Functors in arrow category












4














I am studying Awodey's Category theory book. I have trouble understanding the following line:



Observe that there are two functors in arrow category i.e.
$$
begin{align}
mathscr{C} xleftarrow{textbf{dom}} mathscr{C}^{rightarrow} xrightarrow{textbf{cod}} mathscr{C}
end{align}
$$

where $mathscr{C}^{rightarrow}$ is the arrow category corresponding to $mathscr{C}$.



They have not mentioned what these $textbf{dom}$ and $textbf{cod}$ are? How do we prove that these are functors?



My understanding:
Now, in the diagram given below:



enter image description here



$textbf{dom}:mathscr{C}^{rightarrow} xrightarrow{textbf{dom}} mathscr{C}$. So,



$[f:A to B] mapsto A $ (object mapping of functor $textbf{dom}$) and
if $g=(g_1,g_2):[f:A to B] to [f':A' to B'] $, then $(g_1,g_2) mapsto [f:A to B]$ (the morphism mapping of functor $textbf{dom}$ ).



Using this definition, If I proceed to prove the statement:
(a)
$$
textbf{dom} ( g:f to f' ) = textbf{dom}(g): textbf{dom}(f) to textbf{dom}(f')
$$



LHS = $f:A to B$ and RHS = $textbf{dom}(g): A to A'$ (which seems absurd)
I am not sure if this makes sense.










share|cite|improve this question
























  • domain and codomain. Maps the arrow $f:Ato A'$ to $A$ and to $A'$ respectively.
    – Lord Shark the Unknown
    Dec 31 '18 at 8:26










  • @LordSharktheUnknown I am not able to prove even, first condition of functor i.e. $textbf{dom} ( g:f to f' ) = textbf{dom}(g): textbf{dom}(f) to textbf{dom}(f')$
    – MUH
    Dec 31 '18 at 8:39
















4














I am studying Awodey's Category theory book. I have trouble understanding the following line:



Observe that there are two functors in arrow category i.e.
$$
begin{align}
mathscr{C} xleftarrow{textbf{dom}} mathscr{C}^{rightarrow} xrightarrow{textbf{cod}} mathscr{C}
end{align}
$$

where $mathscr{C}^{rightarrow}$ is the arrow category corresponding to $mathscr{C}$.



They have not mentioned what these $textbf{dom}$ and $textbf{cod}$ are? How do we prove that these are functors?



My understanding:
Now, in the diagram given below:



enter image description here



$textbf{dom}:mathscr{C}^{rightarrow} xrightarrow{textbf{dom}} mathscr{C}$. So,



$[f:A to B] mapsto A $ (object mapping of functor $textbf{dom}$) and
if $g=(g_1,g_2):[f:A to B] to [f':A' to B'] $, then $(g_1,g_2) mapsto [f:A to B]$ (the morphism mapping of functor $textbf{dom}$ ).



Using this definition, If I proceed to prove the statement:
(a)
$$
textbf{dom} ( g:f to f' ) = textbf{dom}(g): textbf{dom}(f) to textbf{dom}(f')
$$



LHS = $f:A to B$ and RHS = $textbf{dom}(g): A to A'$ (which seems absurd)
I am not sure if this makes sense.










share|cite|improve this question
























  • domain and codomain. Maps the arrow $f:Ato A'$ to $A$ and to $A'$ respectively.
    – Lord Shark the Unknown
    Dec 31 '18 at 8:26










  • @LordSharktheUnknown I am not able to prove even, first condition of functor i.e. $textbf{dom} ( g:f to f' ) = textbf{dom}(g): textbf{dom}(f) to textbf{dom}(f')$
    – MUH
    Dec 31 '18 at 8:39














4












4








4







I am studying Awodey's Category theory book. I have trouble understanding the following line:



Observe that there are two functors in arrow category i.e.
$$
begin{align}
mathscr{C} xleftarrow{textbf{dom}} mathscr{C}^{rightarrow} xrightarrow{textbf{cod}} mathscr{C}
end{align}
$$

where $mathscr{C}^{rightarrow}$ is the arrow category corresponding to $mathscr{C}$.



