[Envelopes][1] and [refinements][2] are examples of colimits and limits respectively that play important role in Functional analysis (specifically, in the theory of topological algebras).

The [Arens-Michael envelope][3] is intuitively an operation (a functor) that turns each topological algebra $A$ into its  **nearest from the outside holomorphic algebra** $\text{Env}A$. It can be defined as the colimit of the system of Banach quotient algebras of $A$:
$$
\operatorname{Env}A=\lim_{\leftarrow} A/U,
$$
where $U$ means a *submultiplicative* closed convex balanced neighborhood of zero in $A$
$$
U\cdot U\subseteq U,
$$ 
and 
$$
A/U=(A/\bigcap_{\varepsilon>0}\varepsilon\cdot U)^\blacktriangledown
$$
is the completion of the quotient algebra $A/\bigcap_{\varepsilon>0}\varepsilon\cdot U$ endowed with the norm topology with the unit ball $U+\bigcap_{\varepsilon>0}\varepsilon\cdot U$ (the space $A/U$ will be a Banach algebra with respect to the "multiplication inherited from $A$").

As an illustration, if $A$ is the algebra ${\mathcal P}(M)$ of polynomials on an affine algebraic manifold $M$, then its Arens-Michael envelope $\text{Env}A$ is exactly the algebra ${\mathcal O}(M)$ of holomorphic functions on $M$:
$$
\operatorname{Env}{\mathcal P}(M)={\mathcal O}(M).
$$  
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":
$$
\operatorname{Env}{\mathcal P}_q({\mathbb C}^\times\ltimes{\mathbb C})={\mathcal O}_q({\mathbb C}^\times\ltimes{\mathbb C}),
$$  
for $|q|=1$, but the formula becomes different in the case of $|q|\ne 1$: 
$$
\operatorname{Env}{\mathcal P}_q({\mathbb C}^\times\ltimes{\mathbb C})={\mathcal O}({\mathbb C}^\times)\underset{\delta^q}{\overset{z}{\odot}}{\mathcal
P}^\star({\mathbb C})
$$  
(see details in [my paper][4]). [Alexei Pirkovskii][5], who reanimated this topic after some oblivion, calculates the Arens-Michael envelopes of different algebras as exercises.

Similarly, there are natural functors of taking 

- the [nearest continuous topological algebra][6] (see also [here][7]) with the formula
$$
\operatorname{Env}A={\mathcal C}(M)
$$  
for a subalgebra $A$ in ${\mathcal C}(M)$ having $M$ as the [involutive spectrum][8], and 


- the [nearest smooth topological algebra][6] with the formula
$$
\operatorname{Env}A={\mathcal C}^\infty(M)
$$  
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]: https://en.wikipedia.org/wiki/Envelope_(category_theory)
  [2]: https://en.wikipedia.org/wiki/Refinement_(category_theory)
  [3]: http://arxiv.org/abs/math/0406352
  [4]: http://arxiv.org/abs/0806.3205
  [5]: https://arxiv.org/search/math?searchtype=author&query=Pirkovskii%2C%20A%20Y
  [6]: http://arxiv.org/abs/1303.2424
  [7]: http://arxiv.org/abs/0907.1409
  [8]: http://arxiv.org/abs/1110.2013