# Simplicial object in a category

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${\mathcal C}$

A contravariant functor $X: \Delta \rightarrow {\mathcal C}$( or, equivalently, a covariant functor $X: \Delta ^ {op} \rightarrow {\mathcal C}$) from the category $\Delta$, whose objects are ordered sets $[ n] = \{ 0 \dots n \}$, $n \geq 0$, and whose morphisms are non-decreasing mappings $\mu : [ n] \rightarrow [ m]$, into the category ${\mathcal C}$. A covariant functor $X: \Delta \rightarrow {\mathcal C}$( or, equivalently, a contravariant functor $X: \Delta ^ {op} \rightarrow {\mathcal C}$) is called a co-simplicial object in ${\mathcal C}$.

The morphisms

$$\delta _ {i} = \delta _ {i} ^ {n} : [ n - 1] \rightarrow [ n],\ \ 0 \leq i \leq n,$$

$$\sigma _ {i} = \sigma _ {i} ^ {n} : [ n + 1] \rightarrow [ n],\ 0 \leq i \leq n,$$

of $\Delta$ given by

$$\delta _ {i} ^ {n} ( j) = \ \left \{ \begin{array}{ll} j &\textrm{ if } j < i, \\ j + 1 &\textrm{ if } j \geq i, \\ \end{array} \right .$$

$$\sigma _ {i} ^ {n} ( j) = \left \{ \begin{array}{ll} j &\textrm{ if } j \leq i, \\ j - 1 &\textrm{ if } j > i, \\ \end{array} \right .$$

generate all the morphisms of $\Delta$, so that a simplicial object $X$ is determined by the objects $X ([ n]) = X _ {n}$, $n \geq 0$( called the $n$- fibres or $n$- components of the simplicial object $X$), and the morphisms

$$d _ {i} = X ( \delta _ {i} ): X _ {n} \rightarrow X _ {n - 1 } \ \ \textrm{ and } \ \ s _ {i} = X ( \sigma _ {i} ): X _ {n} \rightarrow X _ {n + 1 }$$

(called boundary operators and degeneracy operators, respectively). In case ${\mathcal C}$ is a category of structured sets, the elements of $X _ {n}$ are usually called the $n$- dimensional simplices of $X$. The mappings $\delta _ {i}$ and $\sigma _ {i}$ satisfy the relations

$$\tag{* } \left. \begin{array}{cll} \delta_{j} \delta_{i} &=& \delta_{i} \delta_{j - 1} \quad\ \ \textrm{ if } i < j, \\ \sigma_{j} \sigma_{i} &=& \sigma_{i} \sigma_{j + 1} \quad \ \ \textrm{ if } i \leq j;\\ \sigma_{j} \delta_{i} &=& \left \{ \begin{array}{ll} \delta_{i} \sigma_{j - 1 } & \textrm{ if } i < j, \\ \mathop{\rm id} & \textrm{ if } i = j \textrm{ or } i = j + 1, \\ \delta_{i - 1}\sigma_{j} & \textrm{ if } i > j + 1; \\ \end{array} \right.\\ \end{array} \right \}$$

and any relation between these mappings is a consequence of the relations (*). This means that a simplicial object $X$ can be identified with a system $\{ X _ {n} , d _ {i} , s _ {i} \}$ of objects $X _ {n}$, $n \geq 0$, of ${\mathcal C}$ and morphisms $d _ {i} : X _ {n} \rightarrow X _ {n - 1 }$ and $s _ {i} : X _ {n} \rightarrow X _ {n + 1 }$, $0 \leq i \leq n$, satisfying the relations

$$d _ {i} d _ {j} = d _ {j - 1 } d _ {i} \ \textrm{ if } i < j;$$

$$s _ {i} s _ {j} = s _ {j + 1 } s _ {i} \ \textrm{ if } i \leq j;$$

$$d _ {i} s _ {j} = \left \{ \begin{array}{ll} s _ {j - 1 } d _ {i} & \textrm{ if } i < j, \\ \mathop{\rm id} & \textrm{ if } i = j \ \textrm{ or } i = j + 1, \\ s _ {j} d _ {i - 1 } & \textrm{ if } i > j + 1. \\ \end{array} \right .$$

