Mixing
Does a categorical product make any restrictions on an object that has no morphisms to one of the factors?
$begingroup$
Suppose we have objects $X,Y$, and product $Xtimes Y$. The definition of the product says that for any other object $Z$, and pair of morphisms $f:Zto X, g:Zto Y$, there is a corresponding product morphism $fcirc g$.
But what if there are no morphisms at all from $Z$ to $Y$ say, but there are many morphisms from $Z$ to $X$? Do we know anything about the morphisms from $Z$ to $Xtimes Y$? Can there be any morphisms? do there have to be any morphisms?
category-theory
asked Jan 30 at 9:48
user56834user56834
3,38621253
$endgroup$
add a comment |
$begingroup$
Suppose we have objects $X,Y$, and product $Xtimes Y$. The definition of the product says that for any other object $Z$, and pair of morphisms $f:Zto X, g:Zto Y$, there is a corresponding product morphism $fcirc g$.
But what if there are no morphisms at all from $Z$ to $Y$ say, but there are many morphisms from $Z$ to $X$? Do we know anything about the morphisms from $Z$ to $Xtimes Y$? Can there be any morphisms? do there have to be any morphisms?
category-theory
asked Jan 30 at 9:48
user56834user56834
3,38621253
$endgroup$
add a comment |
$begingroup$
Suppose we have objects $X,Y$, and product $Xtimes Y$. The definition of the product says that for any other object $Z$, and pair of morphisms $f:Zto X, g:Zto Y$, there is a corresponding product morphism $fcirc g$.
But what if there are no morphisms at all from $Z$ to $Y$ say, but there are many morphisms from $Z$ to $X$? Do we know anything about the morphisms from $Z$ to $Xtimes Y$? Can there be any morphisms? do there have to be any morphisms?
category-theory
asked Jan 30 at 9:48
user56834user56834
3,38621253
$endgroup$
Suppose we have objects $X,Y$, and product $Xtimes Y$. The definition of the product says that for any other object $Z$, and pair of morphisms $f:Zto X, g:Zto Y$, there is a corresponding product morphism $fcirc g$.
But what if there are no morphisms at all from $Z$ to $Y$ say, but there are many morphisms from $Z$ to $X$? Do we know anything about the morphisms from $Z$ to $Xtimes Y$? Can there be any morphisms? do there have to be any morphisms?
category-theory
category-theory
asked Jan 30 at 9:48
user56834user56834
3,38621253
asked Jan 30 at 9:48
user56834user56834
3,38621253
asked Jan 30 at 9:48
user56834user56834
3,38621253
asked Jan 30 at 9:48
user56834user56834
3,38621253
asked Jan 30 at 9:48
user56834user56834
3,38621253
3,38621253
add a comment |
add a comment |
1 Answer
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$begingroup$
There are no morphism from $Z$ to $Xtimes Y$ since for such a morphism $p_Ycirc f:Zrightarrow Y$ and $p_Xcirc f:Zrightarrow X$ are defined where $p_X:Xtimes Yrightarrow X$ is the projection.
answered Jan 30 at 9:51
Tsemo AristideTsemo Aristide
60.2k11446
$endgroup$
add a comment |
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1 Answer
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1 Answer
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$begingroup$
There are no morphism from $Z$ to $Xtimes Y$ since for such a morphism $p_Ycirc f:Zrightarrow Y$ and $p_Xcirc f:Zrightarrow X$ are defined where $p_X:Xtimes Yrightarrow X$ is the projection.
answered Jan 30 at 9:51
Tsemo AristideTsemo Aristide
60.2k11446
$endgroup$
add a comment |
$begingroup$
There are no morphism from $Z$ to $Xtimes Y$ since for such a morphism $p_Ycirc f:Zrightarrow Y$ and $p_Xcirc f:Zrightarrow X$ are defined where $p_X:Xtimes Yrightarrow X$ is the projection.
answered Jan 30 at 9:51
Tsemo AristideTsemo Aristide
60.2k11446
$endgroup$
add a comment |
$begingroup$
There are no morphism from $Z$ to $Xtimes Y$ since for such a morphism $p_Ycirc f:Zrightarrow Y$ and $p_Xcirc f:Zrightarrow X$ are defined where $p_X:Xtimes Yrightarrow X$ is the projection.
answered Jan 30 at 9:51
Tsemo AristideTsemo Aristide
60.2k11446
$endgroup$
There are no morphism from $Z$ to $Xtimes Y$ since for such a morphism $p_Ycirc f:Zrightarrow Y$ and $p_Xcirc f:Zrightarrow X$ are defined where $p_X:Xtimes Yrightarrow X$ is the projection.
answered Jan 30 at 9:51
Tsemo AristideTsemo Aristide
60.2k11446
answered Jan 30 at 9:51
Tsemo AristideTsemo Aristide
60.2k11446
answered Jan 30 at 9:51
Tsemo AristideTsemo Aristide
60.2k11446
answered Jan 30 at 9:51
Tsemo AristideTsemo Aristide
60.2k11446
60.2k11446
add a comment |
add a comment |
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