
    PSh_{(\infty,1)}(C)
    \simeq
    A^{\circ}
    \,.
  $$

Under this correspondence, [[Dugger's theorem]] is precisely the [[model category]]-theoretic analog of the theorem that every [[locally presentable (∞,1)-category]] $D$ is a [[localization of an (∞,1)-category|(∞,1)-categorical localization]] of an [[(∞,1)-category of (∞,1)-presheaves]]

$$
  D \stackrel{\overset{L}{\leftarrow}}{\underset{}{\hookrightarrow}}
  PSh_{(\infty,1)}(C)
  \,.
$$

This [[(∞,1)-category]] theortic interpretation of [[Dugger's theorem]] is proposition A.3.7.6 in

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

category: reference


combinatorial model category

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

# Contents
* table of contents
{:toc}

## Idea

A _combinatorial model category_ is a particularly tractable [[model category]] structure. (Notice however that there is also the, closely related, technical notion of a [[tractable model category]]).

Being combinatorial means that there is very strong control over the cofibrations in these model structures: there is a [[set]] (meaning [[small set]], not a [[proper class]]) of [[cofibrantly generated model category|generating (acyclic) cofibrations]], and all objects, in particular the domains and codomains of these cofibrations, are [[small object]]s.

So as a slogan we have that

+-- {: .standout}

 A combinatorial model structure is one that is generated from _small data_: it is generated from a small set of (acyclic) cofibrations between small objects.

=--

In fact, the combinatoriality condition is a bit stronger than that, as it requires even that _every_ object is small and is the colimit over a small set of generating objects.

There exist large classes of model categories that either are combinatorial or, if not, are [[Quillen equivalence|Quillen equivalent]] to ones that are. See the list of examples below.

The relevance of combinatorial model categories is given more abstractly by the result that

+-- {: .standout}

[[combinatorial simplicial model category|Combinatorial simplicial model categories]] are precisely those model categories that model [[presentable (∞,1)-category|presentable (∞,1)-categories]].

=--

For more see at _[[locally presentable categories - introduction]]_.

There is also a more general class of model structures on locally presentable categories, namely [[accessible model categories]], that share many of the properties of combinatorial ones, and are additionally closed under more constructions such as [[transferred model structures]].  In addition, every combinatorial model category can be enhanced to an [[algebraic model category]].

## Definition

The following definition originates with [[Jeff Smith]]:

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

A [[model category]] $C$ is **combinatorial** if it is

* [[locally presentable category|locally presentable]] as a [[category]]

and

* [[cofibrantly generated model category|cofibrantly generated]] as a [[model category]].

=--

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

Recall from the discussion at [[cofibrantly generated model category]] that this means that $C$ has a [[set]] (i.e. a [[small set]], not a [[proper class]]) $I$ of generating [[cofibrations]] and a set $J$ of generating trivial cofibrations in that

$$
  cof = llp(rlp(I))
$$

$$
  fib = rlp(J)
  \,.
$$

Here $fib, cof \subset Mor(C)$ is the collection of fibrations and cofibration, respectively, and $llp(S), rlp(S)$ is the collection of morphisms satisfying the left or right, respectively, [[lifting property]] with respect to a collection of morphisms $S$.

=--

Jeff Smith's theorem, [below](#SmithTheorem), gives an equivalent characterization that is usually easier to handle.

## Characterization theorems

There are two powerful theorems that characterize combinatorial model categories in terms of data that is often easier to handle:

* [[Jeff Smith]]'s theorem characterizes combinatorial model categories just in terms of weak equivalences and generating cofibrations, hence using only two third of the input data explicitly required. This greatly facilitates identifying combinatorial model category structures. For instance it helps to show that the left [[Bousfield localization of model categories|Bousfield localization]] of certain combinatorial model categories is again a combinatorial model category.

* [[Dugger's theorem]] shows that [[combinatorial model category|combinatorial model categories]] are, up to [[Quillen equivalence]], precisely those model categories that have a _presentation_ in that they are [[Bousfield localization of model categories|Bousfield localization]]s of [[model structure on simplicial presheaves|global model structures on simplicial presheaves]]. This was used by [[Jacob Lurie]] to show that simplicial combinatorial model categories are precisely the models for [[presentable (∞,1)-category|locally presentable (∞,1)-categories]].

### Smith's theorem
 {#SmithTheorem}

A central theorem about combinatorial model categories is **[[Jeff Smith]]'s theorem** which establishes the existence of combinatorial model category structures from a small amount of input data.

+-- {: .num_theorem}
###### Theorem
**(Jeff Smith's theorem)**

For

* $C$ a [[locally presentable category]];

* $Arr_W(C) \subset Arr(C)$ an accessibly embedded [[accessible category|accessible]] [[full subcategory]] of the [[arrow category]] $Arr(C)$ on a [[class]] $W \subset Mor(C)$;

* $I \subset Mor(C)$ a small [[set]] (not a proper class) of morphisms of $C$

such that

* $W$ satisfies [[category with weak equivalences|2-out-of-3]];

* $inj(I) \subset W$

* $cof(I) \cap W$ is closed under [[pushout]] and [[transfinite composition]].

we have that $C$ is a combinatorial model category with

* weak equivalences $W$;

* cofibrations $cof(I)$.

Moreover, every combinatorial model category arises in this way.

=--

Here the notation is as described at [[cofibrantly generated model category]], so: $inj(I) = rlp(I)$ and $cof(I) = llp(rlp(I))$.

This statement was announced by [[Jeff Smith]] in 1998 at a conference in Barcelona and appararently first appeared in print as ([Beke, theorem 1.7](#Beke)).
The above formulation follows ([Barwick, prop 1.7](#Barwick)).

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

To prove the first part of the statement, that the given data encodes a combinatorial model category, it is sufficient to find a small set $J$ such that

$$
  cof(J) = W \cap cof(I)
  \,.
$$

With that the statement follows using the [[small object argument]] to show the existence of the required factorizations.

To find this small set, we make use of the assumption that the subcategory
$Arr_W(C) \subset Arr(C)$ of weak equivalences and commuting squares in $C$ between them is an [[accessible category|accessible]] subcategory of the [[arrow category]] $Arr(C)$.  This means that there is a small set $W_0 \subset W$ such that every element of $W$
is a $\kappa$-[[directed colimit]] over element in $W_0$ in $Arr_W(C)$, for some large enough [[cardinal number]] $\kappa$, such that all elements of $W_0$ are $\kappa$-[[compact object|compact]].

Using the [[small object argument]] factor
every morphism $P \to Q$ in $W_0$ as
$P \stackrel{\in cell(I)}{\to} R \stackrel{\in inj(I)}{\to}Q$.  Note that by [[category with weak equivalences|2-out-of-3]] and $inj(I) \subseteq W$,
the cofibration $P \to R$ is in $W$.
Let $J$ be the set of acyclic cofibratons $P \to R$ so obtained.

