Does a categorical product make any restrictions on an object that has no morphisms to one of the factors?












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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







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    $begingroup$


    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







    share|cite|improve this question









    $endgroup$















      1












      1








      1





      $begingroup$


      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







      share|cite|improve this question









      $endgroup$




      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






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      asked Jan 30 at 9:48









      user56834user56834

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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.






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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.






            share|cite|improve this 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.






              share|cite|improve this answer









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                4












                4








                4





                $begingroup$

                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.






                share|cite|improve this answer









                $endgroup$



                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.







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                share|cite|improve this answer










                answered Jan 30 at 9:51









                Tsemo AristideTsemo Aristide

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