Sanctification means 'to be set apart'. We are set apart as the Lord's chosen to live lives that are different to those who do not have faith. Therefore, God Himself is actually living in us, in the Person of the Holy Spirit.
The Holy Spirit is the One who produces holiness in the believer. He lives in our heart, in the inner man. These are depths of our being and are where the Bible speaks of our soul, our mind, our heart. Where and what is that inner space where we are self-conscious and have thoughts about things, including ourselves? Wherever it is, that is where the Holy Spirit dwells in us.
The Holy Spirit works at this deepest level, in our soul, to bring about change. To bring about that change. The Holy Spirit's work is to make us more Christ-like (Ephesians 4:20-24). Holiness of character, the results of the sanctifying work of the Sprit in us, means the defeat of sin. It is the production of such things as Paul lists in Galatians 5:22-24.