By the choice of $W_0$ every morphism

$$
  \array{
    K &\to& M
    \\
    \downarrow^{\mathrlap{\in I}} && \downarrow^{\mathrlap{\in W}}
    \\
    L &\to& N
  }
$$

in $Arr(C)$ lifts through one of the components in $W_0$ of $W$  (this mechanism is described in detail at [[small object]]) as

$$
  \array{
    K &\to& P &\to& M
    \\
    \downarrow^{\mathrlap{\in I}} && \downarrow^{\mathrlap{\in W_0}} && \downarrow^{\mathrlap{\in W}}
    \\
    L &\to& Q &\to& N
  }
  \,.
$$

We can refine this to a factoring through $J$ as follows:
by construction
the morphism $P \to Q$ factors as
$P \stackrel{\in J}{\to} R \stackrel{\in inj(I)}{\to}Q$. Then $L \to Q$ lifts to $L \to R$ and we obtain the factorization

$$
  \array{
    K &\to& P &\to& M
    \\
    \downarrow^{\mathrlap{\in I}} && \downarrow^{\mathrlap{\in J}} && \downarrow^{\mathrlap{\in W}}
    \\
    L &\to& R &\to& N
  }
$$

of the original square from an element in $I$ to an element in $W$  through an element in $J$.
(In [Beke](http://arxiv.org/abs/math/0102087), following [[Jeff Smith]], this is called the
**solution set condition**: $J$ is "a solution set for $W$ at $I$").
It readily follows that $inj(J) \cap W \subseteq inj(I)$.

By the [[small object argument]] every morphism $A \stackrel{\in W}{\to} B$ in $W$ may be factored as

$$
  A \stackrel{\in cell(J)}{\to} C \stackrel{\in inj(J)}{\to} B
  \,.
$$

Here by [[category with weak equivalences|2-out-of-3]],
the morphism $C \to B$ is in $W$, and hence in
$inj(J) \cap W \subseteq inj(I)$.

This we use to show that every morphism $f \in cof(I) \cap W$ is in $cof(J)$:

since $f \in W$ we may factor $f$ as above and since $f \in cof(I)$ we obtain a lift $\sigma$ in

$$
  \array{
    A &\stackrel{\in cell(J)}{\to}& C
    \\
    \downarrow^{f} &{}^\sigma \nearrow&
    \downarrow^{\mathrlap{\in inj(I)}}
    \\
    B &\stackrel{=}{\to}& B
  }
  \,.
$$

Rearranging this it becomes a [[retract]] diagram in $Arr(C)$

$$
  \array{
    A &\stackrel{=}{\to}& A &\stackrel{=}{\to}& A
    \\
    \downarrow^f && \downarrow^{\mathrlap{\in cell(J)}}
    && \downarrow^f
    \\
    B &\stackrel{\sigma}{\to}&
    C
    &
    \stackrel{\in inj(I)}{\to}
    &
    B
  }
$$

which shows that $f$ is a retract of an element in $cell(J) \subset cof(J)$, hence itself in $cof(J)$.

And the converse statement is immediate: by definition $J \subset cof(I) \cap W$ and $cof(J)$ is the saturation of $J$ under the operation of forming retracts of transfinite compositions of pushouts of elements of $J$, under which $cof(I) \cap W$ is assumed to be closed.

In total we have indeed $cof(J) = cof(I) \cap W$ which shows that the $I$ and $W$ given determine a combinatorial model category.

To see the converse, that every combinatorial model structure arises this way, it is sufficient to show that for every combinatorial model category the category $Arr_W(C) \subset Arr(C)$ is an [[accessible category]].

=--

For applications of this theorem, the following auxiliary statements are useful.

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

For $C$ a combinatorial model category, the full subcategory inclusion

$$
  Mor(C)_W \hookrightarrow Mor(C)
$$

of the [[arrow category]] on the weak equivalences is an [[accessible functor|accessible inclusion]] of an [[accessible category]].

=--

This is due to Smith. A proof appears as [Dugger 01, 7.4](#Dugger01). See also [Barwick, prop. 1.10](#Barwick).

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

Let

$$
  F : Mor(A) \to Mor(B)
$$

be an [[accessible functor]] between [[arrow categories]]. Let $B$ be equipped with [[category with weak equivalences|weak equivalences]] $W$ such that the [[full subcategory]] inclusion

$$
  Mor(W) \hookrightarrow Mor(B)
$$

on the weak equivalences is an [[accessible functor|accessible embedding]] of an [[accessible category]]. Then so is the full subcategory of $Mor(A)$ on the pre-images $F^{-1}(W)$ in $A$.

=--

([Beke, prop. 1.18](#Beke)).

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

By general properties of [[accessible categories]] (see there) the full inverse image along an accessible functor of a full accessible subcategory is again accessible.

=--

### The compactly generated left proper case of Smith's theorem

In the special case when weak equivalences are closed under [[filtered colimits]]
and the model structure is [[left proper]],
the statement of Smith's theorem can be simplified.

The following formulation can be found as Proposition A.2.6.15 in [Lurie](#Lurie).

\begin{theorem}
Suppose $C$ is a [[locally presentable category]]
equipped with a [[class]] $W$ of weak equivalences
that turn it into a [[relative category]]
and a set $I$ of generating cofibrations.
If the following conditions are satisfied, then $C$ admits a [[left proper]] model structure with $I$ as the set of generating cofibrations:

* $inj(I) \subset W$;

* every element of $I$ is an [[h-cofibration]];

* the [[class]] $W$ is _perfect_: it satisfies the [[2-out-of-3 property]], is closed under [[filtered colimits]] (in the category of morphisms in $C$), and is generated under [[filtered colimits]] by a set of morphisms.

\end{theorem}

### Dugger's theorem
 {#DuggerTheorem}

The following theorem is precisely the model-category theory version of the statement that every [[locally presentable (∞,1)-category]] arises as the [[localization of an (∞,1)-category|localization]] of an [[(∞,1)-category of (∞,1)-presheaves]].

+-- {: .num_theorem }
###### Theorem
**([[Dugger's theorem]])**

Every combinatorial model category $C$ is [[Quillen equivalence|Quillen equivalent]] to a left [[Bousfield localization of model categories|Bousfield localization]] $L_S SPSh(K)_{proj}$ of the global projective [[model structure on simplicial presheaves]] $SPSh(K)_{proj}$ on a [[small category]] $K$

$$
  L_S SPSh(K)_{proj} \stackrel{\simeq_{Quillen}}{\to}
  C
  \,.
$$

=--

This is ([Dugger 01, theorem 1.1](#Dugger01)) building on results in ([DuggerUniversalHomotopy](#DuggerUniversalHomotopy)).

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

The proof proceeds (the way Dugger presents it, at least) in roughly three steps:

1. Use that $[C^{op}, sSet_{Quillen}]_{proj}$ is in some precise sense the _homotopy-_ [[free cocompletion]] of $C$. This means that every functor $\gamma : C \to B$ from a plain category $C$ to a model category $B$ factors in an essentially unique way through the [[Yoneda embedding]] $j : C \to [C^{op},sSet]$ by a [[Quillen adjunction]]

   $$
     (\hat \gamma \dashv R) :
     B
     \stackrel{\overset{\hat \gamma}{\leftarrow}}
      {\underset{R}{\to}}
     [C^{op}, sSet_{Quillen}]_{proj}
     \,.
   $$

   The detailed definitions and detailed proof of this are discussed at [[(∞,1)-category of (∞,1)-presheaves]].