They have not mentioned what these $textbf{dom}$ and $textbf{cod}$ are? How do we prove that these are functors?



My understanding:
Now, in the diagram given below:



enter image description here



$textbf{dom}:mathscr{C}^{rightarrow} xrightarrow{textbf{dom}} mathscr{C}$. So,



$[f:A to B] mapsto A $ (object mapping of functor $textbf{dom}$) and
if $g=(g_1,g_2):[f:A to B] to [f':A' to B'] $, then $(g_1,g_2) mapsto [f:A to B]$ (the morphism mapping of functor $textbf{dom}$ ).



Using this definition, If I proceed to prove the statement:
(a)
$$
textbf{dom} ( g:f to f' ) = textbf{dom}(g): textbf{dom}(f) to textbf{dom}(f')
$$



LHS = $f:A to B$ and RHS = $textbf{dom}(g): A to A'$ (which seems absurd)
I am not sure if this makes sense.










share|cite|improve this question















I am studying Awodey's Category theory book. I have trouble understanding the following line:



Observe that there are two functors in arrow category i.e.
$$
begin{align}
mathscr{C} xleftarrow{textbf{dom}} mathscr{C}^{rightarrow} xrightarrow{textbf{cod}} mathscr{C}
end{align}
$$

where $mathscr{C}^{rightarrow}$ is the arrow category corresponding to $mathscr{C}$.



They have not mentioned what these $textbf{dom}$ and $textbf{cod}$ are? How do we prove that these are functors?



My understanding:
Now, in the diagram given below:



enter image description here



$textbf{dom}:mathscr{C}^{rightarrow} xrightarrow{textbf{dom}} mathscr{C}$. So,



$[f:A to B] mapsto A $ (object mapping of functor $textbf{dom}$) and
if $g=(g_1,g_2):[f:A to B] to [f':A' to B'] $, then $(g_1,g_2) mapsto [f:A to B]$ (the morphism mapping of functor $textbf{dom}$ ).



Using this definition, If I proceed to prove the statement:
(a)
$$
textbf{dom} ( g:f to f' ) = textbf{dom}(g): textbf{dom}(f) to textbf{dom}(f')
$$



LHS = $f:A to B$ and RHS = $textbf{dom}(g): A to A'$ (which seems absurd)
I am not sure if this makes sense.







category-theory






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edited Dec 31 '18 at 8:34

























asked Dec 31 '18 at 8:24









MUH

395216




395216












  • domain and codomain. Maps the arrow $f:Ato A'$ to $A$ and to $A'$ respectively.
    – Lord Shark the Unknown
    Dec 31 '18 at 8:26










  • @LordSharktheUnknown I am not able to prove even, first condition of functor i.e. $textbf{dom} ( g:f to f' ) = textbf{dom}(g): textbf{dom}(f) to textbf{dom}(f')$
    – MUH
    Dec 31 '18 at 8:39


















  • domain and codomain. Maps the arrow $f:Ato A'$ to $A$ and to $A'$ respectively.
    – Lord Shark the Unknown
    Dec 31 '18 at 8:26










  • @LordSharktheUnknown I am not able to prove even, first condition of functor i.e. $textbf{dom} ( g:f to f' ) = textbf{dom}(g): textbf{dom}(f) to textbf{dom}(f')$
    – MUH
    Dec 31 '18 at 8:39
















domain and codomain. Maps the arrow $f:Ato A'$ to $A$ and to $A'$ respectively.
– Lord Shark the Unknown
Dec 31 '18 at 8:26




domain and codomain. Maps the arrow $f:Ato A'$ to $A$ and to $A'$ respectively.
– Lord Shark the Unknown
Dec 31 '18 at 8:26












@LordSharktheUnknown I am not able to prove even, first condition of functor i.e. $textbf{dom} ( g:f to f' ) = textbf{dom}(g): textbf{dom}(f) to textbf{dom}(f')$
– MUH
Dec 31 '18 at 8:39




@LordSharktheUnknown I am not able to prove even, first condition of functor i.e. $textbf{dom} ( g:f to f' ) = textbf{dom}(g): textbf{dom}(f) to textbf{dom}(f')$
– MUH
Dec 31 '18 at 8:39










2 Answers
2






active

oldest

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3














Writing: $$fstackrel{(g_1,g_2)}{to}f'tag1$$ where $f,f'$ are objects of arrow category $mathcal C^{to}$ and pair $(g_1,g_2)$ is an element of homset $mathcal C^{to}(f,f')$ represents a commuting diagram pictured in your question.



