Expressing the Lebesgue integral using categories + the difficulty of describing estimates in category theory In this question of mine in a comment to the accepted answer, someone remarked:

There are ways to express even basic things in analysis, such as the
  spectral theorem or the Lebesgue integral, using the language of
  categories. But many of the hard theorems in analysis boil down to
  subtle estimates which (so far!) have not been simplified by clever
  categorical arguments.

I replied to this requesting a reference where the Lebesgue integral is expressed using categories but unfortunately I didn't get an answer. So I'm posting it as a question (I wouldn't mind for a reference concerning the spectral theorem either) with the additional kind request to also provide an explanation why it is difficult to describe estimates using category theory. 
 A: A beautiful ∞-category theoretic formalization of integration of differential forms and of the Stokes theorem has recently been given in (Bunke-Nikolaus-Völkl 13).
I had observed in (Schreiber 13) that if an ∞-category $\mathbf{H}$ is a cohesive ∞-topos then for group objects (deloopable objects) $G$ there is canonically a new object $\flat_{dR} G$ (also written $\flat_{dR}\mathbf{B}G$, but the former convention is more convenient here) which behaves like the moduli stack of $G$-valued differential form data. Dually for $\Pi_{dR}G$ I used this to describe some "unstable" differential cohomology, such as Chern-Weil theory and Wess-Zumino-Witten theory.
The observation in (Bunke-Nikolaus-Völkl 13) is that the axiomatics becomes dramatically stronger when applied to stable objects, hence to spectrum objects in the stable ∞-category $Stab(\mathbf{H})$. 
They observe that on a stable cohesive object $\hat E$ the canonical morphism 
$$
  \Pi_{dR} \hat E \stackrel{\mathbf{d}}{\rightarrow} \flat_{dR}\hat E
$$ 
interprets as the de Rham differential. Moreover they observe that there is canonically a morphism
$$
  \int_{\Delta^1} \colon [\Delta^1, \flat_{dR} \hat E ] \longrightarrow \Pi_{dR} \hat E 
$$
which interprets as fiber integration of differential forms. In particular they show that this satisfies the Stokes theorem in that the compositite $\int \circ \mathbf{d}$ is the difference between the evaluation-at-the-endpoint maps.
This holds fully generally in any cohesive ∞-topos $\mathbf{H}$. When realized in the specific $\mathbf{H} = $ Smooth∞Grpd then this abstract construction reduces to the ordinary integration of smooth differential forms and the ordinary Stokes theorem.
For a review of this with more pointers see on the nLab at 


*

*integration of differential forms -- In cohesive homotopy-type theory.


In closing I remark that (Bunke-Nikolaus-Völkl 13) also show that the de Rham differential morphism $\mathbf{d}$ above with its reverse integration morphism $\int$ is just one part of an exact hexagon that every stable object in a cohesive $\infty$-topos canonically sits in -- and that this hexagon is the differential cohomology hexagon that (Simons-Sullivan 07) observed characterizes ordinary differential cohomology and which they asked/conjectured whether it also characterizes differential generalized cohomology, generally.
The answer is: in a cohesive $\infty$-topos every stable object $\hat E \in Stab(\mathbf{H})$ canonically sits in a hexagon
$$
  \array{
    &&  \Pi_{dR} \hat E && \stackrel{\mathbf{d}}{\longrightarrow} && \flat_{dR}\hat E
    \\
    & \nearrow & & \searrow & & \nearrow_{{\theta_{\hat E}}} && \searrow
    \\
    \flat \Pi_{dR} \hat E  &&  && \hat E &&  && \Pi \flat_{dR} \hat E
    \\
    & \searrow &  & \nearrow & & \searrow && \nearrow_{{ch_{\hat E}}}
    \\
    && \flat \hat E && \longrightarrow && \Pi \hat E
  }
  \,,
$$
which is fully exact in that both outer boundaries a homotopy fiber sequences and both squares are homotopy cartesian; and that this interprets as the differential cohomology hexagon for a differential cohomology refinement $\hat E$ of the generalized cohomology theory $\Pi \hat E$ in that it always has the interpretation explained here.
This answers (Simons-Sullivan 07) in an interesting way: differential generalized cohomology theory effectively is the theory of $\infty$-categories which are cohesive ∞-toposes. As one facet this includes de Rham theory, including de Rham differentiation, integration of differential forms and the Stokes theorem.
This general abstract $\infty$-category theoretic formulation of differential geometry/differential cohomology has some pleasant consequences. I have tried to highlight these recently at 


*

*Differential generalized cohomology in Cohesive homotopy type theory, talk at IHP trimester on Semantics of proofs and certified mathematics, Workshop 1: Formalization of Mathematics, Institut Henri Poincaré, Paris, 5-9 May 2014

A: Sorry to refer to my own work, but I think this answers your question directly: http://www.maths.ed.ac.uk/~tl/glasgowpssl/
That link is to a very short note, but I might as well repeat the result here.  Let's agree that a "map" of Banach spaces is a map of norm $\leq 1$, and let's also agree that when $X$ and $Y$ are Banach spaces, we equip $X \oplus Y$ with the norm $\| (x, y) \| = (\|x\| + \|y\|)/2$.  
Now let $\mathcal{C}$ be the category of triples $(X, \xi, u)$ where $X$ is a Banach space, $\xi$ is a map $X \oplus X \to X$, and $u$ is an element of the closed unit ball of $X$ such that $\xi(u, u) = u$.
Theorem: The initial object of $\mathcal{C}$ is $(L^1[0, 1], \gamma, 1)$ where $1$ is the constant function $1$ and $\gamma$ concatenates two functions then scales the domain by a factor of $1/2$.
Another object of $\mathcal{C}$ is $(\mathbb{R}, \text{mean}, 1)$.  The unique map in $\mathcal{C}$ from the initial object to this object is Lebesgue integration, $\int_0^1: L^1[0, 1] \to \mathbb{R}$.