1. For a given combinatorial model category $B$, choose $C := B_\lambda^{cof}$ the full [[subcategory]] on a [[small set]] (guaranteed to exist since $B$ is [[locally presentable category|locally presentable]]) of cofibrant $\lambda$-[[compact object]]s, for some [[regular cardinal]] $\lambda$, and show that the induced Quillen adjunction

   $$
     B \stackrel{\overset{\hat i}{\leftarrow}}{\underset{R}{\hookrightarrow}}
     [(B_\lambda^{cof})^{op}, sSet]_{proj}
   $$

   induced by the above statement from the inclusion $i : B_\lambda^{cof} \hookrightarrow B$ exhibits $B$ as a homotopy-[[reflective subcategory]] in that the [[derived functor|derived]] counit $ \hat i \circ Q \circ R \stackrel{\simeq}{\to} Id$ ($Q$ some cofibrant replacement functor) is a [[natural transformation|natural]] weak equivalence on fibrant objects (recall from [[adjoint functor]] the characterization of adjoints to full and faithful functors).

1. Define $S$ to be the set of morphisms in $[(C_\lambda^{cof})^{op}, sSet]$ that the left [[derived functor]] $\hat i \circ Q$ of $\hat i$ (here $Q$ is some cofibrant replacement functor) sends to weak equivalences in $B$. Then form the left [[Bousfield localization of model categories|Bousfield localization]] $L_S [(C_\lambda^{cof})^{op},sSet]_{proj}$ with respect to this set of morphisms and prove that this is [[Quillen equivalence|Quillen equivalent]] to $B$.

Carrying this program through requires the following intermediate results.

First recall from the discussion at [[(∞,1)-category of (∞,1)-presheaves]] that to produce the Quilen adjunction $(\hat i \dashv R)$ from $i$, we are to choose a [[cofibrant resolution]] functor

$$
  I : C \to [\Delta,B]
$$

of $i : C= B_\lambda^{cof} \to B$.

The [[adjunct]] of this is a functor $\tilde I : C \times \Delta \to B$. For each object $b \in B$ write $(C \times \Delta \downarrow b)$ for the [[slice category]] induced by this functor.

**Lemma** (Dugger, prop. 4.2)

For every fibrant object $b \in B$ we have that the [[homotopy colimit]] $hocolim (C \times \Delta \downarrow b) \to B)$
is weakly equivalent to $\hat i \circ Q\circ R (b)$.

**Corollary** (Dugger, cor. 4.4) The induced Quillen adjunction

$$
  B \stackrel{\leftarrow}{\to}
  [C^{op}, sSet]
$$

is a homotopy-reflective embedding precisely if the canonical morphisms

$$
  hocolim (C \times \Delta \downarrow b) \to b
$$

are weak equivalences for every fibrant object $b \in B$.

...

=--

Notice that the theorem just mentions plain combinatorial model categories, not [[simplicial model category|simplicial model categories]]. But of course by basic facts of [[enriched category theory]] $Funct(C^{op}, SSet)$ is an [[SSet]]-[[enriched category]] and its projective [[global model structure on functors]] $Func(C^{op}, SSet)_{proj}$ is compatibly a [[simplicial model category]], as are all its [[Bousfield localization of model categories|Bousfield localizations]]. (See [[model structure on simplicial presheaves]] for more details.) Therefore an immediate but very useful corollary of the above statement is

\begin{corollary}\label{EveryCombinatorialModelCatQEquivalentToLeftProper}

Every combinatorial model category is [[Quillen equivalence|Quillen equivalent]] to one which is

* a [[simplicial model category]]

* a left [[proper model category]].

\end{corollary}

### Tractable combinatorial model categories

A combinatorial model category is a [[tractable model category]] precisely if the set $I$ of generating cofibrations can be chosen such that all elements have a cofibrant object as domain.

A [[proper model category|left proper]] combinatorial model category precisely if the set $J$ of generating trivial cofibrations can be chosen with cofibrant domain.

This are corollaries 2.7 and 2..8 in [Bar](http://www.math.harvard.edu/~clarkbar/complete.pdf).

## Properties

### Homotopy colimits {#hocolims}

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

In a combinatorial model category, for every sufficiently large regular [[cardinal number|cardinal]] $\kappa$ the following holds:

* $\kappa$-[[filtered colimits]] preserve weak equivalences;

* hence $\kappa$-[[filtered colimits]] are already [[homotopy colimit]]s.

=--

See also at _[[filtered homotopy colimit]]_.

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

This appears as proposition 7.3 in [Dug00](http://arxiv.org/abs/math/0007068), reproduced for instance as prop. 2.5 in [Bar](http://www.math.harvard.edu/~clarkbar/complete.pdf).

The point is to choose $\kappa$ such that all domains and codomains of the generating cofibrations are $\kappa$-[[compact object]]. This is possible since by assumption that $C$ is a [[locally presentable category]] all its objects are [[small object]]s, hence each a $\lambda$-[[compact object]] for some cardinal $\lambda$. Take $\kappa$ to be the maximum of these.

Let $F, G : J \to C$ be $\kappa$-filtered diagrams in $C$ and $F \to G$ a [[natural transformation]] that is degreewise a weak equivalence. Using the functorial factorization provided by the [[small object argument]] this may be factored as $F \to H \to G$ where the first transformation is objectwise an acyclic cofibration and the second objectwise an acyclic fibration, and by functoriality of the factorization this sits over a factorization

$$
  \lim_\to F \stackrel{\simeq}{\hookrightarrow} \lim_\to H \stackrel{}{\to}\lim_\to G
  \,.
$$

It remains to show that the second morphism is a weak equivalence. But by our factorization and by [[category with weak equivalences|2-out-of-3]] applied to our componentwise weak equivalences, we have that all its components $H(j) \to G(j)$ are acyclic fibrations.

At [[small object]] it is described in detail how $\kappa$-smallness of an object $X$ implies that morphisms from $X$ into a $\kappa$-filtered colimit lift to some component of the colimit

$$
  \array{
    \cdots&\to&H(j-1) &\to& H(j) &\to& H(j+1) &\to& \cdots
    \\
    &&&{}^{\mathllap{\exists \hat f}}\nearrow&\downarrow & \swarrow
    \\
    &&X& \stackrel{\forall f}{\to} &\lim_\to H
  }
  \,.
$$

So given a diagram

$$
  \array{
    X &\to& \lim_\to H
    \\
    \downarrow^{\mathrlap{\in I}} && \downarrow
    \\
    Y &\to& \lim_\to G
  }
$$

we are guaranteed, by the $\kappa$-[[small object|smallness]] of $X$ and $Y$ that we established above, a lift

$$
  \array{
    X &\to& H(j) &\to& \lim_\to H
    \\
    \downarrow^{\mathrlap{\in I}} && \downarrow^{\in \mathrlap{\in rlp(I)}}
    &&
    \downarrow
    \\
    Y &\to& G(j) &\to& \lim_\to G
  }
$$

into some component at $j \in J$ and hence a lift

$$
  \array{
    X &\to& H(j) &\to& \lim_\to H
    \\
    \downarrow^{\mathrlap{\in I}} &
    \nearrow
    & \downarrow^{\in \mathrlap{\in rlp(I)}}
    &&
    \downarrow
    \\
    Y &\to& G(j) &\to& \lim_\to G
  }
  \,.
$$

Thereby $\lim_\to H \to \lim_\to G$ is in $rlp(I) \subset W$.

