Fiat is backed up by the fact that a territorial monopoly of violence demands taxes to be paid in it. Also, a monopoly of violence can decree that certain type of economic units have accounting obligation in the fiat backed up by said monopoly of violence. And a monopoly of violence can and often will refuse to help enforce the repayment of a debt if the repayment is attempted in the backed up fiat. The value of fiat follows from those three facts.

A monopoly of violence is capable of single-handedly giving anything value by demanding the same thing, say pebbles of a certain color, in return for not beating you to pulp. This is the fundamental connection between money and violence. Typically monopolies of violence have demanded things that have been somewhat valued in the first place in return for remaining non-violent for the time being thus boosting their value significantly. Such things include gold or salt.