The [Arens-Michael envelope][1] is an example of a natural colimit in Functional analysis. Intuitively it is an operation (a functor) that turns each topological algebra $A$ into its  **nearest from the outside holomorphic algebra** $\text{Env}A$. 

As an illustration, if $A$ is the algebra ${\mathcal P}(M)$ of polynomials on an affine algebraic manifold $M$, then its Arens-Micael envelope $\text{Env}A$ is exactly the algebra ${\mathcal O}(M)$ of holomorphic functions on $M$:
$$
\text{Env}{\mathcal P}(M)={\mathcal O}(M).
$$  
The non-commutative case is especially interesting, since it generates non-trivial computations in non-commutative geometry. For example, the "polynomial version" of the so-called $az+b$-group is turned into its "holomorphic version":
$$
\text{Env}{\mathcal P}_q({\mathbb C}^\times\ltimes{\mathbb C})={\mathcal O}_q({\mathbb C}^\times\ltimes{\mathbb C}),
$$  
for $|q|=1$, but the formula becomes different in the case of $|q|\ne 1$: 
$$
\text{Env}{\mathcal P}_q({\mathbb C}^\times\ltimes{\mathbb C})={\mathcal O}({\mathbb C}^\times)\underset{\delta^q}{\overset{z}{\odot}}{\mathcal
P}^\star({\mathbb C})
$$  
(see details in [my paper][2]).

Similarly, there are natural functors of taking 

- the [nearest continuous topological algebra][3] with the formula
$$
\text{Env}A={\mathcal C}(M)
$$  
for a subalgebra $A$ in ${\mathcal C}(M)$ having $M$ as the [involutive spectrum][5], and 


- the [nearest smooth topological algebra][4] with the formula
$$
\text{Env}A={\mathcal C}^\infty(M)
$$  
for a subalgebra $A$ in ${\mathcal C}^\infty(M)$ having $M$ as the involutive spectrum and $T_x(M)$ as the involutive tangent space in each point $x\in M$.  


  [1]: http://arxiv.org/abs/math/0406352
  [2]: http://arxiv.org/abs/0806.3205
  [3]: http://arxiv.org/abs/0907.1409
  [4]: http://arxiv.org/abs/1303.2424
  [5]: http://arxiv.org/abs/1110.2013