  We present a notion of $\Delta$-stability and stability filtration in
arbitrary categories which is equivalent to the existence of Harder-Narasimhan
(HN) sequences on objects. Indeed it is equivalent to the existence of a zero
morphism, a partial order on objects, and a collection of some universal
sequences. In additive categories embedded in an ambient triangulated category,
we could obtain a numerical polynomial or central charge of objects by
calculating the Euler characteristic of slope sequences and objects, inducing a
partial order and HN sequences. In the case of normal projective surfaces over
an arbitrary perfect field $k$ we show the existence of $\Delta$-stabilities of
degree one on the relevant bounded derived categories which is equivalent to
the existence of Bridgeland's stabilities on normal surfaces. This result also
leads to new effective restriction theorem of slope semistable sheave on normal
projective varieties over perfect fields. Our approach also gives alternative
proofs of Hodge Index Theorem and Bogomolov Inequality.