=--

### Bousfield localization

Combinatorial model categories, like [[cellular model category|cellular model categories]] have a good theory of [[Bousfield localization of model categories|Bousfield localizations]], at least if in addition they are left [[proper model category|proper]]. See [[Bousfield localization of model categories]] for more on this.

## Examples

### Basic examples

Basic examples are

* [[sSet]] with its standard [[model structure on simplicial sets]];

* [[sSet]] with the Joyal-[[model structure for quasi-categories]];

  notice that this is not directly a [[simplicial model category]], but is enriched over itself. A Quillen equivalent combinatorial simplicial model category is

  * the [[model structure for Cartesian fibrations]] over the point.

* the category of [[dendroidal set]]s with its [[model structure on dendroidal sets]].

* the categories of $(n,r)$-[[Theta space]]s.

### Cisinski model structures

More generally, every _[[Cisinski model structure]]_ is combinatorial.

### Derived examples

Further classes of examples are obtained from such basic examples by localizing presheaf categories with values in these:

* For $V$ a combinatorial model category and $C$ a [[small category]] the injective and projective [[model structure on functors]] $Funct(C,V)_{inj}$ and $Funct(C,V)_{proj}$ are again combinatorial model categories. See there for details.

* If $V$ is a left or right [[proper model category]] then so is $Funct(C,V)_{inj}$ and $Funct(C,V)_{proj}$ and hence the standard results of the theory of [[Bousfield localization of model categories]] applies, which ensures that all left Bousfield localizations $L_S Funct(C,V)$ are again combinatorial model categories. Such local [[model structure on homotopical presheaves|local model structures on homotopical presheaves]] includes notably the local [[model structure on simplicial presheaves]].

### From cofibrantly generated model categories

Not every [[cofibrantly generated model category]] is also a combinatorial model category.

For instance:

+-- {: .num_example}
###### (Counter)example
[[Top]] with the standard [[model structure on topological spaces]] is cofibrantly generated, but not combinatorial. But it is [[Quillen equivalence|Quillen equivalent]] to a combinatorial model structure, namely to the standard [[model structure on simplicial sets]] (see [[homotopy hypothesis]]).
=--

One might therefore ask which cofibrantly generated model categories are Quillen equivalent to combinatorial ones.  It turns out that if one assumes the large-cardinal hypothesis [[Vopenka's principle|Vopěnka's principle]], then *every* cofibrantly generated model category is Quillen equivalent to a combinatorial one.  In fact, if we slightly generalize the notion of "cofibrantly generated," this statement is equivalent to Vopěnka's principle.  For a discussion of this see

* [[Jiří Rosický]], _Are all cofibrantly generated model categories combinatorial?_ ([ps](http://www.math.muni.cz/~rosicky/papers/cof1.ps))

Although Vopěnka's principle cannot be proven from [[ZFC]], and in fact is fairly strong as [[large cardinal]] hypotheses go, this means that looking for cofibrantly generated model categories that are not Quillen equivalent to combinatorial ones is probably a waste of time.  Certainly, all known cofibrantly generated model categories *are* Quillen equivalent to simplicial ones, usually in a fairly natural way.

### Simplicial combinatorial model categories

Those combinatorial model categories that are at the same time [[simplicial model categories]] are precisely those that present [[presentable (∞,1)-categories]].
See [[combinatorial simplicial model category]].

## Related concepts

* [[Ho(CombModCat)]]

* [[Cisinski model structure]]

[[!include locally presentable categories - table]]

[[!include algebraic model structures - table]]

## References

Much of the theory of combinatorial model categories goes back to [[Jeff Smith]]. Apparently Smith will eventually present a book on this subject. To date, however, his ideas and results appear reproduced in articles of other authors.

After [[Jeff Smith]] presented his recognition theorem at a conference in Barcelona, its first appearance in a publication is apparently

* {#Beke} [[Tibor Beke]], Theorem 1.7 in: *Sheafifiable homotopy model categories*,  Math. Proc. Cambridge Philos. Soc. __129__ 3 (2000) 447-475 [[arXiv:math/0102087](http://arxiv.org/abs/math/0102087), [doi:10.1017/S0305004100004722](https://doi.org/10.1017/S0305004100004722)]

An early explicit account of the notion of combinatorial model categories is in Section II of:

* {#Dugger01} [[Daniel Dugger]], *[[Combinatorial model categories have presentations]]*, Adv. Math. **164** 1 (2001) 177-201 [[arXiv:math/0007068](http://arxiv.org/abs/math/0007068), [doi:10.1006/aima.2001.2015](https://doi.org/10.1006/aima.2001.2015)]

(which goes on to state and proof [[Dugger's theorem]], based on results in Dugger's _[[Universal homotopy theories]]_).

The definition of combinatorial model categories is recalled also as:

* {#Barwick10} [[Clark Barwick]], Def. 1.3 in: *On left and right model categories and left and right Bousfield localizations*, Homology, Homotopy and Applications **12** 2 (2010) 245–320
[[doi:10.4310/hha.2010.v12.n2.a9](https://doi.org/10.4310/hha.2010.v12.n2.a9), subsuming:[arXiv:0708.2067](https://arxiv.org/abs/0708.2067), [arXiv:0708.2832](https://arxiv.org/abs/0708.2832), [arXiv:0708.3435](http://arxiv.org/abs/0708.3435)]

* {#Lurie} [[Jacob Lurie]], Def. A.2.6.1 in: _[[Higher Topos Theory]]_ (2009)

Smith's theorem appears as [Lurie, A.2.6.10](#Lurie)  and as [Barwick, prop. 1.7](#Barwick). For more see at *[[Bousfield localization of model categories]]*.

