{
  localUrl: '../page/5k7.html',
  arbitalUrl: 'https://arbital.com/p/5k7',
  rawJsonUrl: '../raw/5k7.json',
  likeableId: '3224',
  likeableType: 'page',
  myLikeValue: '0',
  likeCount: '3',
  dislikeCount: '0',
  likeScore: '3',
  individualLikes: [
    'EricBruylant',
    'KevinClancy',
    'IzaakMeckler'
  ],
  pageId: '5k7',
  edit: '4',
  editSummary: '"I'll" -> "we'll" as per https://arbital.com/p/arbital_style_guide/; also capitalized a letter',
  prevEdit: '3',
  currentEdit: '4',
  wasPublished: 'true',
  type: 'wiki',
  title: 'Exponential notation for function spaces',
  clickbait: 'Why $Y^X$ is good notation for the space of maps from $X$ to $Y$ ',
  textLength: '1502',
  alias: '5k7',
  externalUrl: '',
  sortChildrenBy: 'likes',
  hasVote: 'false',
  voteType: '',
  votesAnonymous: 'false',
  editCreatorId: 'EricRogstad',
  editCreatedAt: '2016-07-25 07:14:47',
  pageCreatorId: 'IzaakMeckler',
  pageCreatedAt: '2016-07-24 20:47:32',
  seeDomainId: '0',
  editDomainId: 'AlexeiAndreev',
  submitToDomainId: '0',
  isAutosave: 'false',
  isSnapshot: 'false',
  isLiveEdit: 'true',
  isMinorEdit: 'false',
  indirectTeacher: 'false',
  todoCount: '0',
  isEditorComment: 'false',
  isApprovedComment: 'true',
  isResolved: 'false',
  snapshotText: '',
  anchorContext: '',
  anchorText: '',
  anchorOffset: '0',
  mergedInto: '',
  isDeleted: 'false',
  viewCount: '28',
  text: 'If $X$ and $Y$ are sets, the set of functions from $X$ to $Y$ (often written $X \\to Y$) is sometimes also written $Y^X$. This latter notation, which we'll call *exponential notation*, is related to the notation for finite powers of sets (e.g., $Y^3$ for the set of triples of elements of $Y$) as well as the notation of exponentiation for numbers.\n\nWithout further ado, here are some reasons this is good notation.\n\n- A function $f : X \\to Y$ can be thought of as an "$X$ wide" tuple of elements of $Y$. That is, a tuple of elements of $Y$ where the positions in the tuple are given by elements of $X$, generalizing the notation $Y^n$ which denotes the set of $n$ wide tuples of elements of $Y$. Note that if $|X| = n$, then $Y^X \\cong Y^n$.\n\n- This notion of exponentiation together with cartesian product as multiplication and disjoint union as addition satisfy the same relations as exponentiation, multiplication, and addition of natural numbers. Namely, \n\n  - $Z^{X \\times Y} \\cong (Z^X)^Y$ (this isomorphism is called currying)\n  - $Z^{X + Y} \\cong Z^X \\times Z^Y$\n  - $Z^1 \\cong Z$ (where $1$ is a one element set, since there is one function into $Z$ for every element of $Z$)\n  - $Z^0 \\cong 1$ (where $0$ is the empty set, since there is one function from the empty set to any set)\n\nMore generally, $Y^X$ is good notation for the exponential object representing $\\text{Hom}_{\\mathcal{C}}(X, Y)$ in an arbitrary cartesian closed category $\\mathcal{C}$ for the first set of reasons listed above.',
  metaText: '',
  isTextLoaded: 'true',
  isSubscribedToDiscussion: 'false',
  isSubscribedToUser: 'false',
  isSubscribedAsMaintainer: 'false',
  discussionSubscriberCount: '3',
  maintainerCount: '3',
  userSubscriberCount: '0',
  lastVisit: '',
  hasDraft: 'false',
  votes: [],
  voteSummary: 'null',
  muVoteSummary: '0',
  voteScaling: '0',
  currentUserVote: '-2',
  voteCount: '0',
  lockedVoteType: '',
  maxEditEver: '0',
  redLinkCount: '0',
  lockedBy: '',
  lockedUntil: '',
  nextPageId: '',
  prevPageId: '',
  usedAsMastery: 'false',
  proposalEditNum: '0',
