Programming arduino with Objective-C
Simple version: No, you cannot program Arduino using Objective-C.

Long version:
The Arduino IDE uses avr-gcc as it's compiler, and avr-gcc is a C/C++ compiler, not an Object-C compiler. The Arduino libraries are written in C/C++.

There are other compilers that exist for AVR processors, and so it is possible to use other languages for the AVR.

However, if you switch to a different compiler for your code, you will not be able to use the Arduino libraries with it, because they are written in C/C++ for the avr-gcc compiler. (It may be possible to compile the Arduino libraries with avr-gcc, and then link them to your code using external language linking methods, but that could be a significant challenge.)

Also, I do not know of any existing Objective-C compilers for AVR; it's possible one may exist, but after a quick web search, I didn't find any.