From http://developer.android.com/tools/support-library/features.html :
Support Library v4:
This library is intended for use with Android 1.6 (API level 4) and higher. It includes the largest set of APIs compared to other libraries, including application component support, user interface functions, accessibility, data processing, network connectivity, and software utilities. Here are a few key classes included in the v4 library:
v7 Library support:
There are several libraries designed for use with Android 2.1 (API level 7) and higher. These libraries provide specific feature sets and can be included in your application independently.
V8 Support Library:
This library is intended for use with Android (API level 8) and higher. It adds support for the RenderScript calculation structure. These APIs are included in the android.support.v8.renderscript package. You should be aware that the steps to include these APIs in your application are very different from other support library APIs. For more information about using these APIs in an application, see the RenderScript Developer's Guide.
Support Library v13:
This library is designed for Android 3.2 (API level 13) and higher. It adds support for the Fragment user interface template with a class (FragmentCompat) and additional fragment support classes. For more information about fragments, see the Fragments Developer's Guide. For more information about the v13 support library APIs, see the android.support.v13 package in the API Help.
If you look at your mistake, you have two conflicting conflicts because you imported both v4 and v13. You can see how they have similar classes:
http://developer.android.com/reference/android/support/v4/app/package-summary.html http://developer.android.com/reference/android/support/v13/app/package-summary.html
You should use v4 for min-sdk = 4-12 and v13 for min-sdk = 13 +
Even though you can use v13 in lower versions of the Android API, your apps will break into anything and 12 APIs. You should be able to use v7 and v13 just fine, until you conflict with v4.
EDIT: Here's a video explaining the Android support libraries that recently appeared http://xamarin.wistia.com/medias/guqtgpdqms
Source code: https://github.com/jamesmontemagno/Xamarin.Android-AppCompat
Source: Xamarin