How to arrive at unique factorization through the limit given naturality compatibility conditions?
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If $alpha : I to C$ from a small category to any category $C$. Define a functor $limlimits_{rightarrow} alpha : X mapsto limlimits_{leftarrow} text{Hom}_C(alpha, X)$ from $C^{op}$ to $text{Set}$.
If $limlimits_{rightarrow} alpha$ is representable then let $Y$ be a representative object. Then we have $text{Hom}_{C}(Y, Y) simeq limlimits_{leftarrow} text{Hom}_C(alpha, Y)$ by definition of representable. So that to $text{id}_Y$ is a associated a natural map $rho$ in $limlimits_{leftarrow} text{Hom}_C(alpha, Y)$ such that $rho_j circ alpha(s) = rho_i$ for any $s : i to j$ in $I$.
Suppose that we are given another family of morphism $f_i : alpha(i) to X$ in $C$ such that $f_j circ alpha(s) = f_i$. I'm seeing how there exists a unique map $g$ in $text{Hom}_C(Y, X)$ but I'm not seeing how $f_i = gcirc rho_i$.
category-theory limits-colimits functors representable-functor natural-transformations
asked Nov 24 '18 at 2:48
Roll up and smoke AdjointRoll up and smoke Adjoint
9,09552458
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add a comment |
$begingroup$
If $alpha : I to C$ from a small category to any category $C$. Define a functor $limlimits_{rightarrow} alpha : X mapsto limlimits_{leftarrow} text{Hom}_C(alpha, X)$ from $C^{op}$ to $text{Set}$.
If $limlimits_{rightarrow} alpha$ is representable then let $Y$ be a representative object. Then we have $text{Hom}_{C}(Y, Y) simeq limlimits_{leftarrow} text{Hom}_C(alpha, Y)$ by definition of representable. So that to $text{id}_Y$ is a associated a natural map $rho$ in $limlimits_{leftarrow} text{Hom}_C(alpha, Y)$ such that $rho_j circ alpha(s) = rho_i$ for any $s : i to j$ in $I$.
Suppose that we are given another family of morphism $f_i : alpha(i) to X$ in $C$ such that $f_j circ alpha(s) = f_i$. I'm seeing how there exists a unique map $g$ in $text{Hom}_C(Y, X)$ but I'm not seeing how $f_i = gcirc rho_i$.
category-theory limits-colimits functors representable-functor natural-transformations
asked Nov 24 '18 at 2:48
Roll up and smoke AdjointRoll up and smoke Adjoint
9,09552458
$endgroup$
1
$begingroup$
The functor $varprojlimtext{Hom}_C(alpha,X)$ is from $C$ to $mathbf{Set}$, not from $C^{op}$.
$endgroup$
– Oskar
Nov 24 '18 at 13:03
add a comment |
$begingroup$
If $alpha : I to C$ from a small category to any category $C$. Define a functor $limlimits_{rightarrow} alpha : X mapsto limlimits_{leftarrow} text{Hom}_C(alpha, X)$ from $C^{op}$ to $text{Set}$.
If $limlimits_{rightarrow} alpha$ is representable then let $Y$ be a representative object. Then we have $text{Hom}_{C}(Y, Y) simeq limlimits_{leftarrow} text{Hom}_C(alpha, Y)$ by definition of representable. So that to $text{id}_Y$ is a associated a natural map $rho$ in $limlimits_{leftarrow} text{Hom}_C(alpha, Y)$ such that $rho_j circ alpha(s) = rho_i$ for any $s : i to j$ in $I$.
Suppose that we are given another family of morphism $f_i : alpha(i) to X$ in $C$ such that $f_j circ alpha(s) = f_i$. I'm seeing how there exists a unique map $g$ in $text{Hom}_C(Y, X)$ but I'm not seeing how $f_i = gcirc rho_i$.
category-theory limits-colimits functors representable-functor natural-transformations
asked Nov 24 '18 at 2:48
Roll up and smoke AdjointRoll up and smoke Adjoint
9,09552458
$endgroup$
If $alpha : I to C$ from a small category to any category $C$. Define a functor $limlimits_{rightarrow} alpha : X mapsto limlimits_{leftarrow} text{Hom}_C(alpha, X)$ from $C^{op}$ to $text{Set}$.
If $limlimits_{rightarrow} alpha$ is representable then let $Y$ be a representative object. Then we have $text{Hom}_{C}(Y, Y) simeq limlimits_{leftarrow} text{Hom}_C(alpha, Y)$ by definition of representable. So that to $text{id}_Y$ is a associated a natural map $rho$ in $limlimits_{leftarrow} text{Hom}_C(alpha, Y)$ such that $rho_j circ alpha(s) = rho_i$ for any $s : i to j$ in $I$.
Suppose that we are given another family of morphism $f_i : alpha(i) to X$ in $C$ such that $f_j circ alpha(s) = f_i$. I'm seeing how there exists a unique map $g$ in $text{Hom}_C(Y, X)$ but I'm not seeing how $f_i = gcirc rho_i$.
category-theory limits-colimits functors representable-functor natural-transformations
category-theory limits-colimits functors representable-functor natural-transformations
asked Nov 24 '18 at 2:48
Roll up and smoke AdjointRoll up and smoke Adjoint
9,09552458
asked Nov 24 '18 at 2:48
Roll up and smoke AdjointRoll up and smoke Adjoint
9,09552458
asked Nov 24 '18 at 2:48
Roll up and smoke AdjointRoll up and smoke Adjoint
9,09552458
asked Nov 24 '18 at 2:48
Roll up and smoke AdjointRoll up and smoke Adjoint
9,09552458
asked Nov 24 '18 at 2:48
Roll up and smoke AdjointRoll up and smoke Adjoint
9,09552458
9,09552458
1
$begingroup$
The functor $varprojlimtext{Hom}_C(alpha,X)$ is from $C$ to $mathbf{Set}$, not from $C^{op}$.
$endgroup$
– Oskar
Nov 24 '18 at 13:03
add a comment |
1
$begingroup$
The functor $varprojlimtext{Hom}_C(alpha,X)$ is from $C$ to $mathbf{Set}$, not from $C^{op}$.
$endgroup$
– Oskar
Nov 24 '18 at 13:03
1
1
$begingroup$
The functor $varprojlimtext{Hom}_C(alpha,X)$ is from $C$ to $mathbf{Set}$, not from $C^{op}$.
$endgroup$
– Oskar
Nov 24 '18 at 13:03
$begingroup$
The functor $varprojlimtext{Hom}_C(alpha,X)$ is from $C$ to $mathbf{Set}$, not from $C^{op}$.
$endgroup$
– Oskar
Nov 24 '18 at 13:03
add a comment |
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$begingroup$
The functor $varprojlimtext{Hom}_C(alpha,X)$ is from $C$ to $mathbf{Set}$, not from $C^{op}$.
$endgroup$
– Oskar
Nov 24 '18 at 13:03