A list of the current iTunes podcast categories in different formats

itunes-podcast-categories.json

[
  {
    "Arts": [
      "Design",
      "Fashion & Beauty",
      "Food",
      "Literature",
      "Performing Arts",
      "Visual Arts"
    ]
  },
  {
    "Business": [
      "Business News",
      "Careers",
      "Investing",
      "Management & Marketing",
      "Shopping"
    ]
  },
  "Comedy",
  {
    "Education": [
      "Educational Technology",
      "Higher Education",
      "K-12",
      "Language Courses",
      "Training"
    ]
  },
  {
    "Games & Hobbies": [
      "Automotive",
      "Aviation",
      "Hobbies",
      "Other Games",
      "Video Games"
    ]
  },
  {
    "Government & Organizations": [
      "Local",
      "National",
      "Non-Profit",
      "Regional"
    ]
  },
  {
    "Health": [
      "Alternative Health",
      "Fitness & Nutrition",
      "Self-Help",
      "Sexuality"
    ]
  },
  "Kids & Family",
  "Music",
  "News & Politics",
  {
    "Religion & Spirituality": [
      "Buddhism",
      "Christianity",
      "Hinduism",
      "Islam",
      "Judaism",
      "Other",
      "Spirituality"
    ]
  },
  {
    "Science & Medicine": [
      "Medicine",
      "Natural Sciences",
      "Social Sciences"
    ]
  },
  {
    "Society & Culture": [
      "History",
      "Personal Journals",
      "Philosophy",
      "Places & Travel"
    ]
  },
  {
    "Sports & Recreation": [
      "Amateur",
      "College & High School",
      "Outdoor",
      "Professional"
    ]
  },
  {
    "Technology": [
      "Gadgets",
      "Tech News",
      "Podcasting",
      "Software How-To"
    ]
  },
  "TV & Film"
]

itunes-podcast-categories.yml

# iTunes Podast Categories
# Last Updated: 2017-02-19
# Source: https://help.apple.com/itc/podcasts_connect/#/itc9267a2f12 
- Arts:
  - Design
  - Fashion & Beauty
  - Food
  - Literature
  - Performing Arts
  - Visual Arts
- Business:
  - Business News
  - Careers
  - Investing
  - Management & Marketing
  - Shopping
- Comedy
- Education:
  - Educational Technology
  - Higher Education
  - K-12
  - Language Courses
  - Training
- Games & Hobbies:
  - Automotive
  - Aviation
  - Hobbies
  - Other Games
  - Video Games
- Government & Organizations:
  - Local
  - National
  - Non-Profit
  - Regional
- Health:
  - Alternative Health
  - Fitness & Nutrition
  - Self-Help
  - Sexuality
- Kids & Family
- Music
- News & Politics
- Religion & Spirituality:
  - Buddhism
  - Christianity
  - Hinduism
  - Islam
  - Judaism
  - Other
  - Spirituality
- Science & Medicine:
  - Medicine
  - Natural Sciences
  - Social Sciences
- Society & Culture:
  - History
  - Personal Journals
  - Philosophy
  - Places & Travel
- Sports & Recreation:
  - Amateur
  - College & High School
  - Outdoor
  - Professional
- Technology:
  - Gadgets
  - Tech News
  - Podcasting
  - Software How-To
- TV & Film

I extracted those just now from the website:

{
  "Arts": [
    "Books",
    "Design",
    "Fashion & Beauty",
    "Food",
    "Performing Arts",
    "Visual Arts"
  ],
  "Business": [
    "Careers",
    "Entrepreneurship",
    "Investing",
    "Management",
    "Marketing",
    "Non-Profit"
  ],
  "Comedy": [
    "Comedy Interviews",
    "Improv",
    "Stand-Up"
  ],
  "Education": [
    "Courses",
    "How To",
    "Language Learning",
    "Self-Improvement"
  ],
  "Fiction": [
    "Comedy Fiction",
    "Drama",
    "Science Fiction"
  ],
  "Government": [],
  "History": [],
  "Health & Fitness": [
    "Alternative Health",
    "Fitness",
    "Medicine",
    "Mental Health",
    "Nutrition",
    "Sexuality"
  ],
  "Kids & Family": [
    "Education for Kids",
    "Parenting",
    "Pets & Animals",
    "Stories for Kids"
  ],
  "Leisure": [
    "Animation & Manga",
    "Automotive",
    "Aviation",
    "Crafts",
    "Games",
    "Hobbies",
    "Home & Garden",
    "Video Games"
  ],
  "Music": [
    "Music Commentary",
    "Music History",
    "Music Interviews"
  ],
  "News": [
    "Business News",
    "Daily News",
    "Entertainment News",
    "News Commentary",
    "Politics",
    "Sports News",
    "Tech News"
  ],
  "Religion & Spirituality": [
    "Buddhism",
    "Christianity",
    "Hinduism",
    "Islam",
    "Judaism",
    "Religion",
    "Spirituality"
  ],
  "Science": [
    "Astronomy",
    "Chemistry",
    "Earth Sciences",
    "Life Sciences",
    "Mathematics",
    "Natural Sciences",
    "Nature",
    "Physics",
    "Social Sciences"
  ],
  "Society & Culture": [
    "Documentary",
    "Personal Journals",
    "Philosophy",
    "Places & Travel",
    "Relationships"
  ],
  "Sports": [
    "Baseball",
    "Basketball",
    "Cricket",
    "Fantasy Sports",
    "Football",
    "Golf",
    "Hockey",
    "Rugby",
    "Running",
    "Soccer",
    "Swimming",
    "Tennis",
    "Volleyball",
    "Wilderness",
    "Wrestling"
  ],
  "Technology": [],
  "True Crime": [],
  "TV & Film": [
    "After Shows",
    "Film History",
    "Film Interviews",
    "Film Reviews",
    "TV Reviews"
  ]
}

It seems "running" has been removed: https://podcasters.apple.com/support/1691-apple-podcasts-categories