Tom Leinster has a note here about how you can realize L^1[0,1] as the initial object of a certain category. You should really read his note because it is only 2.5 pages and is much more charming than what I am going to write below as background, but if you don't want to click on the link here is the idea:

We work in the category of Banach spaces with contractive maps, where we are defining $X \oplus Y$ to have the norm $|| (x,y) || = \frac{1}{2}(||x|| + ||y||)$. Consider triples $(X, \xi, u)$ where $X$ is a Banach space, $u \in X$ has norm at most 1, $\xi:X \oplus X \to X$ is a map of Banach spaces with $\xi(u,u) = u$. A morphism of such triples is a map of Banach spaces commuting with all structure in sight. The it turns out that the initial object in this category is $(L^1[0,1], \gamma, 1)$ where $\gamma(f,g)$ smushes $f$ and $g$ by a factor of two horizontally and then puts them side by side. Essentially this is because once you know where the constant function 1 goes, you can determine where any piecewise constant function whose discontinuities are at dyadic rationals goes, and then by density you get a unique map out of $L^1$.

Leinster mentions that there are some abstract results which actually construct an initial object for a category like this. I have looked through Barr and Wells, but I do not see what exactly I should be using here. The only general initial object construction I know (The one at the beginning of the adjoint functor theorem chapter of Categories for the Working Mathematician), doesn't seem to apply (maybe it does but I am not seeing it). Does anyone know such a general construction which applies here? How does that construction look when you apply it to this situation? Does it look anything like the usual construction of $L^1[0,1]$?

Tom Leinster has a note here about how you can realize L^1[0,1] as the initial object of a certain category. You should really read his note because it is only 2.5 pages and is much more charming than what I am going to write below as background, but if you don't want to click on the link here is the idea:

We work in the category of Banach spaces with contractive maps, where we are defining $X \oplus Y$ to have the norm $|| (x,y) || = \frac{1}{2}(||x|| + ||y||)$. Consider triples $(X, \xi, u)$ where $X$ is a Banach space, $u \in X$ has norm at most 1, $\xi:X \oplus X \to X$ is a map of Banach spaces with $\xi(u,u) = u$. A morphism of such triples is a map of Banach spaces commuting with all structure in sight. The it turns out that the initial object in this category is $(L^1[0,1], \gamma, 1)$ where $\gamma(f,g)$ smushes $f$ and $g$ by a factor of two horizontally and then puts them side by side. Essentially this is because once you know where the constant function 1 goes, you can determine where any piecewise constant function whose discontinuities are at dyadic rationals goes, and then by density you get a unique map out of $L^1$.

Leinster mentions that there are some abstract results which actually construct an initial object for a category like this. I have looked through Barr and Wells, but I do not see what exactly I should be using here. The only general initial object construction I know (The one at the beginning of the adjoint functor theorem chapter of Categories for the Working Mathematician), doesn't seem to apply (maybe it does but I am not seeing it). Does anyone know such a general construction which applies here? How does that construction look when you apply it to this situation? Does it look anything like the usual construction of $L^1[0,1]$?

Tom Leinster has a note here about how you can realize L^1[0,1] as the initial object of a certain category. You should really read his note because it is only 2.5 pages and is much more charming than what I am going to write below as background, but if you don't want to click on the link here is the idea:

Consider triples $(X, \xi, u)$ where $X$ is a Banach space, $u \in X$ has norm at most 1, $\xi:X \oplus X \to X $ is a map of Banach spaces* with $\xi(u,u) = u$. A morphism of such triples is a map of Banach spaces* commuting with all structure in sight. The it turns out that the initial object in this category is $(L^1[0,1], \gamma, 1)$ where $\gamma(f,g)$ smushes $f$ and $g$ by a factor of two horizontally and then puts them side by side. Essentially this is because once you know where the constant function 1 goes, you can determine where any piecewise constant function whose discontinuities are at dyadic rationals goes, and then by density you get a unique map out of $L^1$.

Leinster mentions that there are some abstract results which actually construct an initial object for a category like this. I have looked through Barr and Wells, but I do not see what exactly I should be using here. The only general initial object construction I know (The one at the beginning of the adjoint functor theorem chapter of Categories for the Working Mathematician), doesn't seem to apply (maybe it does but I am not seeing it). Does anyone know such a general construction which applies here? How does that construction look when you apply it to this situation? Does it look anything like the usual construction of $L^1[0,1]$?

