In the Nava-Durga the sixth form is of Goddess Katyayini & I have heard that Goddess Durga became the daughter of Rishi Katyayan that is why she was given this name. I want to know who the rishi was and how could he obtain the Supreme Goddess as his daughter?
I was reading the Katha-Sarit-Sagar and came across this name so am sharing with you the details of Rishi Katyayan as per this particular story. I couldn't find an online english version of the text so am sharing the scrrenshots of the relevant portion here along with a summary:
Rishi Katyayana was originally a Shiva ganna named Pushpadanta who had earlier been a human named Devdatta who managed to please Lord Shiva and ask for his blessings to become his attendant. He used to live on Kailash with his wife Jaya serving Lord Shiva and Goddess Parvati. Parvati, who had wanted Shiva to tell her a story that no one else knew but Pushpadanta, managed to overhear it.
The Goddess did not like the idea of Pushpadanta knowing a story that was meant only for her ears and therefore cursed him to become a mortal on earth. Pushpadanta's friend Malyavan who had been his brother in the human form tried to intercede on his behalf but he was also cursed similarly.
Pushpadanta lived his life as Vararuchi also known as Katyayan and faced many ups and downs as a human. He worshipped Goddess Durga and finally got to meet the Pishach Kanabhuti who told him the way to get rid of the curse:
This story that Katyayana narrates to Kanabhuti is the same one that he had heard from Lord Shiva and is believed to have been in seven parts. But only one of them survived and is now known as the Kathasaritsagar.
After narrating the story he goes back to the forest to worship the goddess to help him give up his body. Though the Kathasaritsagar doesn't mention the incident of him siring the goddess or shaping the energy beam as mentioned in the other answer, his connection to the goddess Durga/Parvati is unmistakably there throughout the text.
EDIT: I came across another reference in the Agni Purana that mentions Skanda or Kartikeya teaching Katyayan the rules of Sanskrit grammar:
O Kātyāyana! I shall describe the essence of grammar [i.e., vyākaraṇa-sāra] that is of the form of perfected words for imparting to the beginners..
This and the further chapters with below mentioned titled=s cover many rules followed in Sanskrit and give the sense of Katyayan possibly being a grammarian:
Chapter 349 - The pratyāhāras and fundamental rules in Grammar (vyākaraṇa) Chapter 350 - The forms of completed euphonic combinations (sandhi) Chapter 351 - Perfected forms of inflection in the nouns Chapter 352 - Narration of the finished forms of the substantives in the feminine Chapter 353 - The finished forms of substantives in the neuter Chapter 354 - The relation between a noun and a verb in a sentence (kāraka) Chapter 355 - Different kinds of compounds Chapter 356 - The formation of taddhita (secondary nominal bases) Chapter 357 - Adding primary affixes known as uṇādi Chapter 358 - The completed forms of the verbs after adding the conjugational signs Chapter 359 - Adding the kṛt affixes (primary affixes added to verbs) Chapter 360 - Synonyms denoting the celestial region and the nether world
I hope this helped you know a little more about the rishi's origins.
We find a brief story of emergence of goddess Katyayani at the ashrama of hermit Katyayana In Vamana Purana- Chapter 18 - Description of greatness of Goddess-Verses 6-13. And from that we come to know him and how he gave birth to Goddess Katyayani.
Brief story Of Birth Of Devi Katyayani from Vamana Purana-
Once demon Mahishasura defeated all the gods from heaven. All the gods then went to Lord Vishnu for help with the lead of Brahmadeva. There they saw Lord Vishnu and Lord Shiva being seated together. Gods told all the deeds of Mahishasura to them and asked for their help. Vishnu-Shiva-Indra then got filled with anger. As a result of their anger a ray (splendor or Teja )came out from their mouths. And that ray got crstallised like ridge of the mountain in the ashrama of Maharshi Katyayana . He then processed that splender and it gave birth to an goddess called Katyayani.