Futher details are discussed in:

* [[Jiří Rosický]], *On combinatorial model categories*, Appl. Cat. Str. 17 (2009), 303-316 ([arXiv:0708.2185](https://arxiv.org/abs/0708.2185))

Review:

* [[George Raptis]], *Notes on combinatorial model categories* 2014/2020 ([pdf](https://graptismath.net/files/notes-comb-model-cat.pdf), [[RaptisCombinatorialModelCategories.pdf:file]])

On the [[localization of a 2-category]][[Ho(CombModCat)]] of combinatorial model categories at the [[Quillen equivalences]] and its equivalence to the [[homotopy 2-category]] of (locally) presentable [[derivators]]:

* {#Renaudin06} [[Olivier Renaudin]], *Plongement de certaines théories homotopiques de Quillen dans les dérivateurs*, Journal of Pure and Applied Algebra Volume 213, Issue 10, October 2009, Pages 1916-1935
([arXiv:math/0603339](https://arxiv.org/abs/math/0603339), [doi:10.1016/j.jpaa.2009.02.014](https://doi.org/10.1016/j.jpaa.2009.02.014))

See also

* {#Low14} [[Zhen Lin Low]], _The heart of a combinatorial model category_, Theory and Applications of Categories, Vol. 31, 2016, No. 2, pp 31-62 ([arXiv:1402.6659](http://arxiv.org/abs/1402.6659))

[[!redirects Smith's theorem]]
[[!redirects Smith theorem]]
[[!redirects Smith's theorem]]
[[!redirects Jeff Smith's theorem]]
[[!redirects Jeff Smith theorem]]
[[!redirects Jeff Smith's theorem]]
[[!redirects Smith recognition theorem]]
[[!redirects Smith's recognition theorem]]

[[!redirects combinatorial model categories]]


combinatorial representation theory

The philosophy of the subfield and explicit approach to representation theory which is sometimes called combinatorial representation theory and the main examples of problems in it is described in the influential survey

* Hélène Barcelo, [[Arun Ram]], _Combinatorial representation theory_, Combinatorics at MSRI in Berkeley: New perspectives in algebraic combinatorics (Berkeley, CA, 1996--97), 23--90, Math. Sci. Res. Inst. Publ. __38__, Cambridge Univ. Press 1999.[pdf](http://www.ms.unimelb.edu.au/~ram/Publications/1999MSRIpubv38p23.pdf), [math.RT/9707221](http://arxiv.org/abs/math/9707221)

Related entries include: [[representation theory]], [[symmetric function]], [[permutation representation]], [[crystal basis]], [[Kazhdan-Lusztig theory]], [[Jack polynomial]], [[Macdonald polynomial]], [[Littlewood-Richardson coefficient]], [[Schur-Weyl duality]], [[Schur function]], [[Schur functor]], [[Young diagram]]

Some other references close in spirit

* P. Littelmann, _Paths and root operators in representation theory_, Ann. Math. 142, 1995, pp. 499--525.
* Fulton, J. Harris, _Representation theory_,
* [[Claudio Procesi]], _Lie groups, an approach through invariants and representations_, Universitext, Springer 2006, [gBooks](http://books.google.co.in/books?id=Sl8OAGYRz_AC&printsec=frontcover&hl=hr&source=gbs_atb)

category: combinatorics


combinatorial simplicial model category

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

#Contents#
* table of contents
{:toc}

## Idea

Combinatorial simplicial model categories, i.e. [[sSet-enriched category|sSet-enriched]] [[model categories]] that are both

* [[combinatorial model categories]]

* [[simplicial model categories]]

are precisely those [[model category|model categories]] that present [[presentable (infinity,1)-category|presentable $\infty$-categories]] (see there for details).

## Related concepts

* [[Ho(CombModCat)]]

[[!redirects combinatorial simplicial model categories]]
[[!redirects simplicial combinatorial model category]]
[[!redirects simplicial combinatorial model categories]]


This is a page about a major book on combinatorial [[species]] and about its sequel.

* F. Bergeron, G. Labelle and P. Leroux, __Combinatorial species and tree-like structures__, Enc. of Math. and its Appl., Camb. Univ. Press 1997

which is a corrected translation from French

* François Bergeron, Gilbert Labelle, Pierre Leroux, _Théorie des espèces et combinatoire des structures arborescentes_, LaCIM, Montréal (1994).

* [bookpage](http://bergeron.math.uqam.ca/Species/especes.html)
* Preface by [[Gian-Carlo Rota]] ([pdf](http://bergeron.math.uqam.ca/Species/foreword.pdf), [ps.gz](http://bergeron.math.uqam.ca/Species/foreword.ps.gz))

Table of Contents

1 Introduction to Species of Structures

    1.1 Notion of Species of Structures
    1.2 Associated Series
    1.3 Addition and Multiplication
    1.4 Substitution and Differentiation

2 Complements on Species of Structures

    2.1 Pointing and Cartesian Product
    2.2 Functorial Composition
    2.3 Weighted Species
    2.4 Extension to Multisort Context
    2.5 Virtual Species
    2.6 Molecular and Atomic Species

3 Combinatorial Functional Equations

    3.1 Lagrange Inversion
    3.2 Implicit Species Theorem
    3.3 Quadratic Iterative Methods
    3.4 Elements of Asymptotic Analysis

4 Complements on Unlabeled Enumeration

    4.1 The Dissymmetry Theorem for Trees
    4.2 Connected Graphs and Blocks
    4.3 Proof of the Substitution Formula
    4.4 Asymmetric Structures

5 Species on Totally Ordered Sets

    5.1 L-Species
    5.2 Combinatorial Differential Equations

Appendix 1: Group Action and Polya Theory
Appendix 2: Miscellaneous Tables

New book

* F. Bergeron, G. Labelle, P. Leroux, _Introduction to the Theory of
Species of Structures_ 2008, [pdf](http://bergeron.math.uqam.ca/Site/bergeron_anglais_files/livre_combinatoire.pdf)

category:reference


combinatorial spectrum

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

#Contents#
* table of contents
{:toc}

## Idea

A _combinatorial spectrum_ is to a [[spectrum]] of [[topological space]]s as a [[simplicial set]] is to a [[topological space]]: it is a [[graded set]] that behaves like a set of simplices constituting a space, where the special property is that the simplices are not just in non-negative degree $n \in \mathbb{N}$ but in all integral degrees $n \in \mathbb{Z}$.

## Definition

A **combinatorial spectrum** is ([Kan 63, def. 4.1](#Kan63))

* a sequence $E = \{E_n\}_{n \in \mathbb{Z}}$ of [[pointed set]]s

* equipped for each $n \in \mathbb{Z}$ and $i \in \mathbb{N}$ with

  * morphisms of [[pointed set]]s $d_i : E_n \to E_{n-1}$ called **face maps**;

  * morphisms of [[pointed set]]s $s_i : E_n \to E_{n+1}$ called **degeneracy maps**

* such that

  * the usual [[simplicial set|simplicial identities]] are satisfied;

  * each simplex has only finitely many faces different from the point of $E_{n-1}$: i.e. for every $x \in E_n$ there are only finitely many $i \in \mathbb{N}$ for which $d_i(x)$ is not the point.

## Examples

The standard [[simplicial set]]s corresponding to the standard [[simplex|simplices]] have their analogs for simplicial spectra. . The difference is that regarded as a spectrum the $k$-simplex may sit in any degree $n \in \mathbb{N}$, not necessarily in degree $k$.

**The $(k+1)$-simplex in degree $n$.** For each integer $n \in \mathbb{Z}$ and $k \in \mathbb{N}$ there is a spectrum

$$
  \Delta^{k}[n-k]
$$

which is _generated_ from a single element $x \in E_n$ subject to the relation that $d_i x = *$ for $i \gt k$.  So this is something with $(k+1)$-faces, hence looking like a $k$-[[simplex]], but sitting in degree $n$.

