Here is the entry, with my translation, in the Martyrologium Romanum for the Feast of the Exaltation of the Holy Cross:
Festum exaltationis Sanctae Crucis, quae, postridie dedicationis basilicae Resurrectionis super sepulcrum Christi erectae, exaltatur et honoratur, sicut victoriae eius paschalis tropaeum et signum in caelo appariturum, alterum adventum eius iam universis praenuntians.
The Feast of the Exaltation of the Holy Cross, which, the day after the dedication of the Basilica of the Resurrection raised over the tomb of Christ, is exalted and honored, in the manner of a memorial of His paschal victory and the sign which is to appear in the sky, already announcing in advance His second coming.
Our wonderful Lewis & Short says that a tropaeum is "a sign and memorial of victory, a trophy; orig. a trunk of a tree, on which were fixed the arms, shields, helmets, etc., taken from the enemy; afterwards made of stone and ornamented in the same manner". So, a tropaeum is a kind of war memorial.
To my mind there are echoes here of the magnificent hymn of Venantius Fortunatus, the Vexilla Regis prodeunt …
Vexilla regis prodeunt,
fulget crucis mysterium,
quo carne carnis conditor
suspensus est patibulo.
I think there is also a "once and future" reference to the vision Constantine had of the Cross before his victory over Maxentius. Constantine would later build the Basilica of the Holy Sepulchur. The future dimension is, of course, the appearance of the Lord in the East at the Second Coming (a great reason to celebrate Mass ad orientem).