Implement a URL shortener

JavaScript (ES6), 149 bytes

u=>u.split`/`.map((p,i)=>i?/^\d+$/.test(p)?(+p).toString(36):p[0]:(d=p.split`.`).slice(0,-1).map((s,j)=>s[l=j,0]).join``+"."+d[l].slice(1,3)).join`/`

Explanation

I made this independent of @Neil's solution but it ended up looking very similar.

u=>
  u.split`/`.map((p,i)=>       // for each part p at index i
    i?                         // if this is not the first part
      /^\d+$/.test(p)?         // if p is only digits
        (+p).toString(36)      // return p as a base-36 number
      :p[0]                    // else return the first letter
    :
      (d=p.split`.`)           // d = domain parts
      .slice(0,-1).map((s,j)=> // for each domain part before the last
        s[l=j,0]               // return the first letter, l = index of last domain part
      ).join``
      +"."+d[l].slice(1,3)     // add the 2 letters as the final domain
  )
  .join`/`                     // output each new part separated by a slash

Test

var solution = u=>u.split`/`.map((p,i)=>i?/^\d+$/.test(p)?(+p).toString(36):p[0]:(d=p.split`.`).slice(0,-1).map((s,j)=>s[l=j,0]).join``+"."+d[l].slice(1,3)).join`/`

<input type="text" id="input" value="math.stackexchange.com/a/2231/" />
<button onclick="result.textContent=solution(input.value)">Go</button>
<pre id="result"></pre>