  permissions: {
    edit: {
      has: 'false',
      reason: 'You don't have domain permission to edit this page'
    },
    proposeEdit: {
      has: 'true',
      reason: ''
    },
    delete: {
      has: 'false',
      reason: 'You don't have domain permission to delete this page'
    },
    comment: {
      has: 'false',
      reason: 'You can't comment in this domain because you are not a member'
    },
    proposeComment: {
      has: 'true',
      reason: ''
    }
  },
  summaries: {},
  creatorIds: [
    'IzaakMeckler',
    'EricRogstad'
  ],
  childIds: [],
  parentIds: [
    'math'
  ],
  commentIds: [
    '5kr'
  ],
  questionIds: [],
  tagIds: [],
  relatedIds: [],
  markIds: [],
  explanations: [],
  learnMore: [],
  requirements: [],
  subjects: [],
  lenses: [],
  lensParentId: '',
  pathPages: [],
  learnMoreTaughtMap: {},
  learnMoreCoveredMap: {},
  learnMoreRequiredMap: {},
  editHistory: {},
  domainSubmissions: {},
  answers: [],
  answerCount: '0',
  commentCount: '0',
  newCommentCount: '0',
  linkedMarkCount: '0',
  changeLogs: [
    {
      likeableId: '0',
      likeableType: 'changeLog',
      myLikeValue: '0',
      likeCount: '0',
      dislikeCount: '0',
      likeScore: '0',
      individualLikes: [],
      id: '17482',
      pageId: '5k7',
      userId: 'EricRogstad',
      edit: '0',
      type: 'newParent',
      createdAt: '2016-07-25 07:16:04',
      auxPageId: 'math',
      oldSettingsValue: '',
      newSettingsValue: ''
    },
    {
      likeableId: '0',
      likeableType: 'changeLog',
      myLikeValue: '0',
      likeCount: '0',
      dislikeCount: '0',
      likeScore: '0',
      individualLikes: [],
      id: '17479',
      pageId: '5k7',
      userId: 'EricRogstad',
      edit: '4',
      type: 'newEdit',
      createdAt: '2016-07-25 07:14:47',
      auxPageId: '',
      oldSettingsValue: '',
      newSettingsValue: '"I'll" -> "we'll" as per https://arbital.com/p/arbital_style_guide/; also capitalized a letter'
    },
    {
      likeableId: '0',
      likeableType: 'changeLog',
      myLikeValue: '0',
      likeCount: '0',
      dislikeCount: '0',
      likeScore: '0',
      individualLikes: [],
      id: '17461',
      pageId: '5k7',
      userId: 'IzaakMeckler',
      edit: '3',
      type: 'newEdit',
      createdAt: '2016-07-24 21:57:06',
      auxPageId: '',
      oldSettingsValue: '',
      newSettingsValue: ''
    },
    {
      likeableId: '0',
      likeableType: 'changeLog',
      myLikeValue: '0',
      likeCount: '0',
      dislikeCount: '0',
      likeScore: '0',
      individualLikes: [],
      id: '17458',
      pageId: '5k7',
      userId: 'IzaakMeckler',
      edit: '2',
      type: 'newEdit',
      createdAt: '2016-07-24 20:48:06',
      auxPageId: '',
      oldSettingsValue: '',
      newSettingsValue: ''
    },
    {
      likeableId: '0',
      likeableType: 'changeLog',
      myLikeValue: '0',
      likeCount: '0',
      dislikeCount: '0',
      likeScore: '0',
      individualLikes: [],
      id: '17457',
      pageId: '5k7',
      userId: 'IzaakMeckler',
      edit: '1',
      type: 'newEdit',
      createdAt: '2016-07-24 20:47:32',
      auxPageId: '',
      oldSettingsValue: '',
      newSettingsValue: ''
    }
  ],
  feedSubmissions: [],
  searchStrings: {},
  hasChildren: 'false',
  hasParents: 'true',
  redAliases: {},
  improvementTagIds: [],
  nonMetaTagIds: [],
  todos: [],
  slowDownMap: 'null',
  speedUpMap: 'null',
  arcPageIds: 'null',
  contentRequests: {}
}