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| In [[mathematics]], a '''simplicial set''' is a construction in [[category (mathematics)|categorical]] [[homotopy theory]] which is a purely algebraic model of the notion of a "[[well-behaved]]" topological space. Historically, this model arose from earlier work in [[combinatorial topology]] and in particular from the notion of [[simplicial complex]]es.
| | Catrina Le is what's constructed on her birth credentials though she doesn't really like being called prefer that. Software raising is where her primary income comes from remember, though , soon her husband and consequently her will start their personal own business. What the woman loves doing is to get to karaoke but jane is thinking on starting something totally new. For years she's been living with Vermont. She is running and looking after a blog here: http://[http://Www.Google.com/search?q=Prometeu&btnI=lucky Prometeu].net/<br><br> |
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| ==Motivation==
| | my homepage ... [http://Prometeu.net/ clash of clans unlimited gems] |
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| A simplicial set is a categorical (that is, purely algebraic) model capturing those [[topological space]]s which can be built up (or faithfully represented up to homotopy) from [[simplex|simplices]] and their incidence relations. This is similar to the approach of [[CW complex]]es to modeling topological spaces, with the crucial difference that simplicial sets are purely algebraic and do not carry any actual topology (this will become clear in the formal definition).
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| To get back to actual topological spaces, there is a ''geometric realization'' [[functor]] available which takes values in the category of [[compactly generated Hausdorff space]]s. Most classical results on CW complexes in homotopy theory have analogous versions for simplicial sets which generalize these results. While algebraic topologists largely continue to prefer CW complexes, there is a growing contingent of researchers interested in using simplicial sets for applications in [[algebraic geometry]] where CW complexes do not naturally exist.
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| ==Formal definition==
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| Using the language of [[category theory]], a '''simplicial set''' ''X'' is a [[functor#Covariance and contravariance|contravariant functor]]
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| :''X'': Δ → '''Set'''
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| where Δ denotes the [[simplex category]] whose objects are finite strings of ordinal numbers of the form
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| :''' ''n'' ''' = 0 → 1 → ... → ''n''
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| (or in other words non-empty [[totally ordered]] [[finite set]]s) and whose morphisms are order-preserving functions between them, and '''Set''' is the category of [[small set (category theory)|small sets]].
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| It is common to define simplicial sets as a [[functor#Covariance and contravariance|covariant functor]] from the [[Dual (category theory)|opposite category]], as
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| : ''X'': Δ<sup>op</sup> → '''Set'''
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| That is, as a [[presheaf (category theory)|presheaf]]. This definition is clearly equivalent to the one immediately above.<!--Thus, the simplicial set is an example of a [[presheaf (category theory)|presheaf]].-->
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| Alternatively, one can think of a simplicial set as a '''simplicial object''' (see below) in the category '''Set''', but this is only different language for the definition just given. If we use a ''covariant'' functor ''X'' instead of a contravariant one, we arrive at the definition of a '''cosimplicial set'''.
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| Simplicial sets form a category usually denoted s'''Set''' or just '''S''' whose objects are simplicial sets and whose morphisms are [[natural transformations]] between them. There is a corresponding category for cosimplicial sets as well, denoted by c'''Set'''.
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| These definitions arise from the relationship of the conditions imposed on the ''face maps'' and ''degeneracy maps'' to the category Δ.
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| ==Face and degeneracy maps==
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| In Δ<sup>op</sup>, there are two particularly important classes of maps called '''face maps''' and '''degeneracy maps''' which capture the underlying combinatorial structure of simplicial sets.
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| The face maps
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| ''d<sub>i</sub> :''
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| ''' ''n'' '''
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| →
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| ''' ''n'' − 1'' '''
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| are given by
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| :''d<sub>i</sub>'' (0 → … → ''n'') = (0 → … → ''i'' − 1 → ''i'' + 1 → … → ''n'').
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| The degeneracy maps
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| ''s<sub>i</sub>'' :
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| ''' ''n'' '''
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| →
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| ''' ''n'' + 1 '''
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| are given by
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| :''s<sub>i</sub>'' (0 → … → ''n'') = (0 → … → ''i'' − 1 → ''i'' → ''i'' → ''i'' + 1 → … → ''n'').