We have the functor $mathbf{dom}:mathcal C^{to}tomathcal C$ prescribed by:$$[fstackrel{(g_1,g_2)}{to}f']mapsto[mathsf{dom}fstackrel{g_1}{to}mathsf{dom}f']$$



And we have the functor $mathbf{cod}:mathcal C^{to}tomathcal C$ prescribed by:$$[fstackrel{(g_1,g_2)}{to}f']mapsto[mathsf{cod}fstackrel{g_2}{to}mathsf{cod}f']$$



In order to prove that $mathbf{dom}$ and $mathbf{cod}$ are functors it must be shown both of them respect identities and composition.



If $(1)$ stands for an identity then $f=f'$ and $g_1,g_2$ are both identities in $mathcal C$. This guarantees that identities are respected.



By composition we must expand $(1)$ to: $$fstackrel{(g_1,g_2)}{to}f'text{ and }f'stackrel{(g'_1,g'_2)}{to}f''tag2$$with commuting squares.



Then we have: $$(g'_1,g'_2)circ(g_1,g_2)=(g'_1circ g_1,g'_2circ g_2)$$assuring that composition is respected.






share|cite|improve this answer





























    3














    The functor $mathbf{dom}$ takes the object $f$ to $A,$ the object $f'$ to $A';$ and takes the arrow $(g_1,g_2):fto f'$ to $g_1.$ Notice $g_1$ is an arrow $Ato A'$ as it should be.



    The identity morphism $mathrm{id}_f$ is simply $(mathrm{id}_A, mathrm{id}_{A'}),$ so $mathbf{dom}(mathrm{id}_f) = mathrm{id}_A$ as required.



    The composition of two morphisms $(g_1,g_2)circ (h_1,h_2)$ is $(g_1circ h_1, g_2circ h_2),$ as can be seen by drawing two commuting squares side by side. So $mathbf{dom}((g_1,g_2)circ (h_1,h_2)) = g_1circ h_1 = mathbf{dom}((g_1,g_2))circmathbf{dom}((h_1circ h_2))$






    share|cite|improve this answer





















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      2 Answers
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      active

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      2 Answers
      2






      active

      oldest

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      active

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      3














      Writing: $$fstackrel{(g_1,g_2)}{to}f'tag1$$ where $f,f'$ are objects of arrow category $mathcal C^{to}$ and pair $(g_1,g_2)$ is an element of homset $mathcal C^{to}(f,f')$ represents a commuting diagram pictured in your question.



      We have the functor $mathbf{dom}:mathcal C^{to}tomathcal C$ prescribed by:$$[fstackrel{(g_1,g_2)}{to}f']mapsto[mathsf{dom}fstackrel{g_1}{to}mathsf{dom}f']$$



      And we have the functor $mathbf{cod}:mathcal C^{to}tomathcal C$ prescribed by:$$[fstackrel{(g_1,g_2)}{to}f']mapsto[mathsf{cod}fstackrel{g_2}{to}mathsf{cod}f']$$



      In order to prove that $mathbf{dom}$ and $mathbf{cod}$ are functors it must be shown both of them respect identities and composition.



      If $(1)$ stands for an identity then $f=f'$ and $g_1,g_2$ are both identities in $mathcal C$. This guarantees that identities are respected.



      By composition we must expand $(1)$ to: $$fstackrel{(g_1,g_2)}{to}f'text{ and }f'stackrel{(g'_1,g'_2)}{to}f''tag2$$with commuting squares.



      Then we have: $$(g'_1,g'_2)circ(g_1,g_2)=(g'_1circ g_1,g'_2circ g_2)$$assuring that composition is respected.






      share|cite|improve this answer


























        3














        Writing: $$fstackrel{(g_1,g_2)}{to}f'tag1$$ where $f,f'$ are objects of arrow category $mathcal C^{to}$ and pair $(g_1,g_2)$ is an element of homset $mathcal C^{to}(f,f')$ represents a commuting diagram pictured in your question.



