Difference between revisions of "Simplicial object in a category"
Ulf Rehmann (talk | contribs) m (tex encoded by computer) |
Ulf Rehmann (talk | contribs) m (Undo revision 48710 by Ulf Rehmann (talk)) Tag: Undo |
||
Line 1: | Line 1: | ||
− | < | + | ''<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085380/s0853801.png" />'' |
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | + | A contravariant functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085380/s0853802.png" /> (or, equivalently, a covariant functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085380/s0853803.png" />) from the category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085380/s0853804.png" />, whose objects are ordered sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085380/s0853805.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085380/s0853806.png" />, and whose morphisms are non-decreasing mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085380/s0853807.png" />, into the category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085380/s0853808.png" />. A covariant functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085380/s0853809.png" /> (or, equivalently, a contravariant functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085380/s08538010.png" />) is called a co-simplicial object in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085380/s08538011.png" />. | |
− | |||
− | |||
− | |||
− | |||
− | A contravariant functor | ||
− | or, equivalently, a covariant functor | ||
− | from the category | ||
− | whose objects are ordered sets | ||
− | |||
− | and whose morphisms are non-decreasing mappings | ||
− | into the category | ||
− | A covariant functor | ||
− | or, equivalently, a contravariant functor | ||
− | is called a co-simplicial object in | ||
The morphisms | The morphisms | ||
− | + | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085380/s08538012.png" /></td> </tr></table> | |
− | |||
− | 0 | ||
− | |||
− | + | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085380/s08538013.png" /></td> </tr></table> | |
− | |||
− | |||
− | of | + | of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085380/s08538014.png" /> given by |
− | given by | ||
− | + | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085380/s08538015.png" /></td> </tr></table> | |
− | |||
− | |||
− | + | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085380/s08538016.png" /></td> </tr></table> | |
− | |||
− | generate all the morphisms of | + | generate all the morphisms of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085380/s08538017.png" />, so that a simplicial object <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085380/s08538018.png" /> is determined by the objects <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085380/s08538019.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085380/s08538020.png" /> (called the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085380/s08538022.png" />-fibres or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085380/s08538024.png" />-components of the simplicial object <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085380/s08538025.png" />), and the morphisms |
− | so that a simplicial object | ||
− | is determined by the objects | ||
− | |||
− | called the | ||
− | fibres or | ||
− | components of the simplicial object | ||
− | and the morphisms | ||
− | + | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085380/s08538026.png" /></td> </tr></table> | |
− | |||
− | |||
− | |||
− | |||
− | (called boundary operators and degeneracy operators, respectively). In case | + | (called boundary operators and degeneracy operators, respectively). In case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085380/s08538027.png" /> is a category of structured sets, the elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085380/s08538028.png" /> are usually called the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085380/s08538030.png" />-dimensional simplices of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085380/s08538031.png" />. The mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085380/s08538032.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085380/s08538033.png" /> satisfy the relations |
− | is a category of structured sets, the elements of | ||
− | are usually called the | ||
− | dimensional simplices of | ||
− | The mappings | ||
− | and | ||
− | satisfy the relations | ||
− | + | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085380/s08538034.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table> | |
− | |||
− | and any relation between these mappings is a consequence of the relations (*). This means that a simplicial object | + | and any relation between these mappings is a consequence of the relations (*). This means that a simplicial object <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085380/s08538035.png" /> can be identified with a system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085380/s08538036.png" /> of objects <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085380/s08538037.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085380/s08538038.png" />, of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085380/s08538039.png" /> and morphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085380/s08538040.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085380/s08538041.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085380/s08538042.png" />, satisfying the relations |
− | can be identified with a system | ||
− | of objects | ||
− | |||
− | of | ||
− | and morphisms | ||
− | and | ||
− | |||
− | satisfying the relations | ||
− | + | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085380/s08538043.png" /></td> </tr></table> | |
− | |||
− | |||
− | + | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085380/s08538044.png" /></td> </tr></table> | |
− | |||
− | |||
− | + | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085380/s08538045.png" /></td> </tr></table> | |
− | |||
− | Similarly, a co-simplicial object | + | Similarly, a co-simplicial object <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085380/s08538046.png" /> can be identified with a system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085380/s08538047.png" /> of objects <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085380/s08538048.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085380/s08538049.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085380/s08538051.png" />-co-fibres) and morphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085380/s08538052.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085380/s08538053.png" /> (co-boundary operators), and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085380/s08538054.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085380/s08538055.png" /> (co-degeneracy operators), satisfying the relations (*) (with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085380/s08538056.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085380/s08538057.png" />). |
