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A topological space with a co-multiplication; the dual notion is an [[H-space|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022700/c0227002.png" />-space]].
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A topological space with a co-multiplication; the dual notion is an [[H-space| $  H $-
 +
space]].
  
 
====Comments====
 
====Comments====
The sum of two objects <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022700/c0227003.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022700/c0227004.png" /> in the category of pointed topological spaces is the disjoint union of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022700/c0227005.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022700/c0227006.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022700/c0227007.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022700/c0227008.png" /> identified, and this point serves as base point; it can be realized (and visualized) as the subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022700/c0227009.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022700/c02270010.png" />. A co-<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022700/c02270011.png" />-space thus is a pointed topological space with a continuous mapping of pointed spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022700/c02270012.png" />, termed co-multiplication, such that the composites <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022700/c02270013.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022700/c02270014.png" /> are homotopic to the identity. Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022700/c02270015.png" /> is the mapping which sends all of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022700/c02270016.png" /> to the base point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022700/c02270017.png" />. If the two composites <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022700/c02270018.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022700/c02270019.png" /> are homotopic to each other, the co-multiplication is called homotopy co-associative (or homotopy associative). A continuous mapping of pointed spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022700/c02270020.png" /> is a homotopy co-inverse for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022700/c02270021.png" /> if the two composites <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022700/c02270022.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022700/c02270023.png" /> are both homotopic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022700/c02270024.png" />. Here for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022700/c02270025.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022700/c02270026.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022700/c02270027.png" /> is the mapping determined by the defining property of the sum in the category of pointed topological spaces, i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022700/c02270028.png" /> restricted to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022700/c02270029.png" /> is equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022700/c02270030.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022700/c02270031.png" /> restricted to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022700/c02270032.png" /> is equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022700/c02270033.png" />. A co-<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022700/c02270034.png" />-space with co-associative co-multiplication which admits a homotopy co-inverse is called an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022700/c02270036.png" />-co-group. Thus, an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022700/c02270037.png" />-co-group is a co-group object in the category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022700/c02270038.png" /> of pointed topological spaces and homotopy classes of mappings.
+
The sum of two objects $  ( X, x _ {0} ) $
 +
and $  ( Y, y _ {0} ) $
 +
in the category of pointed topological spaces is the disjoint union of $  X $
 +
and $  Y $
 +
with $  x _ {0} $
 +
and $  y _ {0} $
 +
identified, and this point serves as base point; it can be realized (and visualized) as the subset $  X \times \{ y _ {0} \} \cup \{ x _ {0} \} \times Y $
 +
of $  X \times Y $.  
 +
A co- $  H $-
 +
space thus is a pointed topological space with a continuous mapping of pointed spaces $  \mu : Q \rightarrow Q \lor Q $,  
 +
termed co-multiplication, such that the composites $  Q \rightarrow  ^  \mu  Q \lor Q \rightarrow ^ { \mathop{\rm id} \lor \epsilon } Q \lor \{ q _ {0} \} \simeq Q $
 +
and $  Q \rightarrow  ^  \mu  Q \lor Q \rightarrow ^ {\epsilon \lor  \mathop{\rm id} } Q $
 +
are homotopic to the identity. Here $  \epsilon $
 +
is the mapping which sends all of $  Q $
 +
to the base point $  q _ {0} $.  
 +
If the two composites $  Q \rightarrow  ^  \mu  Q \lor Q \rightarrow ^ { \mathop{\rm id} \lor \mu } Q \lor Q \lor Q $
 +
and $  Q \rightarrow  ^  \mu  Q \lor Q \rightarrow ^ {\mu \lor  \mathop{\rm id} } Q \lor Q \lor Q $
 +
are homotopic to each other, the co-multiplication is called homotopy co-associative (or homotopy associative). A continuous mapping of pointed spaces $  r: Q \rightarrow Q $
 +
is a homotopy co-inverse for $  \mu $
 +
if the two composites $  Q \rightarrow  ^  \mu  Q \lor Q \rightarrow ^ {(  \mathop{\rm id} , r) } Q $
 +
and $  Q \rightarrow  ^  \mu  Q \lor Q \rightarrow ^ {( r,  \mathop{\rm id} ) } Q $
 +
are both homotopic to $  \epsilon : Q \rightarrow Q $.  
 +
Here for $  f: X \rightarrow Z $,  
 +
$  g: Y \rightarrow Z $,
 +
$  ( f, g) $
 +
is the mapping determined by the defining property of the sum in the category of pointed topological spaces, i.e. $  ( f, g) $
 +
restricted to $  X $
 +
is equal to $  f $,  
 +
and $  ( f, g) $
 +
restricted to $  Y $
 +
is equal to $  g $.  
 +
A co- $  H $-
 +
space with co-associative co-multiplication which admits a homotopy co-inverse is called an $  H $-
 +
co-group. Thus, an $  H $-
 +
co-group is a co-group object in the category $  {\mathcal H} {\mathcal t} {\mathcal p} $
 +
of pointed topological spaces and homotopy classes of mappings.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  E.H. Spanier,  "Algebraic topology" , McGraw-Hill  (1966)  pp. Chapt. I, Sect. 6</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  E.H. Spanier,  "Algebraic topology" , McGraw-Hill  (1966)  pp. Chapt. I, Sect. 6</TD></TR></table>