        We have the functor $mathbf{dom}:mathcal C^{to}tomathcal C$ prescribed by:$$[fstackrel{(g_1,g_2)}{to}f']mapsto[mathsf{dom}fstackrel{g_1}{to}mathsf{dom}f']$$



        And we have the functor $mathbf{cod}:mathcal C^{to}tomathcal C$ prescribed by:$$[fstackrel{(g_1,g_2)}{to}f']mapsto[mathsf{cod}fstackrel{g_2}{to}mathsf{cod}f']$$



        In order to prove that $mathbf{dom}$ and $mathbf{cod}$ are functors it must be shown both of them respect identities and composition.



        If $(1)$ stands for an identity then $f=f'$ and $g_1,g_2$ are both identities in $mathcal C$. This guarantees that identities are respected.



        By composition we must expand $(1)$ to: $$fstackrel{(g_1,g_2)}{to}f'text{ and }f'stackrel{(g'_1,g'_2)}{to}f''tag2$$with commuting squares.



        Then we have: $$(g'_1,g'_2)circ(g_1,g_2)=(g'_1circ g_1,g'_2circ g_2)$$assuring that composition is respected.






        share|cite|improve this answer
























          3












          3








          3






          Writing: $$fstackrel{(g_1,g_2)}{to}f'tag1$$ where $f,f'$ are objects of arrow category $mathcal C^{to}$ and pair $(g_1,g_2)$ is an element of homset $mathcal C^{to}(f,f')$ represents a commuting diagram pictured in your question.



          We have the functor $mathbf{dom}:mathcal C^{to}tomathcal C$ prescribed by:$$[fstackrel{(g_1,g_2)}{to}f']mapsto[mathsf{dom}fstackrel{g_1}{to}mathsf{dom}f']$$



          And we have the functor $mathbf{cod}:mathcal C^{to}tomathcal C$ prescribed by:$$[fstackrel{(g_1,g_2)}{to}f']mapsto[mathsf{cod}fstackrel{g_2}{to}mathsf{cod}f']$$



          In order to prove that $mathbf{dom}$ and $mathbf{cod}$ are functors it must be shown both of them respect identities and composition.



          If $(1)$ stands for an identity then $f=f'$ and $g_1,g_2$ are both identities in $mathcal C$. This guarantees that identities are respected.



          By composition we must expand $(1)$ to: $$fstackrel{(g_1,g_2)}{to}f'text{ and }f'stackrel{(g'_1,g'_2)}{to}f''tag2$$with commuting squares.



          Then we have: $$(g'_1,g'_2)circ(g_1,g_2)=(g'_1circ g_1,g'_2circ g_2)$$assuring that composition is respected.






          share|cite|improve this answer












          Writing: $$fstackrel{(g_1,g_2)}{to}f'tag1$$ where $f,f'$ are objects of arrow category $mathcal C^{to}$ and pair $(g_1,g_2)$ is an element of homset $mathcal C^{to}(f,f')$ represents a commuting diagram pictured in your question.



          We have the functor $mathbf{dom}:mathcal C^{to}tomathcal C$ prescribed by:$$[fstackrel{(g_1,g_2)}{to}f']mapsto[mathsf{dom}fstackrel{g_1}{to}mathsf{dom}f']$$



          And we have the functor $mathbf{cod}:mathcal C^{to}tomathcal C$ prescribed by:$$[fstackrel{(g_1,g_2)}{to}f']mapsto[mathsf{cod}fstackrel{g_2}{to}mathsf{cod}f']$$



          In order to prove that $mathbf{dom}$ and $mathbf{cod}$ are functors it must be shown both of them respect identities and composition.



          If $(1)$ stands for an identity then $f=f'$ and $g_1,g_2$ are both identities in $mathcal C$. This guarantees that identities are respected.



          By composition we must expand $(1)$ to: $$fstackrel{(g_1,g_2)}{to}f'text{ and }f'stackrel{(g'_1,g'_2)}{to}f''tag2$$with commuting squares.