The sub-spectrum

$$
  \partial (\Delta^{k}[n-k])
$$

of $\Delta^{k}[n-k]$ generated by the faces $d_i x$. This is the **boundary** of the $k$-simplex in degree $n$.

The spectrum

$$
  S^n
$$

generated by a single simplex $x \in E_n$ subject to the relation $d_i x = 0$ for all $i$. This is the **$n$-sphere** as a spectrum.

The sub-spectrum $\Lambda^{k}_j[n-k]$ for $0 \leq j \leq k$ is the sub-spectrum of $\Delta^{k}[n-k]$ generated from all the faces $d_i x$ except $d_j x$. This is the  $j$th **horn** of the $k$-simplex in degree $n$. Compare with the [[horn]] of a [[simplex]].

As for [[simplex|simplices]], there are canonical [[horn]] inclusion morphisms of combinatorial spectra

$$
  \Lambda^k_j[n-k] \hookrightarrow \Delta^k[n-k]
  \,.
$$

A condition entirely analogous to the [[Kan fibration]] condition on [[Kan complex|Kan simplicial sets]] leads to the notion of **Kan combinatorial spectrum**.

## Properties

### Model category structure
 {#ModelCategoryStructure}

The category of combinatorial spectra admits a [[model
structure]] ([Brown 73](#Brown73)), whose cofibrations are monomorphisms
and weak equivalences are maps that induce isomorphisms
on homotopy groups.

Generating (acyclic) cofibrations are given by inclusions
of horns respectively boundaries into stable simplices,
as defined in the previous section, in complete analogy
with the usual Kan-Quillen model structure
on simplicial sets.
(Acyclic) fibrations are maps satisfying
the corresponding lifting properties.

This is indeed a [[model structure for spectra]], related by a [[zig-zag]] of [[Quillen equivalences]] to the [[Bousfield-Friedlander model structure]] on [[sequential spectra]]  ([Bousfield-Friedlander 78, section 2.5](#BousfieldFriedlander78)). See also [below](#RelationshipToOtherSpectra).

### Smash product

In ([Brown 73, Appendix A](#Brown73)) is defined a  [[smash product of spectra]] on the [[homotopy category]] of combinatorial spectra.
The main idea is to define the smash product
of a stable $m$-simplex and a stable $n$-simplex
to be the stable $(m+n)$-simplex whose face maps
are defined using the face maps of the two simplices
involved using a formula that somewhat resembles
the formula for the differential of the tensor
product of two chain complexes.

Whether or not it is possible to introduce a symmetric
monoidal smash product on the category of combinatorial
spectra obtaining a [[monoidal model category]]
that is Quillen equivalent to the
monoidal model category of (say)
symmetric simplicial spectra is currently
an open problem.

### Relationship to other spectra
 {#RelationshipToOtherSpectra}

From the perspective of a combinatorial spectrum, an "intuitive spectrum" is supposed to be some sort of space-like object having "cells in all integer dimensions," while a "space" (or simplicial set) has cells only in nonnegative dimensions.  The traditional definitions of spectra approximate this intuition by using a *sequence* of spaces $\{E_n\}$ with maps $E_n \to \Omega E_{n+1}$ or $\Sigma E_n \to E_{n+1}$, where we think of the space $E_n$ as being "shifted down by $n$ dimensions."  Thus, for instance, the $(-2)$-cells of the spectrum can come from 0-cells of $E_2$, or 1-cells in $E_3$, or 2-cells in $E_4$, etc.  The structure maps $\Sigma E_n \to E_{n+1}$ support this intuition, since the suspension $\Sigma$ shifts things up by one dimension; thus it maps the $k$-cells of $E_n$ into the $k+1$-cells of $E_{n+1}$.

In fact, this can be made precise: starting from a spectrum of simplicial sets, in the sense of a sequence of spaces with maps $\Sigma E_n \to E_{n+1}$, one can construct a combinatorial spectrum by "piecing together" the cells in all dimensions.  This construction can be found in Kan's original article; it provides an equivalence
of [[homotopy theories]] between combinatorial spectra and ordinary spectra built from simplicial sets. In ([Bousfield-Friedlander 78, section 2.5](#BousfieldFriedlander78)) this is lifted to a [[zig-zag]] of [[Quillen equivalence]] between the model structure for Kan's combinatorial spectra and the [[Bousfield-Friedlander model structure]] on [[sequential spectra]] in [[sSet]] to

[[Mike Shulman|I]] don't know whether anyone has gone back to treat combiantorial from a "modern" standpoint, such as by putting a [[monoidal model category]] structure on combinatorial spectra.  They do seem less interesting and useful from a modern standpoint, because no one has ever managed to give them a [[smash product]] which is associative and unital on the point-set level; thus they don't provide a good framework for talking about ($A_\infty$ or $E_\infty$) ring spectra, module spectra, and other aspects of [[brave new algebra]].  It's also not clear how hard anyone has tried, though.  Presumably one would have to modify the definition by incorporating the "symmetries" somehow, as is done for example by passing from ordinary simplicial-set spectra to [[symmetric spectrum|symmetric spectra]].

### Stable Dold-Kan correspondence

Combinatorial spectra [[internalization|internal]] to [[abelian groups]] ([Kan 63, def. 51](#Kan63)) are [[equivalence of categories|equivalent]] to [[chain complexes]] ([Kan 63, prop.5.8](#Kan63)). See at _[[stable Dold-Kan correspondence]]_ for more on this.

## References

An early reference is

* {#Kan63} [[Daniel Kan]], _Semisimplicial spectra_, Illinois J. Math. Volume 7, Issue 3 (1963), 463-478. ([Euclid](http://projecteuclid.org/euclid.ijm/1255644953))

see also at _[[stable Dold-Kan correspondence]]_.

The definition is recalled in part II, section 7 of

* {#Brown73} [[Kenneth Brown]], _[[BrownAHT|Abstract homotopy theory and generalized sheaf cohomology]]_, 1973

Discussion of the relation (equivalence of [[homotopy categories]]) to the [[Bousfield-Friedlander model structure]] on [[sequential spectra]] in [[simplicial sets]] is in

* {#BousfieldFriedlander78} [[Aldridge Bousfield]], [[Eric Friedlander]], section 2.5 of _Homotopy theory of $\Gamma$-spaces, spectra, and bisimplicial sets_, Springer Lecture Notes in Math., Vol. 658, Springer, Berlin, 1978, pp. 80-130. ([pdf](https://www.math.rochester.edu/people/faculty/doug/otherpapers/bousfield-friedlander.pdf))

See also

* Ruian Chen, [[Igor Kriz]] and Ales Pultr, _Kan's combinatorial spectra and their sheaves revisited_, Theory and Applications of Categories, Vol. 32, 2017, No. 39, pp 1363-1396. ([TAC](http://www.tac.mta.ca/tac/volumes/32/39/32-39abs.html))

[[!redirects combinatorial spectra]]

[[!redirects model structure on combinatorial spectra]]


combinatorial weak factorization system

# Combinatorial weak factorization systems

* table of contents
{: toc}

## Definition

A [[weak factorization system]] $(L,R)$ on a [[locally presentable category]] is **combinatorial** if it is cofibrantly generated by a set of morphisms.  That is, there is a set $I$ of morphisms such that for any morphism $f$, we have $f\in R$ if and only if $f$ has the right lifting property with respect to all $i\in I$.