ततोऽनुकोपान्मधुसूदनस्य सशंकरस्यापि पितामहस्य।
तथैव शक्रादिषु दैवतेषु महर्द्धि तेजो वदनाद् विनिःसृतम् ।। 18.6 ।।
तच्चैकतां पर्वतकूटसन्निभं जगाम तेजः प्रवराश्रमे मुने।
कात्यायनस्याप्रतिमस्य तेन महर्षिणा तेज उपाकृतं च ।। 18.7 ।।
तेनार्षिसृष्टेन च तेजसा वृतं ज्वलत्प्रकाशार्कसहस्रतुल्यम्।
तस्माच्च जाता तरलायताक्षी कात्यायनी योगविशुद्धदेहा ।। 18.8 ।।
As a result of anger built up in the minds of Madhusudana , Shankara , Pitamaha and Indra etc. gods , rays came out from their mouths. O hermit ! that ray was crystallised like the ridge of a mountain in the ashrama of Katyayana hermit. That hermit processed (Upabrhana) that splendor. That crystallised splendor of all the gods and duly processed start glowing like several thousand suns. Such combination of splendor gave birth to caprice and to a huge eyes and purified body , a goddess called Katyayani.
How could he obtain the Supreme Goddess as his daughter?
From the above story we came to know that Rishi katyayana was having very great merit (splendor) and powers . So that he could process beams of light radiating from the bodies of the gods and from that processing goddess was born. So he was technically did not gave birth to the goddess , but he made the appearing of the goddess possible through his powers , by processing the teja (splendor) of gods like Vishnu , Shiva , Brahma and Indra and that is why he is considered as father of devi katyayani.
Why she was given this name.?
The answer is also provided in vamana purana shloka 13.
तथार्षितेजोत्तममुत्तमं महन्नाम्ना पृथिव्यामभवत् प्रसिद्धम् ।
कात्यायनीत्येव तदा बभौ सा नाम्ना च तेनैव जगत्प्रसिद्धा ।। 18.13 ।।
The great and supreme splendor of hermit was popularly known as Katyayani and the goddess so originated became famous in this universe with the same name Viz. Katyayani.
The goddess Katyayani is called so because she was born from the great merit or splendor of rishi Katyayana. i.e. because he processed the tejas of gods from his merit and from that the goddess was born. So the goddess is called as Katyayani after the name of Rishi Katyayana.
I want to know who the Rishi Katyayana was.?
We find several personalities called as Katyayana who wrote or are author of various works like Vartikas on Paniniya sutras and a son of Somadatta. Another is author of Chandra-Vākyās and several names from Tamil Nadu and kerala.
But since we are interested in Rishi Katyayana who is associated with hinduism and mentioned in Hinduism scriptures lets see one Rishi Katyayana mentioned in Garuda Purana -Achara Kanda - Chapter 205 to whom Skanda or kumara narrated the rules of grammar.
कुमार उवाच ।
अथ व्याकरणं वक्ष्ये कात्यायन समासतः ।
सिद्धशब्दविवेकाय बालव्युत्पत्तिहेतवे ॥ GP 1.205.1 ॥
Kumara said : O Katyayana , I shall briefly expound the rules of grammar which will enable infants to easily comprehend that subject and to understand the formation of word.
So the above mentioned Rishi Katyayana is a grammarian.
David Frawley in his book Vedic Yoga: The Path of the Rishi is also quoting from Rishi Katyayana
Other internet sources are mentioning that he is related with clan of Vishwamitra and gave birth to Shukla yajurveda Katyayana Branch. Unfortunately we are not very much sure about who exactly among these this story of Katyayani Devi is associated. He name is just briefly mentioned in Mahabharata but apart from this we won't get any stories related to Rishi katyayana from Hinduism scriptures. .
@SwiftPushkar has given a nice answer. Here some more detail extracted from the 13th chapter of the Guhyakālīkhaṇḍa of the Mahākālasaṃhitā is added.
