Simplicial object in a category

$ {\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)