## Construction

By the [[small object argument]], any set $I$ of morphisms in a locally presentable category generates a combinatorial weak factorization system.

## Properties

* Combinatorial wfs can be both right- and left-lifted along adjunctions; see [[transferred model structure]].

## Related concepts

* A [[combinatorial model category]] is a [[model category]] whose two constituent weak factorization systems are combinatorial.

* A more general notion is an [[accessible weak factorization system]].

* Any combinatorial weak factorization system can be equipped with the structure of an [[algebraic weak factorization system]] (generally in more than one way).

## References

* {#MR13} [[M. Makkai]], [[J. Rosický]], _Cellular categories_, 2013, [arXiv:1304.7572](https://arxiv.org/abs/1304.7572)

[[!include algebraic model structures - table]]

[[!redirects combinatorial weak factorization systems]]
[[!redirects combinatorial wfs]]
[[!redirects combinatorial WFS]]


combinatorics

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

#Contents#
* table of contents
{:toc}

## Idea

**Combinatorics** is a field whose main subject (defining its central subfield of [[enumerative combinatorics]]) is the study of cardinality of finite sets of data, which are described by elementary rules, and reducing the question of cardinality of more complicated data to simpler such "combinatorial" problems. The basic method of combinatorics is to find bijections between two sets of data defined in manifestly different way, and in particular to find the bijections between the sets of more complicated and less explicit origin, with the sets whose definition and structure is more explicit, and considered more manageable (by already known "combinatorial" means). For example, a typical bijection the isomorphisms classes of [[representation]]s of some object (e.g. of a group) and "combinatorial" data like [[Young tableaux]], graphs of some kind and so on (this comprises [[combinatorial representation theory]]).

More widely, combinatorics is also studying various *deterministic* algorithms on finite sets with structure, involving counting, coloring and simple-minded elementary steps/decisions. When studying some other structure, which is not necessarily a finite structure, one can use "combinatorial arguments' at some step, or develop a construction which is partly combinatorial. Some subclasses of examples of structures in mathematics are fully combinatorial in their nature. For example, there are so called **combinatorial games** in [[game theory]].

Some of the questions in [[set theory|set]] and [[model theory]] concerning the fine questions on cardinality of infinite sets (and their independence from various axioms in [[set theory]]) are often called the "infinitary combinatorics".

## Related entries

* [[graph]], [[quiver]]

* [[matroid]], [[oriented matroid]]

* [[permutation]]

* [[symmetric function]], [[noncommutative symmetric function]]

* [[Schur functor]], [[Schur function]], [[plethysm]]

* (combinatorial) [[species]]

* [[combinatorial functor]]

* [[Catalan number]], [[Motzkin number]], [[Moebius function]]

* [[game theory]]

* [[combinatorial representation theory]]

* [[enumerative combinatorics]]

## References

* {#Stanley11} [[Richard Stanley]], *[Enumerative combinatorics](http://www-math.mit.edu/~rstan/ec/)* -- [Volume 1](http://www-math.mit.edu/~rstan/ec/ec1/), 2011 ([pdf](http://www-math.mit.edu/~rstan/ec/ec1.pdf))

* {#Stanley99} [[Richard Stanley]], *Enumerative combinatorics 2*, Cambridge University Press (1999, 2010) ([doi:10.1017/CBO9780511609589](https://doi.org/10.1017/CBO9780511609589), [webpage](http://www-math.mit.edu/~rstan/ec/))

* [[Bruce Sagan]], *Combinatorics: The Art of Counting*, Graduate studies in Mathematics **210**, AMS  ([ISBN:978-1-4704-6032-7](https://bookstore.ams.org/gsm-210) [pdf](https://users.math.msu.edu/users/bsagan/Books/Aoc/final.pdf))

* Martin Aigner, *A Course in Enumeration*, Springer 2007 ([doi:10.1007/978-3-540-39035-0](https://link.springer.com/book/10.1007/978-3-540-39035-0))

On combinatorics in [[algebraic topology]]:

* [[Dmitry N. Kozlov]], *Trends in topological combinatorics* [[arXiv:math/0507390](https://arxiv.org/abs/math/0507390)]

* [[Dmitry Kozlov]], *Combinatorial Algebraic Topology*,  Algorithms and Computation in Mathematics **21**, Springer (2008) [[doi:10.1007/978-3-540-71962-5](https://doi.org/10.1007/978-3-540-71962-5)]

An application of [[Hodge theory]]

* {#Huh22} [[June Huh]], _Combinatorics and Hodge theory_, Proc. Int. Cong. Math. **1** (2022)
 [[doi:10.4171/ICM2022/205](), [pdf](https://www.mathunion.org/fileadmin/IMU/Prizes/Fields/2022/jh.pdf), [[Huh-CombinatoricsAndHodgeTheory.pdf:file]]]

A categorification of many counting procedures may be viewed via so called objective linear algebra. Some ideas can be found in

* [[Imma Gálvez-Carrillo]], [[Joachim Kock]], [[Andrew Tonks]], _Decomposition spaces, incidence algebras and Möbius inversion_, [arXiv:1404.3202](https://arxiv.org/abs/1404.3202)

[[!redirects combinatorial]]
[[!redirects combinatorialist]]

category: combinatorics


combinatorics - contents

**[[combinatorics]]**

[[enumerative combinatorics]]

[[graph theory]]

[[rewriting]]

**Basic structures**

* [[binary linear code]]
* [[chord diagram]]
* [[combinatorial design]]
* [[graph]]
* [[Latin square]]
* [[matroid]]
* [[partition]]
* [[permutation]]
* [[shuffle]]
* [[tree]]
* [[Young diagram]]

**Generating functions**

* [[combinatorial species]]
* [[generating function]]
* [[power series]]

**Proof techniques**

* [[bijective proof]]
* [[Lagrange inversion]]
* [[Möbius inversion]]
  * [[order polynomial]]
  * [[zeta polynomial]]
* [[Pólya enumeration theorem]]

**Combinatorial identities**

* [[binomial theorem]]
* [[Catalan number]]
* [[Chu–Vandermonde identity]]

**Polytopes**

* [[associahedron]]

category: combinatorics


combinatory categorial grammar

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

# Contents
* table of contents
{:toc}

## Idea

Combinatory categorial grammars are a kind of [[categorial grammar]] inspired by [[combinatory logic]].
They are [[mildly context-sensitive grammar|mildly context-sensitive]].

## References

* {#Steedman87} Steedman, Mark. _Combinatory grammars and parasitic gaps._ Natural Language & Linguistic Theory 5.3 (1987)


combinatory logic

# Combinatory logic

* table of contents
{: toc}

## Idea

> Combinator algebra has always struck me as a subject almost impossible to watch: it's a lousy spectator sport but has a reputation of habit-forming as a participator sport. P. Freyd ([1989](#Freyd89), p.63)

**Combinatory logic** is a rephrasing of the [[lambda calculus]] that avoids explicit mention of [[variables]] and of [[lambda abstraction]].  Instead it uses [[combinators]] -- in the most common case, they are traditionally called $S$, $K$, and $I$ -- with the only other operation being [[application]].

