Terminal and Initial objects in the category of functors












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Let $C$ and $D$ are 2 non empty categories and $[C,D]$ is the category of functors between $C$ and $D$. I assume that the constant functor $Delta_t$ (that maps all objects from $C$ into a single object $t$ from $D$) is the terminal object in the category. I also think that it always exists. Am I right?



Another question is about initial object in the category. What it is?






category-theory







share|cite|improve this question











$endgroup$

















    1












    $begingroup$


    Let $C$ and $D$ are 2 non empty categories and $[C,D]$ is the category of functors between $C$ and $D$. I assume that the constant functor $Delta_t$ (that maps all objects from $C$ into a single object $t$ from $D$) is the terminal object in the category. I also think that it always exists. Am I right?



    Another question is about initial object in the category. What it is?






    category-theory







    share|cite|improve this question











    $endgroup$















      1












      1








      1





      $begingroup$


      Let $C$ and $D$ are 2 non empty categories and $[C,D]$ is the category of functors between $C$ and $D$. I assume that the constant functor $Delta_t$ (that maps all objects from $C$ into a single object $t$ from $D$) is the terminal object in the category. I also think that it always exists. Am I right?



      Another question is about initial object in the category. What it is?






      category-theory







      share|cite|improve this question











      $endgroup$




      Let $C$ and $D$ are 2 non empty categories and $[C,D]$ is the category of functors between $C$ and $D$. I assume that the constant functor $Delta_t$ (that maps all objects from $C$ into a single object $t$ from $D$) is the terminal object in the category. I also think that it always exists. Am I right?



      Another question is about initial object in the category. What it is?






      category-theory




      category-theory






      share|cite|improve this question















      share|cite|improve this question













      share|cite|improve this question




      share|cite|improve this question








      edited Dec 10 '18 at 12:51







      Ivan

















      asked Dec 10 '18 at 12:16









      IvanIvan

      1076




      1076






















          1 Answer
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          $begingroup$

          It depends on the constant functor! since all your natural transformations actually happen on the image category, you need that the image if your object is the terminal respectively final object. Hence, the terminal object is the constant functor onto a $underline{textrm{terminal}}$ object, and dually the initial object is the constant functor onto the $underline{textrm{initial}}$ object (assuming that $C$ admits those)! which is something you need, otherwise you run into problems fast.






          share|cite|improve this answer









          $endgroup$









          • 1




            $begingroup$
            I.e. if $i$ is the initial object of $D$, $t$ is the terminal object of $D$ then constant functor $Delta_i$ that maps all objects from $C$ into $i$, is the initial object of $[C,D]$ and $Delta_t$ is the terminal one?
            $endgroup$
            – Ivan
            Dec 10 '18 at 12:43








          • 2




            $begingroup$
            exactly! a lot of such stuff happens due to your natural transofrmation jsut happening in the target and since all your maps have to be unique (by assumtion on $t$ and $i$) this is again initial and terminal.
            $endgroup$
            – Enkidu
            Dec 10 '18 at 12:44











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          $begingroup$

          It depends on the constant functor! since all your natural transformations actually happen on the image category, you need that the image if your object is the terminal respectively final object. Hence, the terminal object is the constant functor onto a $underline{textrm{terminal}}$ object, and dually the initial object is the constant functor onto the $underline{textrm{initial}}$ object (assuming that $C$ admits those)! which is something you need, otherwise you run into problems fast.






          share|cite|improve this answer









          $endgroup$









          • 1




            $begingroup$
            I.e. if $i$ is the initial object of $D$, $t$ is the terminal object of $D$ then constant functor $Delta_i$ that maps all objects from $C$ into $i$, is the initial object of $[C,D]$ and $Delta_t$ is the terminal one?
            $endgroup$
            – Ivan
            Dec 10 '18 at 12:43








          • 2




            $begingroup$
            exactly! a lot of such stuff happens due to your natural transofrmation jsut happening in the target and since all your maps have to be unique (by assumtion on $t$ and $i$) this is again initial and terminal.
            $endgroup$
            – Enkidu
            Dec 10 '18 at 12:44
















          2












          $begingroup$

          It depends on the constant functor! since all your natural transformations actually happen on the image category, you need that the image if your object is the terminal respectively final object. Hence, the terminal object is the constant functor onto a $underline{textrm{terminal}}$ object, and dually the initial object is the constant functor onto the $underline{textrm{initial}}$ object (assuming that $C$ admits those)! which is something you need, otherwise you run into problems fast.