          Then we have: $$(g'_1,g'_2)circ(g_1,g_2)=(g'_1circ g_1,g'_2circ g_2)$$assuring that composition is respected.







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered Dec 31 '18 at 8:53









          drhab

          98.3k544129




          98.3k544129























              3














              The functor $mathbf{dom}$ takes the object $f$ to $A,$ the object $f'$ to $A';$ and takes the arrow $(g_1,g_2):fto f'$ to $g_1.$ Notice $g_1$ is an arrow $Ato A'$ as it should be.



              The identity morphism $mathrm{id}_f$ is simply $(mathrm{id}_A, mathrm{id}_{A'}),$ so $mathbf{dom}(mathrm{id}_f) = mathrm{id}_A$ as required.



              The composition of two morphisms $(g_1,g_2)circ (h_1,h_2)$ is $(g_1circ h_1, g_2circ h_2),$ as can be seen by drawing two commuting squares side by side. So $mathbf{dom}((g_1,g_2)circ (h_1,h_2)) = g_1circ h_1 = mathbf{dom}((g_1,g_2))circmathbf{dom}((h_1circ h_2))$






              share|cite|improve this answer


























                3














                The functor $mathbf{dom}$ takes the object $f$ to $A,$ the object $f'$ to $A';$ and takes the arrow $(g_1,g_2):fto f'$ to $g_1.$ Notice $g_1$ is an arrow $Ato A'$ as it should be.



                The identity morphism $mathrm{id}_f$ is simply $(mathrm{id}_A, mathrm{id}_{A'}),$ so $mathbf{dom}(mathrm{id}_f) = mathrm{id}_A$ as required.



                The composition of two morphisms $(g_1,g_2)circ (h_1,h_2)$ is $(g_1circ h_1, g_2circ h_2),$ as can be seen by drawing two commuting squares side by side. So $mathbf{dom}((g_1,g_2)circ (h_1,h_2)) = g_1circ h_1 = mathbf{dom}((g_1,g_2))circmathbf{dom}((h_1circ h_2))$






                share|cite|improve this answer
























                  3












                  3








                  3






                  The functor $mathbf{dom}$ takes the object $f$ to $A,$ the object $f'$ to $A';$ and takes the arrow $(g_1,g_2):fto f'$ to $g_1.$ Notice $g_1$ is an arrow $Ato A'$ as it should be.



                  The identity morphism $mathrm{id}_f$ is simply $(mathrm{id}_A, mathrm{id}_{A'}),$ so $mathbf{dom}(mathrm{id}_f) = mathrm{id}_A$ as required.



                  The composition of two morphisms $(g_1,g_2)circ (h_1,h_2)$ is $(g_1circ h_1, g_2circ h_2),$ as can be seen by drawing two commuting squares side by side. So $mathbf{dom}((g_1,g_2)circ (h_1,h_2)) = g_1circ h_1 = mathbf{dom}((g_1,g_2))circmathbf{dom}((h_1circ h_2))$






                  share|cite|improve this answer












                  The functor $mathbf{dom}$ takes the object $f$ to $A,$ the object $f'$ to $A';$ and takes the arrow $(g_1,g_2):fto f'$ to $g_1.$ Notice $g_1$ is an arrow $Ato A'$ as it should be.



                  The identity morphism $mathrm{id}_f$ is simply $(mathrm{id}_A, mathrm{id}_{A'}),$ so $mathbf{dom}(mathrm{id}_f) = mathrm{id}_A$ as required.



                  The composition of two morphisms $(g_1,g_2)circ (h_1,h_2)$ is $(g_1circ h_1, g_2circ h_2),$ as can be seen by drawing two commuting squares side by side. So $mathbf{dom}((g_1,g_2)circ (h_1,h_2)) = g_1circ h_1 = mathbf{dom}((g_1,g_2))circmathbf{dom}((h_1circ h_2))$







                  share|cite|improve this answer












                  share|cite|improve this answer



                  share|cite|improve this answer










                  answered Dec 31 '18 at 8:57









                  spaceisdarkgreen

                  32.5k21753




                  32.5k21753






























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