Latest revision as of 17:45, 4 June 2020


A topological space with a co-multiplication; the dual notion is an $ H $- space.

Comments

The sum of two objects $ ( X, x _ {0} ) $ and $ ( Y, y _ {0} ) $ in the category of pointed topological spaces is the disjoint union of $ X $ and $ Y $ with $ x _ {0} $ and $ y _ {0} $ identified, and this point serves as base point; it can be realized (and visualized) as the subset $ X \times \{ y _ {0} \} \cup \{ x _ {0} \} \times Y $ of $ X \times Y $. A co- $ H $- space thus is a pointed topological space with a continuous mapping of pointed spaces $ \mu : Q \rightarrow Q \lor Q $, termed co-multiplication, such that the composites $ Q \rightarrow ^ \mu Q \lor Q \rightarrow ^ { \mathop{\rm id} \lor \epsilon } Q \lor \{ q _ {0} \} \simeq Q $ and $ Q \rightarrow ^ \mu Q \lor Q \rightarrow ^ {\epsilon \lor \mathop{\rm id} } Q $ are homotopic to the identity. Here $ \epsilon $ is the mapping which sends all of $ Q $ to the base point $ q _ {0} $. If the two composites $ Q \rightarrow ^ \mu Q \lor Q \rightarrow ^ { \mathop{\rm id} \lor \mu } Q \lor Q \lor Q $ and $ Q \rightarrow ^ \mu Q \lor Q \rightarrow ^ {\mu \lor \mathop{\rm id} } Q \lor Q \lor Q $ are homotopic to each other, the co-multiplication is called homotopy co-associative (or homotopy associative). A continuous mapping of pointed spaces $ r: Q \rightarrow Q $ is a homotopy co-inverse for $ \mu $ if the two composites $ Q \rightarrow ^ \mu Q \lor Q \rightarrow ^ {( \mathop{\rm id} , r) } Q $ and $ Q \rightarrow ^ \mu Q \lor Q \rightarrow ^ {( r, \mathop{\rm id} ) } Q $ are both homotopic to $ \epsilon : Q \rightarrow Q $. Here for $ f: X \rightarrow Z $, $ g: Y \rightarrow Z $, $ ( f, g) $ is the mapping determined by the defining property of the sum in the category of pointed topological spaces, i.e. $ ( f, g) $ restricted to $ X $ is equal to $ f $, and $ ( f, g) $ restricted to $ Y $ is equal to $ g $. A co- $ H $- space with co-associative co-multiplication which admits a homotopy co-inverse is called an $ H $- co-group. Thus, an $ H $- co-group is a co-group object in the category $ {\mathcal H} {\mathcal t} {\mathcal p} $ of pointed topological spaces and homotopy classes of mappings.

References

[a1] E.H. Spanier, "Algebraic topology" , McGraw-Hill (1966) pp. Chapt. I, Sect. 6
How to Cite This Entry:
Co-H-space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Co-H-space&oldid=46369