{ 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: {} }