− | can be identified with a system | ||
− | of objects | ||
− | |||
− | |||
− | co-fibres) and morphisms | ||
− | |||
− | co-boundary operators), and | ||
− | |||
− | co-degeneracy operators), satisfying the relations (*) (with | ||
− | |||
− | A simplicial mapping | + | A simplicial mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085380/s08538058.png" /> between simplicial objects (in the same category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085380/s08538059.png" />) is a transformation (morphism) of functors from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085380/s08538060.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085380/s08538061.png" />, that is, a family of morphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085380/s08538062.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085380/s08538063.png" />, of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085380/s08538064.png" /> such that |
− | between simplicial objects (in the same category | ||
− | is a transformation (morphism) of functors from | ||
− | into | ||
− | that is, a family of morphisms | ||
− | |||
− | of | ||
− | such that | ||
− | + | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085380/s08538065.png" /></td> </tr></table> | |
− | |||
− | 0 | ||
− | |||
− | + | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085380/s08538066.png" /></td> </tr></table> | |
− | |||
− | |||
− | The simplicial objects of | + | The simplicial objects of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085380/s08538067.png" /> and their simplicial mappings form a category, denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085380/s08538068.png" />. |
− | and their simplicial mappings form a category, denoted by | ||
− | A simplicial homotopy | + | A simplicial homotopy <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085380/s08538069.png" /> between two simplicial mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085380/s08538070.png" /> between simplicial objects in a category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085380/s08538071.png" /> is a family of morphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085380/s08538072.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085380/s08538073.png" />, of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085380/s08538074.png" /> such that |
− | between two simplicial mappings | ||
− | between simplicial objects in a category | ||
− | is a family of morphisms | ||
− | |||
− | of | ||
− | such that | ||
− | + | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085380/s08538075.png" /></td> </tr></table> | |
− | |||
− | |||
− | + | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085380/s08538076.png" /></td> </tr></table> | |
− | |||
− | |||
− | + | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085380/s08538077.png" /></td> </tr></table> | |
− | |||
− | + | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085380/s08538078.png" /></td> </tr></table> | |
− | s | ||
− | On the basis of this definition one can reproduce in essence the whole of ordinary homotopy theory in the category | + | On the basis of this definition one can reproduce in essence the whole of ordinary homotopy theory in the category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085380/s08538079.png" />, for any category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085380/s08538080.png" />. In the case of the category of sets or topological spaces, the geometric realization functor (see [[Simplicial set|Simplicial set]]) carries this "simplicial" theory into the usual one. |
− | for any category | ||
− | In the case of the category of sets or topological spaces, the geometric realization functor (see [[Simplicial set|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. | 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 | + | Every simplicial Abelian group can be made into a chain complex with boundary operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085380/s08538081.png" />. |
====References==== | ====References==== | ||
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> P. Gabriel, M. Zisman, "Calculus of fractions and homotopy theory" , Springer (1967)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> J.P. May, "Simplicial objects in algebraic topology" , v. Nostrand (1967)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> K. Lamotke, "Semisimpliziale algebraische Topologie" , Springer (1968)</TD></TR></table> | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> P. Gabriel, M. Zisman, "Calculus of fractions and homotopy theory" , Springer (1967)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> J.P. May, "Simplicial objects in algebraic topology" , v. Nostrand (1967)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> K. Lamotke, "Semisimpliziale algebraische Topologie" , Springer (1968)</TD></TR></table> |
Revision as of 14:53, 7 June 2020
A contravariant functor (or, equivalently, a covariant functor
) from the category
, whose objects are ordered sets
,
, and whose morphisms are non-decreasing mappings
, into the category
. A covariant functor
(or, equivalently, a contravariant functor
) is called a co-simplicial object in
.
The morphisms
![]() |
![]() |
of given by
![]() |
![]() |
generate all the morphisms of , so that a simplicial object
is determined by the objects
,
(called the
-fibres or
-components of the simplicial object
), and the morphisms
![]() |
(called boundary operators and degeneracy operators, respectively). In case is a category of structured sets, the elements of
are usually called the
-dimensional simplices of
. The mappings
and
satisfy the relations
![]() | (*) |
and any relation between these mappings is a consequence of the relations (*). This means that a simplicial object can be identified with a system
of objects
,
, of
and morphisms
and
,
, satisfying the relations
![]() |
![]() |
![]() |
Similarly, a co-simplicial object can be identified with a system
of objects
,
(
-co-fibres) and morphisms
,
(co-boundary operators), and
,
(co-degeneracy operators), satisfying the relations (*) (with
,
).
A simplicial mapping between simplicial objects (in the same category
) is a transformation (morphism) of functors from
into
, that is, a family of morphisms
,
, of
such that
![]() |
![]() |
The simplicial objects of and their simplicial mappings form a category, denoted by
.
A simplicial homotopy between two simplicial mappings
between simplicial objects in a category
is a family of morphisms
,
, of
such that
![]() |
![]() |
![]() |
![]() |
On the basis of this definition one can reproduce in essence the whole of ordinary homotopy theory in the category , for any category
. 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 .
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) |
Simplicial object in a category. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Simplicial_object_in_a_category&oldid=49427