While I'm at it, I'll add another result that isn't in that note (or written up anywhere yet).  This characterizes Lebesgue integrability and integration on arbitrary finite measure spaces.
Let $\mathbf{Meas}$ be the category of finite measure spaces and "embeddings" (by which I mean maps that are isomorphisms to their images).  Let $\mathbf{Ban}$ be the category of Banach spaces (with maps as above).  
Let $\mathcal{D}$ be the category of pairs $(F, u)$, where $F$ is a functor $\mathbf{Meas} \to \mathbf{Ban}$ and $u$ assigns to each measure space $X = (X, \mu)$ an element $u_X \in F(X, \mu)$, subject to two laws: first, $\|u_X\| \leq \mu(X)$, and second, whenever
$$
Y \stackrel{i}{\longrightarrow} X \stackrel{j}{\longleftarrow} Z
$$
in $\mathbf{Meas}$ with $X = iY \sqcup jZ$ (disjoint union) then $(Fi)u_Y + (Fj)u_Z = u_X$.
Theorem The initial object of $\mathcal{D}$ is $(L^1, I)$, where $I_X \in L^1(X)$ is the constant function $1$.
(In the case of this initial object, the equation "$(Fi)u_Y + (Fj)u_Z = u_X$" says that when $X$ is partitioned into subsets $Y$ and $Z$, the indicator function of $Y$ plus the indicator function of $Z$ is the constant function $1$.) 
Another object of $\mathcal{D}$ is $(K, t)$, where $K$ has constant value $\mathbb{R}$ (or $\mathbb{C}$, depending on our choice of ground field) and $t_X = \mu(X)$ for a measure space $X = (X, \mu)$.  The unique map in $\mathcal{D}$ from the initial object to this object is integration.  To spell that out a bit more: the maps in $\mathcal{D}$ are natural transformations satisfying the obvious condition, and in this case, the $X$-component of the unique map $(L^1, I) \to (K, t)$ is $\int_X: L^1(X) \to \mathbb{R}$.
A: I am not sure what you are looking for, but maybe the Giry monad (the page needs some work), for measure theory, and (categorical) Gelfand duality, for the spectral theorem, can help to phrase a more precise question.
A: Your question is, by its nature, rather vague and here is a subjective take on it in addition to the answers you have already received.  Whether this is the type of information that you are looking for is something only you can decide.
I will begin with the spectral theorem since this evinced only cursory replies.  The two basic dualities in analysis are probably the Riesz representation theorem (between $C(K)$ and the Radon measures on the compact space $K$) and Gelfand Naimark duality.  One can extend the latter be replacing the real line as the range space of the latter by more exotic algebras.  The spectral theorem for self-adjoint operators can be expressed in the form that the space of such operators with spectra in the compact subet $K$ of the reals (the complex plane for normal operators) is identifiable with the generalised spectrum that one gets by using the algebra $L(H)$ of operators.  This is basically a formulation of the functional calculus and has a fairly elementary proof---set it up for polynomials, derive one of the estimates you mention and extend by continuity and the Weierstraß theorem). One can then get the version involving spectral measures by invoking a suitable extension of the RRT for vector-valued measures.
This can be extended to the case of unbounded operators using the space of bounded continuous functions on the line but is a bit more delicate to state and prove since one has to use more involved structures on both spaces---not the norm but the so-called strict or mixed topologies (Buck, Orlicz, Wiweger, et al.)
As regards the question on integration there is a simple and transparent construction of the Banach space $L^1([0,1])$ in the language of category theory.  Denote by $\cal F_n$ the finite algebra generated by the dyadic intervals of length $2^{-n}$.  Then the corresponding $\ell^1$-space can be identified with $L^1(\cal F_n)$.  These spaces form an inductive spectrum in the category of Banach spaces with linear contractions as morphisms and we can define $L^1$ to be its inductive limit. (As mentioned in a comment, this is the proof given by Tom Leinster).
But the decisive point, which hasn't been mentioned above, is that we can identify  this space with A SPACE OF FUNCTIONS (better, equivalence classes thereof) on the interval.  Without this fact, these considerations remain a mere game with glass beads ("Glasperlenspiel").
It is for this reason that the treatment to be found in the textbooks of analysis and probablity are more elaborate (rather than just saying, say, take the space of continuous functions with the the norm defined using the Riemann integral and complete it).  Remarkably, the definition of a Lebesgue integrable function can be compressed into one sentence as was shown by Jan Mikusinski: a function such that there is a sequence $\alpha_k$ of real numbers and a sequence of intervals $I_k$ in the union of the $\cal F_n$ such that the series $\sum_k \alpha_k \lambda (I_k)$
is absolutely summable and $f(x)=\sum \alpha_k \chi_{I_k}(x)$ whenever the RHS converges absolutely.  
Of course, one then has to show that the basic properties follow from this definition---without Fatou, Beppo-Levi, dominated convergence, Radon Nikodym, etc., analysts and probabilists would be at rather a loss.  This is where the estimates come in and is done by Mikusinski in his texts.