*A map of banach spaces in this case is a contractible linear map

Tom Leinster has a note here about how you can realize L^1[0,1] as the initial object of a certain category. You should really read his note because it is only 2.5 pages and is much more charming than what I am going to write below as background, but if you don't want to click on the link here is the idea:

We work in the category of Banach spaces with contractive maps, where we are defining $X \oplus Y$ to have the norm $|| (x,y) || = \frac{1}{2}(||x|| + ||y||)$. Consider triples $(X, \xi, u)$ where $X$ is a Banach space, $u \in X$ has norm at most 1, $\xi:X \oplus X \to X$ is a map of Banach spaces with $\xi(u,u) = u$. A morphism of such triples is a map of Banach spaces commuting with all structure in sight. The it turns out that the initial object in this category is $(L^1[0,1], \gamma, 1)$ where $\gamma(f,g)$ smushes $f$ and $g$ by a factor of two horizontally and then puts them side by side. Essentially this is because once you know where the constant function 1 goes, you can determine where any piecewise constant function whose discontinuities are at dyadic rationals goes, and then by density you get a unique map out of $L^1$.

Leinster mentions that there are some abstract results which actually construct an initial object for a category like this. I have looked through Barr and Wells, but I do not see what exactly I should be using here. The only general initial object construction I know (The one at the beginning of the adjoint functor theorem chapter of Categories for the Working Mathematician), doesn't seem to apply (maybe it does but I am not seeing it). Does anyone know such a general construction which applies here? How does that construction look when you apply it to this situation? Does it look anything like the usual construction of $L^1[0,1]$?

Tom Leinster has a note here about how you can realize L^1[0,1] as the initial object of a certain category. You should really read his note because it is only 2.5 pages and is much more charming than what I am going to write below as background, but if you don't want to click on the link here is the idea:

Consider triples $(X, \xi, u)$ where $X$ is a Banach space, $\xi:X \oplus X \to X $ is a map of Banach spaces*, and $u \in X$ has norm at most 1. A morphism of such triples is a map of Banach spaces* commuting with all structure in sight. The it turns out that the initial object in this category is $(L^1[0,1], \gamma, 1)$ where $\gamma(f,g)$ smushes $f$ and $g$ by a factor of two horizontally and then puts them side by side. Essentially this is because once you know where the constant function 1 goes, you can determine where any piecewise constant function whose discontinuities are at dyadic rationals goes, and then by density you get a unique map out of $L^1$.

Leinster mentions that there are some abstract results which actually construct an initial object for a category like this. The only general initial object construction I know (The one at the beginning of the adjoint functor theorem chapter of Categories for the Working Mathematician), doesn't seem to apply (maybe it does but I am not seeing it). Does anyone know such a general construction which applies here? How does that construction look when you apply it to this situation? Does it look anything like the usual construction of $L^1[0,1]$?

*A map of banach spaces in this case is a contractible linear map

Tom Leinster has a note here about how you can realize L^1[0,1] as the initial object of a certain category. You should really read his note because it is only 2.5 pages and is much more charming than what I am going to write below as background, but if you don't want to click on the link here is the idea:

Consider triples $(X, \xi, u)$ where $X$ is a Banach space, $u \in X$ has norm at most 1, $\xi:X \oplus X \to X $ is a map of Banach spaces* with $\xi(u,u) = u$. A morphism of such triples is a map of Banach spaces* commuting with all structure in sight. The it turns out that the initial object in this category is $(L^1[0,1], \gamma, 1)$ where $\gamma(f,g)$ smushes $f$ and $g$ by a factor of two horizontally and then puts them side by side. Essentially this is because once you know where the constant function 1 goes, you can determine where any piecewise constant function whose discontinuities are at dyadic rationals goes, and then by density you get a unique map out of $L^1$.

Leinster mentions that there are some abstract results which actually construct an initial object for a category like this. I have looked through Barr and Wells, but I do not see what exactly I should be using here. The only general initial object construction I know (The one at the beginning of the adjoint functor theorem chapter of Categories for the Working Mathematician), doesn't seem to apply (maybe it does but I am not seeing it). Does anyone know such a general construction which applies here? How does that construction look when you apply it to this situation? Does it look anything like the usual construction of $L^1[0,1]$?

*A map of banach spaces in this case is a contractible linear map