Similarly, a co-simplicial object $X$ can be identified with a system $\{ X _ {n} , d ^ {i} , s ^ {i} \}$ of objects $X ^ {n}$, $n \geq 0$( $n$- co-fibres) and morphisms $d _ {i} : X ^ {n - 1 } \rightarrow X ^ {n}$, $0 \leq i \leq n$( co-boundary operators), and $s ^ {i} : X ^ {n + 1 } \rightarrow X ^ {n}$, $0 \leq i \leq n$( co-degeneracy operators), satisfying the relations (*) (with $\delta _ {i} = d ^ {i}$, $\sigma _ {i} = s ^ {i}$).

A simplicial mapping $f: X \rightarrow Y$ between simplicial objects (in the same category ${\mathcal C}$) is a transformation (morphism) of functors from $X: \Delta \rightarrow {\mathcal C}$ into $Y: \Delta \rightarrow {\mathcal C}$, that is, a family of morphisms $f _ {n} : X _ {n} \rightarrow Y _ {n}$, $n \geq 0$, of ${\mathcal C}$ such that

$$d _ {i} f _ {n + 1 } = f _ {n} d _ {i} ,\ \ 0 \leq i \leq n + 1,$$

$$s _ {i} f _ {n} = f _ {n + 1 } s _ {i} ,\ 0 \leq i \leq n.$$

The simplicial objects of ${\mathcal C}$ and their simplicial mappings form a category, denoted by $\Delta ^ {0} {\mathcal C}$.

A simplicial homotopy $h: f \simeq g$ between two simplicial mappings $f, g: X \rightarrow Y$ between simplicial objects in a category ${\mathcal C}$ is a family of morphisms $h _ {i} : X _ {n} \rightarrow Y _ {n + 1 }$, $0 \leq i \leq n$, of ${\mathcal C}$ such that

$$d _ {0} h _ {0} = f _ {n} ;$$

$$d _ {n} h _ {n} = g _ {n} ;$$

$$d _ {i} h _ {j} = \left \{ \begin{array}{ll} h _ {j - 1 } d _ {i} &\textrm{ if } i < j, \\ d _ {j} h _ {j - 1 } &\textrm{ if } i = j > 0, \\ h _ {j} d _ {i - 1 } &\textrm{ if } i > j + 1; \\ \end{array} \right .$$

$$s _ {i} h _ {j} = \left \{ \begin{array}{ll} h _ {j + 1 } s _ {i} &\textrm{ if } i \leq j, \\ h _ {j} s _ {i - 1 } &\textrm{ if } i > j. \\ \end{array} \right .$$

On the basis of this definition one can reproduce in essence the whole of ordinary homotopy theory in the category $\Delta ^ {0} {\mathcal C}$, for any category ${\mathcal C}$. In the case of the category of sets or topological spaces, the geometric realization functor (see Simplicial set) carries this "simplicial" theory into the usual one.

Examples of simplicial objects are a simplicial set, a simplicial topological space, a simplicial algebraic variety, a simplicial group, a simplicial Abelian group, a simplicial Lie algebra, a simplicial smooth manifold, etc.

Every simplicial Abelian group can be made into a chain complex with boundary operator $d = \sum (- 1) ^ {i} d _ {i}$.

#### References

 [1] P. Gabriel, M. Zisman, "Calculus of fractions and homotopy theory" , Springer (1967) [2] J.P. May, "Simplicial objects in algebraic topology" , v. Nostrand (1967) [3] K. Lamotke, "Semisimpliziale algebraische Topologie" , Springer (1968)
How to Cite This Entry:
Simplicial object in a category. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Simplicial_object_in_a_category&oldid=49679
This article was adapted from an original article by S.N. MalyginM.M. Postnikov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article