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| By definition, these maps satisfy the following '''simplicial identities''':
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| #''d<sub>i</sub> d<sub>j</sub>'' = ''d''<sub>''j''−1</sub> ''d''<sub>''i''</sub> if ''i'' < ''j''
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| #''d<sub>i</sub> s<sub>j</sub>'' = ''s''<sub>''j''−1</sub> ''d<sub>i</sub>'' if ''i'' < ''j''
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| #''d<sub>i</sub> s<sub>j</sub>'' = id if ''i'' = ''j'' or ''i'' = ''j'' + 1
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| #''d<sub>i</sub> s<sub>j</sub>'' = ''s''<sub>''j''</sub> ''d''<sub>''i''−1</sub> if ''i'' > ''j'' + 1
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| #''s<sub>i</sub> s<sub>j</sub>'' = ''s''<sub>''j''+1</sub> ''s''<sub>''i''</sub> if ''i'' ≤ ''j''.
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| The simplicial category Δ has as its morphisms the monotonic non-decreasing functions. Since the morphisms are generated by those that 'skip' or 'add' a single element, the detailed relations written out above underlie the topological applications. It can be shown that these relations suffice.
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| ==The standard ''n''-simplex and the simplex category==
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| Categorically, the '''standard ''n''-simplex''', denoted Δ<sup>''n''</sup>, is the functor ''hom(-, '''''n''''')'' where '''''n''''' denotes the string 0 → 1 → ... → ''n'' of the first (''n'' + 1) nonnegative integers and the homset is taken in the category Δ. In many texts, it is written instead as ''hom('''''n''''',-)'' where the homset is understood to be in the opposite category Δ<sup>op</sup>.<ref>S. Gelfand, Yu. Manin, "Methods of Homological Algebra"</ref>
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| The geometric realization |Δ<sup>''n''</sup>| is just defined to be the standard topological ''n''-[[simplex]] in general position given by
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| :<math>|\Delta^n| = \{(x_0, \dots, x_n) \in \mathbb{R}^{n+1}: 0\leq x_i \leq 1, \sum x_i = 1 \}.</math>
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| By the [[Yoneda lemma]], the ''n''-simplices of a simplicial set ''X'' are classified by natural transformations in hom(Δ<sup>''n''</sup>, ''X'').<ref>
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| Specifically, consider <math>\Delta^n=\Delta^{\mathrm{op}}(\mathbf{n},-)</math>, then the Yoneda lemma gives <math>\mathrm{Nat}(\Delta^{\mathrm{op}}(\mathbf{n},-), X) \cong X(\mathbf{n})</math>
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| </ref> The ''n''-simplices of ''X'' are then collectively denoted by ''X<sub>n</sub>''. Furthermore, there is a [[simplex category]], denoted by <math>\Delta\downarrow{X}</math> whose objects are maps (''i.e.'' natural transformations) Δ<sup>''n''</sup> → ''X'' and whose morphisms are natural transformations Δ<sup>''n''</sup> → Δ<sup>''m''</sup> over ''X'' arising from maps '''''n''''' ''→'' '''''m''''' in Δ. That is, <math>\Delta\downarrow{X}</math> is a [[slice category]] of Δ over ''X''. The following isomorphism shows that a simplicial set ''X'' is a [[colimit]] of its simplices:<ref>Goerss & Jardine, p.7</ref>
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| : <math>X \cong \varinjlim_{\Delta^n \to X} \Delta^n</math>
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| where the colimit is taken over the simplex category of ''X''.
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| ==Geometric realization==
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| There is a functor |•|: '''S''' ''→'' '''CGHaus''' called the '''geometric realization''' taking a simplicial set ''X'' to its corresponding realization in the category of [[compactly-generated space|compactly-generated]] [[Hausdorff topological space]]s.
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| This larger category is used as the target of the functor because, in particular, a product of simplicial sets
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| :<math>X \times Y</math>
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| is realized as a product
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| :<math>|X| \times_{Ke} |Y|</math>
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| of the corresponding topological spaces, where <math>\times_{Ke}</math> denotes the [[Kelley space product]]. To define the realization functor, we first define it on n-simplices ''Δ<sup>n</sup>'' as the corresponding topological n-simplex |''Δ<sup>n</sup>''|. The definition then naturally extends to any simplicial set ''X'' by setting
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| :|X| = lim<sub>''Δ<sup>n</sup> → X''</sub> |''Δ<sup>n</sup>''| | |
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| where the colimit is taken over the n-simplex category of ''X''. The geometric realization is functorial on '''S'''.
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| ==Singular set for a space==
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| The '''singular set''' of a topological space ''Y'' is the simplicial set defined by ''S(Y):'' '''n''' ''→'' ''hom(''|''Δ<sup>n</sup>''|, ''Y)'' for each object '''n''' ''∈ Δ'', with the obvious functoriality condition on the morphisms. This definition is analogous to a standard idea in [[singular homology]] of "probing" a target topological space with standard topological n-simplices. Furthermore, the '''singular functor''' ''S'' is [[adjoint functor|right adjoint]] to the geometric realization functor described above, i.e.:
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| :hom<sub>'''Top'''</sub>(|''X''|, ''Y'') ≅ hom<sub>'''S'''</sub>(''X'', ''SY'')
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| for any simplicial set ''X'' and any topological space ''Y''.
