"Reduce" function for Series

Vectorized but slow

You can use NumPy's concatenate:

import numpy as np

list(np.concatenate(categories.values))

Performance

But we have lists, i.e. Python objects already. So the vectorization has to switch back and forth between Python objects and NumPy data types. This make things slow:

categories = pd.Series([['a', 'b'], ['c', 'd', 'e']] * 1000)

%timeit list(np.concatenate(categories.values))
100 loops, best of 3: 7.66 ms per loop

%timeit np.concatenate(categories.values)
100 loops, best of 3: 5.33 ms per loop

%timeit list(chain.from_iterable(categories.values))
1000 loops, best of 3: 231 µs per loop

You can try your luck with business["categories"].str.join(''), but I am guessing that Pandas uses Pythons string functions. I doubt you can do better tha what Python already offers you.

With itertools.chain() on the values

This could be faster:

from itertools import chain
categories = list(chain.from_iterable(categories.values))

Performance

from functools import reduce
from itertools import chain

categories = pd.Series([['a', 'b'], ['c', 'd', 'e']] * 1000)

%timeit list(chain.from_iterable(categories.values))
1000 loops, best of 3: 231 µs per loop

%timeit list(chain(*categories.values.flat))
1000 loops, best of 3: 237 µs per loop

%timeit reduce(lambda l1, l2: l1 + l2, categories)
100 loops, best of 3: 15.8 ms per loop

For this data set the chaining is about 68x faster.

Vectorization?

Vectorization works when you have native NumPy data types (pandas uses NumPy for its data after all). Since we have lists in the Series already and want a list as result, it is rather unlikely that vectorization will speed things up. The conversion between standard Python objects and pandas/NumPy data types will likely eat up all the performance you might get from the vectorization. I made one attempt to vectorize the algorithm in another answer.