Since this move circumvents the necessity of variable management, combinatory logic is important for applications in [[computer science]] and [[linguistics]].

Under the [[propositions as types]] correspondence, combinatory logic corresponds to the presentation of [[logic]] using a [[Hilbert system]] (instead of [[natural deduction]] or [[sequent calculus]]).  There are versions of combinatory logic corresponding to [[linear logic]] and [[affine logic]], generally called **BCI logic** and **BCK logic** (after the combinators they use).

## Historical remarks

The 1920s gave birth to several formal systems that permitted to base mathematics on the notion of function like e.g. the lambda calculus or von Neumann 'set' theory that eventually morphed into [[NBG|Neumann-Bernays-Gödel set theory]]. Notably the first of such systems was combinatory logic which was introduced by [[Moses Schönfinkel]] in 1920 ([Schönfinkel 1924](#Schoen24)) and developed (partly independently) by [[Haskell Curry]] by the end of the decade.

Of course, a later such function based approach to mathematics is [[category theory]] and there has been in fact some interaction between combinatory logic and category theory ([Bunder 1984](#Bunder84)). In 1972 when the connection between Cartesian closed categories and lambda calculus was an object of active research, [[Jim Lambek]] wrote:

> Schönfinkel's original program to base all of mathematics on combinatory logic can now be said to have been carried out by Lawvere in his many articles pursuing a categorical formulation of mathematics. ([1972](#Lambek72), p.57)

## The combinators S, K, and I

The basic combinators can be defined in terms of $\lambda$-calculus as

* $S = \lambda x. \lambda y. \lambda z. (x z)(y z)$
* $K = \lambda x. \lambda y. x$
* $I = \lambda x. x$

In [[simply typed lambda-calculus]], these combinators are [[polymorphic]] with types

* $S : (A \to (B\to C)) \to ((A\to B) \to (A\to C))$
* $K : A\to (B\to A)$
* $I : A\to A$

for arbitrary types $A,B,C$.

When combinatory logic is presented on its own rather than in terms of $\lambda$-calculus, the combinators are characterized by reduction rules that mirror their definition in terms of $\lambda$-abstractions:

* $S x y z \equiv (x z)(y z)$
* $K x y \equiv x$
* $I x \equiv x$

Sometimes the combinator $I$ is omitted, as it can be defined from $S$ and $K$ as $I=S K K$.

## Encoding abstraction

The combinators $S$, $K$, and $I$ are used to encode $\lambda$-abstraction in the following way.  If $M$ is a term in combinatory logic (perhaps containing variables), we define $\lambda x.M$ by induction over the structure of the term $M$.

* If $M=x$, then $\lambda x.x = I$.
* If $M$ is a variable $y\neq x$, then $\lambda x.y = K y$.
* If $M = N P$ is an application of one term to another, then $\lambda x.M = S (\lambda x.N) (\lambda x.P)$, so we can recursively translate $\lambda x.N$ and $\lambda x.P$.

(It is common to see this described with the second case given as "$\lambda x.M = K M$ whenever $M$ does not contain $x$ as a [[free variable]]".  However, this overlaps with the third case and is not structurally recursive.  The above definition translates $\lambda x.y z$ to $S(K y)(K z)$, while the one with the free-variable-check produces $K(y z)$; both reduce to $y z$ when applied to any argument.)

Using this construction of an "abstraction operator" in the combinatory logic, we can then proceed to translate an arbitrary term of $\lambda$-calculus into combinatory logic by translating all the abstractions "from the inside out".  Formally, we crawl over the term and whenever we find an abstraction, we first recurse into its body, then we apply the above abstraction operator.

The construction of an abstraction operator in combinatory logic is analogous to a [[cut admissibility]] theorem; the resulting translation of $\lambda$-calculus into combinatory logic is then analogous to the corresponding [[cut-elimination]] theorem.

## Linear (BCI) and affine (BCK) combinatory logic
 {#LinearAndAffine}

Versions of combinatory logic that correspond to [[linear logic]] and [[affine logic]] have been studied under the names **BCI logic** and **BCK logic**  respectively, with the letters indicating the combinators used, where

* $B = \lambda x.\lambda y.\lambda z. x(yz) : (B\to C) \to (A\to B) \to (A\to C)$
* $C = \lambda x.\lambda y.\lambda z. (xz)y : (A\to (B\to C)) \to (B \to (A\to C))$

The combinator $I$ is also definable as $CKK$, so it is also available in BCK logic (which might therefore be called BCKI logic).  Similarly, traditional combinatory logic could be called "SKI logic", or just "SK logic".

Note that $B$ has the same type as the "composition" operation in a [[closed category]], while $C$ has the type of a symmetry/exchange for a closed category.  Similarly, $K$ is a sort of [[weakening rule]] while $S$ combines composition, symmetry, and [[contraction rule|contraction]].

## Remark

A _partial combinatory algebra_ , also called a _Schönfinkel algebra_ ,  is an algebraic structure that abstracts the structure of (partial) combinatory logic. They play an important role in the theory of [[realizability topos|realizability toposes]].

## Related pages

* [[partial combinatory algebra]]
* [[combinator]]
* [[lambda-calculus]]
* [[categorial grammar]]

## References

* {#Schoen24}[[Moses Schönfinkel]], _Über die Bausteine der mathematischen Logik_ , Math. Ann. **92** (1924) pp.305-316. ([gdz](http://gdz.sub.uni-goettingen.de/dms/load/img/?PID=GDZPPN002270110)) English translation with comments by Quine pp.355-366 in van Heijenoort (ed.), _From Frege to Gödel_ , Harvard UP 1967.

* {#Bunder84}M. W. Bunder, Category Theory Based on Combinatory Logic , Arch. Math. Logik **24** (1984) pp.1-15. ([gdz](http://gdz.sub.uni-goettingen.de/dms/load/img/?PPN=GDZPPN002045680))

* {#Freyd89}[[Peter Freyd]], _Combinators_ , Contemp. Math. **92** (1989) pp.63-66. ([pdf](http://www.math.upenn.edu/~pjf/combinators.pdf))

* [[J. Roger Hindley]], [[Jonathan P. Seldin]], *Lambda-Calculus and Combinators -- An Introduction*, Cambridge University Press (2008) [[doi:10.1017/CBO9780511809835](https://doi.org/10.1017/CBO9780511809835)]

* {#Lambek72}[[Jim Lambek]], _Deductive systems and categories III. Cartesian closed categories, intuitionist propositional calculus, and combinatory logic_ , pp.57–82 LNM **274** Springer Heidelberg 1972.

[[!redirects combinatory logics]]
[[!redirects BCI logic]]
[[!redirects BCI-logic]]
[[!redirects BCK logic]]
[[!redirects BCK-logics]]

[[!redirects combinator]]
[[!redirects combinators]]

[[!redirects combinator logic]]
[[!redirects combinator logics]]