Some info but nothing concrete can be found at https://www.etymonline.com/word/category
In short, the sense category was borrowed all the way from Greek. The rest would be matter for Latin.SE (where Ancient Greek is handled too).
Discussion. You had probably understood as much already. This solves nothing really. There is a polysemic definition for the Greek word, that is a circular definition for the most important part, and not very illuminating about the rest for they explain further down that the cycle had apparently started with Aristotle but was never really understood--which directly contravenes and thus invalidates the introduction that tries to show an understanding of the development.
It is assumed that the Aristotelean sense "category" of kategoria had to have developed as specialization from the other meanings, "accusation, prediction" after the verb kategorein "to speak against; to accuse, assert, predicate".
The verb's original sense of "accuse" had weakened to "assert, name" by the time Aristotle applied kategoria to his 10 classes of things that can be named.
This seems shortsighted. The opposite direction is theoretically plausible, cp. to implicate sb., notwithstanding the practicle limit of finding evidence pre Aristotle.
The root "*ger- 'to gather'" reconstructed from agora "public assembly" would match that sense of category as assemblage of items under one concept.
Compare collection for example with simmilar semantics, already so in Mycenaean Greek 𐀀𐀒𐀨 (a-ko-ra) "1. collection 2. flock" (cf. wiki: agora).
A modern reconstruction is Proto-Indo-European *h2ger- instead, *h2 accounting for the a-. Initial *H leaves barely any trace elsewhere, thus the only evidence would be Greek itself. A circular argument. One cognate under this root is gaṇá in Sanskrit where it already could mean "7. (grammar) a series of roots or words following the same rule and called after the first word of the series [...])". There is also the modern Hindi for "set theory". (cf. wiki: गण)
The supposed prefix kat(a)- is notably uncertain in Harper's dictionary, to which the wiktionary has nothing much to add. Its interpretation should therefore be taken with a pinch of salt, as well as the medial a. The cognates' semantics do not exactly match either (if you follow the above link). It needs repeating that no rigid method for the reconstruction of semantics exists, such that any pledge for, say, "flock" + "down" (i.e. kata) underlying a proper noun cannot alone carry conviction. This leaves room for doubt.
Notable analogies to "assembly" exist with other roots.1. So the fact remains that a contraction would require too much guesswork in face of doubt. In that sense, the question might be a better fit for linguistics.SE if need be.
Finally, what I can say is that German kategorisch in the collocation kategorisch ablehnen "to reject summarily" continues the negative connotation alleged above. Cp. summary judgement, and for analogies sake rubrum (from the rube "red" color of a verdicts summary), protocol (from the proto "first" page attached to a file, summarizing the content), or simply set (versus Ger. Satz "sentence").
1: Without any hope to be complete
*sem- "together" as in assembly itself (cp. Ger. Versammlung), which incidently reflects in a Greek prefix a- "same" (cp. Sanskrit sa-).
*k'om- "with" as it seems to appear in Latin comes, comitia, the analogue to the Agora in Republican Rome, also seen in Greek kai "and"
*ghedh- as in together, gather, gathering, Slovene *zgodovína f. "history" (maybe uncertain but cp. Serbo-Croatian zgoda "event").
- Here it is not quite clear to me where the to- came from, though there is comparison in Ger. zusammen "together", zu erst "at first", apparently from *de-, cp. Latin de- with quite opposite semantic; There might be precedent in PIE *dʰǵʰyes- "yesterday" (cp. today, cf. *k'e- "here", cp. Lat. hanc hodie "today", Ger. heute "today") or *dk'em- "ten" (cp. *k'mt- "hundred"), *dwo- "2," (cf. *dwis- "appart, in two", *wi- "appart"); There is copious precedent for verbal suffixes from *dheh1 "do, put, place", *deH- "share, give", etc. and a more recent theory of ditropic particles (syntactically prefixed, but phonologically attached to any preceding word, which is decisive in Germanic for Verner's law at least, note s-mobile, cp. *steh2- "stand, set", systeme, histemi, cf. caste system)
*Ges- ("take" or "collect", if I remember correctly) ToDo.