Almost all operatives have "Manifestations".
When you first get an operative, they start out at Manifestation level 0.
You can upgrade their Manifestations once you have enough Manifestation Shards to do so.
You can get more Manifestation Shards from either:
Pulling dupes of the operative (you get 60 Shards per dupe)
Doing that operatives Personal File
You can 12 Memory Chip each day.
5* operatives use 3 Memory Chip per Manifestation Shard, and each specific 5* operative's Personal File is limited to a max of 2 Manifestation Shards gained each day.
4* operatives use 2 Memory Chip per Manifestation Shard, and you can do that as many times as you like.
Although you'll get all the 4*'s maxed anyway, so you should never use your Memory Chips on a 4* - always use them on 5*'s.
Each Manifestation level requires 9 individual upgrades, with the 9th costing twice as much, so the total cost of a Manifestation level is 10x the individual level upgrade cost.
These total costs per Manifestation are:
M1 - 60 Manifestation Shards
M2 - 60 Manifestation Shards
M3 - 90 Manifestation Shards
M4 - 120 Manifestation Shards
M5 - 150 Manifestation Shards
Total: 480 Manifestation Shards (240 days of the Personal File, or 8 dupes)
The costs are the same for both 4* and 5* operatives.
Tess has the first of her five Manifestatiosn
She has three out of the nine levels needed for her second Manifestation.
The next level of her 2nd Manifestation costs 6 shards, of which three are owned.
Katya has maxed all 5 of her Manifestations
Some operatives do not have Manifestations, and instead have Auctus Drives instead.
To upgrade a Manifestation, you upgrade that Manifestations individual levels 9 times, but for Auctus Drives you just upgrade the Auctus Drive - there are no sub-levels.
Upgrading an Auctus Drive requires the Auctus Drive upgrade itself, which can't be earned from the Personal File like how Manifestation Shards work.
Whilst the max Manifestations an operative can have is 5, the max Auctus Drives is 4.
So far, all of the operatives that have an Auctus Drive have been released for free during that operatives initial release.
During their initial release, they have also been given away with:
Two copies of their signature 5* weapon (so you can get it to T2)
A fixed-stat logistics set (both stats are that operatives ideal stats, fixed to around 70% of a max-roll).
These fixed-stat logistics officers cannot be re-rolled.
All four of their Auctus Drives
These rewards are usually spread out over a few weeks, such that you can't get them all immediately and instantly max the operative.
These free items mean these operatives are given away with pretty much everything you need to max them.
This also means that a lot of players will have these operatives and all of the above, as they collected them all during that operatives initial release.
The reasoning for gi.ving away a mostly-maxed operative for free is of course a combination of multiple reasons, including:
It helps new players get into endgame or difficult content, by immediately providing them a strong operative they can use, without having to wait to pull 5* operatives and get their operatives and maybe some Manifestation.
It allows meta-oriented players to get a new operative that patch, instead of not pulling for a new operative and instead saving up their pulls a previously-released operative, thus going a patch or several without a new operative.
It guarantees players will get an operative that patch - if you don't have enough pulls to pull the events main operative, you'll still get an operative that patch.
... and having an operative means that you can buy an outfit for that operative.
This is the main reason - players don't buy outfits for operatives they don't have, so if a player doesn't get the new operative, they won't buy any of their outfits.
Similarly, players are less likely to buy previously released outfits, and outfits for older weaker operatives.
By just giving players a new operative, it ensures that everybody has the opportunity to buy that operatives new outfit.
Similarly, giving the Auctus Drive operative a lot of investment means that that are decently strong, and thus usable, and thus it's worth buying an outfit for them.
Auctus Drive operatives have thus far all been relatively weak - none of them are massively strong, and all of them are generally considered mediocre in the games current meta.
This is by-design - if the free Auctus Drive operative were super strong, then you wouldn't need to actually [spend money to] pull for any other operative would you?
The Auctus Drive operatives release thus far are:
Activating an Auctus Drive only requires one Auctus Drive item.
Mauxir - Lotus has all four of her Auctus Drives activated.