Although there are contradictory opinions on whether the use of crossbows should be termed archery or not, there are a significant number of reasons pointing towards the fact that crossbows are very similar to other archery equipment.
The argument put forward by the purists is that archery is a sport that uses simple vertical recurve bows that shoot arrows, not compound or recurve crossbows that fire bolts, and in the case of compound bows other adjustments and accessories that improve shooting accuracy of a target). This is particularly true of the Olympics archery events where only recurve vertical bows are allowed.
However, besides the Olympics, there are other notable archery events that allow the use of compound bows. For example, World Archery Championships. So, in some contexts, the use of crossbows is widely accepted as archery.
Furthermore, crossbows of all types and different types of archery equipment utilize a similar essential framework to drive comparative shots at a target. The mechanical mechanisms by which arrows/ bolts are shot vary between recurve, compound and longbows. However, all of them operate on the same basic methodology: Potential energy is stored in their limbs, and bowstring which is then relayed and converted to kinetic energy resulting in arrow propulsion from the bow. The shooting missile (arrow/bolt) have a tip/broadhead, embed, shaft, nock, and fletching, very much like a compound or conventional bows bolts do.
The two of them have a similar range, too, and these are often the reasons crossbow enthusiasts classify it as a part of archery.