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| ==Homotopy theory of simplicial sets==
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| In the category of simplicial sets one can define [[fibration]]s to be [[Kan fibration]]s. A map of simplicial sets is defined to be a [[weak equivalence (homotopy theory)|weak equivalence]] if the geometric realization is a weak equivalence of spaces. A map of simplicial sets is defined to be a [[cofibration]] if it is a [[monomorphism]] of simplicial sets. It is a difficult theorem of [[Daniel Quillen]] that the category of simplicial sets with these classes of morphisms satisfies the axioms for a [[proper model category|proper]] [[closed model category|closed]] [[simplicial model category|simplicial]] [[model category]].
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| A key turning point of the theory is that the realization of a Kan fibration is a [[Serre fibration]] of spaces. With the model structure in place, a homotopy theory of simplicial sets can be developed using standard [[homotopical algebra|homotopical]] [[abstract nonsense]]. Furthermore, the geometric realization and singular functors give a [[Quillen adjunction|Quillen equivalence]] of [[closed model category|closed model categories]] inducing an equivalence of homotopy categories
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| :|•|: ''Ho''('''S''') ↔ ''Ho''('''Top''') : ''S''
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| between the [[homotopy category]] for simplicial sets and the usual homotopy category of CW complexes with homotopy classes of maps between them. It is part of the general
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| definition of a Quillen adjunction that the right adjoint functor (in this case, the singular set functor) carries fibrations (resp. trivial fibrations) to fibrations (resp. trivial fibrations).
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| ==Simplicial objects==
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| A '''simplicial object''' ''X'' in a category ''C'' is a contravariant functor
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| :''X: Δ → C''
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| or equivalently a covariant functor
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| :''X'': Δ<sup>op</sup> → ''C''
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| When ''C'' is the [[category of sets]], we are just talking about simplicial sets. Letting ''C'' be the [[category of groups]] or [[category of abelian groups]], we obtain the categories s'''Grp''' of simplicial [[group (mathematics)|group]]s and s'''Ab''' of simplicial [[abelian group]]s, respectively.
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| [[Simplicial group]]s and simplicial abelian groups also carry closed model structures induced by that of the underlying simplicial sets.
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| The homotopy groups of simplicial abelian groups can be computed by making use of the [[Dold-Kan correspondence]] which yields an equivalence of categories between simplicial abelian groups and bounded [[chain complex]]es and is given by functors
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| :''N:'' s'''Ab''' ''→ Ch<sub>+</sub>''
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| and
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| :''Γ: Ch<sub>+</sub> → ''s'''Ab'''.
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| ==See also==
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| *[[Dendroidal set]], a generalization of simplicial set.
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| *[[Simplicial presheaf]]
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| *[[infinity-category]]
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| *[[Homotopy type theory]]
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| *[[Kan complex]]
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| == Notes ==
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| <references/>
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| == References ==
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| * {{Cite book | last1=Goerss | first1=P. G. | last2=Jardine | first2=J. F. | title=Simplicial Homotopy Theory | publisher=Birkhäuser | location=Basel, Boston, Berlin | series=Progress in Mathematics | isbn=978-3-7643-6064-1 | year=1999 | volume=174 | postscript=<!--None-->}}
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| * {{Cite book | last1=Gelfand | first1=S. | last2=Manin | first2=Yu. | title=Methods of homological algebra}}
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| * Dylan G.L. Allegretti, [http://www.math.uchicago.edu/~may/VIGRE/VIGRE2008/REUPapers/Allegretti.pdf ''Simplicial Sets and van Kampen's Theorem''] ''(An elementary introduction to simplicial sets)''.
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| *{{nlab|id=simplicial+set|title=simplicial set}}
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| {{DEFAULTSORT:Simplicial Set}}
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| [[Category:Algebraic topology]]
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| [[Category:Homotopy theory]]
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| [[Category:Simplicial sets| ]]
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| [[Category:Functors]]
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Catrina Le is what's constructed on her birth credentials though she doesn't really like being called prefer that. Software raising is where her primary income comes from remember, though , soon her husband and consequently her will start their personal own business. What the woman loves doing is to get to karaoke but jane is thinking on starting something totally new. For years she's been living with Vermont. She is running and looking after a blog here: http://Prometeu.net/
my homepage ... clash of clans unlimited gems