Kātyāyana was the son of Yāgñavalkya through his 2nd wife Kātyāyanī (the first being the celebrated Maitreyī) as per Skandapurāṇa : Nāgarakhaṇḍa : Chapter 31. His āśrama was situated in the Himalayas. Once Mahiṣāsura happened to pass by his āśrama situated in the Himālaya while Kātyāyana was out collecting wood, leaving his śiṣya Kapota to protect the āśrama through his Yogic powers. Being the troublemaker he was, Mahiṣasura assumed the form of a woman & successfully seduced Kapota. Devoid of Yogic protection, the Dānava army accompanying Mahiṣasura desecrated the āśrama & left. When Kātyāyana came back, he saw his āśrama wrecked & his śiṣya Kapota in a compromising position with a woman in open air nearby. Kātyāyana was able free Kapota from the clutches of the woman & cursed Mahiṣasura to be killed by a woman in battle.
Mahiṣasura, at first laughed off at the concept of a woman being able to fight with him in battle, but Kātyāyana made him conscious about the powers of austerities (in Kātyāyana's own words, ṛṣi's don't carry weapons for their curses are their daggers & their anger is their khaṛga) & the greatness of ṛṣis (Gautama making Indra's body covered with vaginas, Durvāsā's curse making the Lakṣmī abandon Svarga to the great distress of the devas, Agastya's drinking of the ocean, Kapila's burning of the thousand troublemaking sons of Sagara, Visvāmitra's creation of a 2nd Svarga, Parāśara's creation of the submarine fire, Indra's harassment at the hands of the kṛtyā created by Cyavana were cited as examples). Kātyāyana then goes to inform Mahiṣasura about the existence of a woman (Bhagavatī Ādyāśakti) whose beauty outshines even that of the devīs like Lakṣmī, Rati & Śacī, putting even the likes of heavenly courtesans like Rambhā, Ghṛtācī, Menakā, Urvaśī & Tilottamā to shame. After describing her beauty, Kātyāyana then describes her attributes (in his words, Her playfulness excels that of Kāmadeva, Her forgiving nature exceeds that of Bhūdevī, Her strength exceeds that of Vāyu, Her seriousness exceeds that of Sarasvatī, Her firmness exceeds that of Meru, Her prowess in gaining victory in battles exceeds that of Viṣṇu, Her weight-lifting capabilites exceeds that of the divine tortoise who supports Ananta, the carrier of the earth & Her anger exceeds that of Kāla & Yama).
When Mahiṣasura inquires about the whereabouts of such a woman, Kātyāyana replies that She is omnipresent & then proceeds to harshly rebuke the tyrant by stating that the stature of Mahiṣasura (who has conquered the all the 7 underworld realms, earthly realm & Svargaloka) before Her is like that of a frog before Śeṣanāga, a rabbit before a lion, a drain before the river Gaṅgā, a firefly before the moon, a yāma (unit of time equal to ¼th of daytime) before a yuga & a single night before the end of a kalpa.
Incensed at being insulted in this way, Mahiṣasura tells Kātyāyana to make that woman manifest before him or else he will kill him. So Kātyāyana proceeds to a nearby bilva tree & prays to Jagadambā present in it to protect him. Bhagavatī answered his prayers & manifested out of that bilva tree, wielding bow & arrows, seated on a lion, surrounded by 4 maids, in midst of a a blinding mass of light comparable to a crore of suns. Mahiṣasura was eager to fight on the spot but was dissuaded by his advisors, who saw the turn of events as an ill-omen. So Mahiṣasura then left the āśrama.
Now as per Kālikapurāṇa, when the devas were at Svetadvīpa to complain about Mahiṣasura to Bhagavāna Viṣṇu, Bhagavatī Ādyāśakti manifested before them & directed them to assemble at ṛṣi Kātyāyana's āśrama in order to solve their problems. Here She manifested out of the energies of all the devas (combined by Kātyāyana) as famously described in the Devīmāhātmya.