Show buffer name relative to git folder

autoload/airline/extensions.vim

@@ -229,6 +229,12 @@ function! airline#extensions#load()
     call add(loaded_ext, 'bufferline')
   endif
 
+  if get(g:, 'airline#extensions#fugitiveline#enabled', 1)
+        \ && exists('*fugitive#head')
+    call airline#extensions#fugitiveline#init(s:ext)
+    call add(loaded_ext, 'fugitiveline')
+  endif
+
   if (get(g:, 'airline#extensions#virtualenv#enabled', 1) && (exists(':VirtualEnvList') || isdirectory($VIRTUAL_ENV)))
     call airline#extensions#virtualenv#init(s:ext)
     call add(loaded_ext, 'virtualenv')

autoload/airline/extensions/fugitiveline.vim

@@ -0,0 +1,37 @@
+" MIT License. Copyright (c) 2017 Cimbali.
+" vim: et ts=2 sts=2 sw=2
+
+scriptencoding utf-8
+
+if !exists('*fugitive#head')
+  finish
+endif
+
+
+if exists("+autochdir") && &autochdir == 1
+  let s:fmod = ':p'
+else
+  let s:fmod = ':.'
+endif
+
+function! airline#extensions#fugitiveline#bufname()
+  try
+    let buffer = fugitive#buffer()
+    if buffer.type('blob')
+      return fnamemodify(buffer.repo().translate(buffer.path()), s:fmod)
+    endif
+  catch
+  endtry
+
+  return fnamemodify(bufname('%'), s:fmod)
+endfunction
+
+function! airline#extensions#fugitiveline#init(ext)
+  if exists("+autochdir") && &autochdir == 1
+    " if 'acd' is set, vim-airline uses the path section, so we need to redefine this here as well
+    call airline#parts#define_raw('path', '%<%{airline#extensions#fugitiveline#bufname()}%m')
+  else
+    call airline#parts#define_raw('file', '%<%{airline#extensions#fugitiveline#bufname()}%m')
+  endif
+endfunction
+

doc/airline.txt

@@ -400,6 +400,13 @@ vim-bufferline <https://github.com/bling/vim-bufferline>
 * determine whether bufferline will overwrite customization variables >
   let g:airline#extensions#bufferline#overwrite_variables = 1
 <
+-------------------------------------                   *airline-fugitiveline*
+This extension hides the fugitive://**// part of the buffer names, to only
+show the file name as if it were in the current working tree.
+
+* enable/disable bufferline integration >
+  let g:airline#extensions#fugitiveline#enabled = 1
+<
 -------------------------------------                       *airline-branch*
 
 vim-airline will display the branch-indicator together with the branch name in