          share|cite|improve this answer









          $endgroup$









          • 1




            $begingroup$
            I.e. if $i$ is the initial object of $D$, $t$ is the terminal object of $D$ then constant functor $Delta_i$ that maps all objects from $C$ into $i$, is the initial object of $[C,D]$ and $Delta_t$ is the terminal one?
            $endgroup$
            – Ivan
            Dec 10 '18 at 12:43








          • 2




            $begingroup$
            exactly! a lot of such stuff happens due to your natural transofrmation jsut happening in the target and since all your maps have to be unique (by assumtion on $t$ and $i$) this is again initial and terminal.
            $endgroup$
            – Enkidu
            Dec 10 '18 at 12:44














          2












          2








          2





          $begingroup$

          It depends on the constant functor! since all your natural transformations actually happen on the image category, you need that the image if your object is the terminal respectively final object. Hence, the terminal object is the constant functor onto a $underline{textrm{terminal}}$ object, and dually the initial object is the constant functor onto the $underline{textrm{initial}}$ object (assuming that $C$ admits those)! which is something you need, otherwise you run into problems fast.






          share|cite|improve this answer









          $endgroup$



          It depends on the constant functor! since all your natural transformations actually happen on the image category, you need that the image if your object is the terminal respectively final object. Hence, the terminal object is the constant functor onto a $underline{textrm{terminal}}$ object, and dually the initial object is the constant functor onto the $underline{textrm{initial}}$ object (assuming that $C$ admits those)! which is something you need, otherwise you run into problems fast.







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered Dec 10 '18 at 12:23









          EnkiduEnkidu

          1,32619




          1,32619








          • 1




            $begingroup$
            I.e. if $i$ is the initial object of $D$, $t$ is the terminal object of $D$ then constant functor $Delta_i$ that maps all objects from $C$ into $i$, is the initial object of $[C,D]$ and $Delta_t$ is the terminal one?
            $endgroup$
            – Ivan
            Dec 10 '18 at 12:43








          • 2




            $begingroup$
            exactly! a lot of such stuff happens due to your natural transofrmation jsut happening in the target and since all your maps have to be unique (by assumtion on $t$ and $i$) this is again initial and terminal.
            $endgroup$
            – Enkidu
            Dec 10 '18 at 12:44














          • 1




            $begingroup$
            I.e. if $i$ is the initial object of $D$, $t$ is the terminal object of $D$ then constant functor $Delta_i$ that maps all objects from $C$ into $i$, is the initial object of $[C,D]$ and $Delta_t$ is the terminal one?
            $endgroup$
            – Ivan
            Dec 10 '18 at 12:43








          • 2




            $begingroup$
            exactly! a lot of such stuff happens due to your natural transofrmation jsut happening in the target and since all your maps have to be unique (by assumtion on $t$ and $i$) this is again initial and terminal.
            $endgroup$
            – Enkidu
            Dec 10 '18 at 12:44








          1




          1




          $begingroup$
          I.e. if $i$ is the initial object of $D$, $t$ is the terminal object of $D$ then constant functor $Delta_i$ that maps all objects from $C$ into $i$, is the initial object of $[C,D]$ and $Delta_t$ is the terminal one?
          $endgroup$
          – Ivan
          Dec 10 '18 at 12:43






          $begingroup$
          I.e. if $i$ is the initial object of $D$, $t$ is the terminal object of $D$ then constant functor $Delta_i$ that maps all objects from $C$ into $i$, is the initial object of $[C,D]$ and $Delta_t$ is the terminal one?
          $endgroup$
          – Ivan
          Dec 10 '18 at 12:43






          2




          2




          $begingroup$
          exactly! a lot of such stuff happens due to your natural transofrmation jsut happening in the target and since all your maps have to be unique (by assumtion on $t$ and $i$) this is again initial and terminal.
          $endgroup$
          – Enkidu
          Dec 10 '18 at 12:44




          $begingroup$
          exactly! a lot of such stuff happens due to your natural transofrmation jsut happening in the target and since all your maps have to be unique (by assumtion on $t$ and $i$) this is again initial and terminal.
          $endgroup$
          – Enkidu
          Dec 10 '18 at 12:44


















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