{ localUrl: '../page/gen_elt.html', arbitalUrl: 'https://arbital.com/p/gen_elt', rawJsonUrl: '../raw/61q.json', likeableId: '3582', likeableType: 'page', myLikeValue: '0', likeCount: '2', dislikeCount: '0', likeScore: '2', individualLikes: [ 'EricBruylant', 'KevinClancy' ], pageId: 'gen_elt', edit: '3', editSummary: '', prevEdit: '2', currentEdit: '3', wasPublished: 'true', type: 'wiki', title: 'Generalized element', clickbait: 'A category-theoretic generalization of the notion of element of a set.', textLength: '3695', alias: 'gen_elt', externalUrl: '', sortChildrenBy: 'likes', hasVote: 'false', voteType: '', votesAnonymous: 'false', editCreatorId: 'DylanHendrickson', editCreatedAt: '2016-10-07 17:59:07', pageCreatorId: 'LukeSciarappa', pageCreatedAt: '2016-08-31 04:42:05', seeDomainId: '0', editDomainId: 'AlexeiAndreev', submitToDomainId: '0', isAutosave: 'false', isSnapshot: 'false', isLiveEdit: 'true', isMinorEdit: 'false', indirectTeacher: 'false', todoCount: '6', isEditorComment: 'false', isApprovedComment: 'true', isResolved: 'false', snapshotText: '', anchorContext: '', anchorText: '', anchorOffset: '0', mergedInto: '', isDeleted: 'false', viewCount: '43', text: 'In [-4c7], a **generalized element** of an object $X$ of a category is any morphism $x : A \\to X$ with [3lg codomain] $X$. In this situation, $A$ is called the **shape**, or **domain of definition**, of the element $x$. We'll unpack this.\n\n## Generalized elements generalize elements ##\n We'll need a set with a single element: for concreteness, let us denote it $I$, and say that its single element is $*$. That is, let $I = \\{*\\}$.\n For a given set $X$, there is a natural correspondence between the following notions: an element of $X$, and a function from the set $I$ to the set $X$. On the one hand, if you have an element $x$ of $X$, you can define a function from $I$ to $X$ by setting $f(i) = x$ for any $i \\in I$; that is, by taking $f$ to be the constant function with value $x$. On the other hand, if you have a function $f : I \\to X$, then since $*$ is an element of $I$, $f(*)$ is an element of $X$. So in the category of sets, generalized elements of a set $X$ that have shape $I$, which are by definition maps $I \\to X$, are the same thing (at least up to isomorphism, which as usual is all we care about).\n\n## Generalized elements in sets ##\n In the category of sets, if a set $A$ has $n$ elements, a generalized element of shape $A$ of a set $X$ is an $n$-tuple of elements of $X$. \n[todo: is there more to say here? or less?]\n\n## Sometimes there is no `best shape' ##\nBased on the case of sets, you might initially think that it suffices to consider generalized elements whose shape is the terminal object [todo: add link] $1$. However, in the category of groups, since the terminal object is also initial [todo: explain this somewhere], each object has a unique generalized element of shape $1$. However, in this case, there is a single shape that suffices, namely the integers $\\mathbb{Z}$. A generalized element of shape $\\mathbb{Z}$ of an abelian group $A$ is just an ordinary element of $A$. \n\nHowever, sometimes there is no single object whose generalized elements can distinguish everything up to isomorphism. For example, consider $\\text{Set} \\times \\text{Set}$ [todo: link to a page about the product of two categories]. If we use generalized elements of shape $(X,Y)$, then they won't be able to distinguish between the objects $(2^A, 2^{X + B})$ and $(2^{Y + A}, 2^{B})$, up to isomorphism, since maps from $(X,Y)$ into the first are the same as elements of $(2^A)^X\\times(2^{X+B})^Y \\cong 2^{X\\times A + Y \\times (X + B)} \\cong 2^{X \\times A + Y \\times B + X \\times Y}$, and maps from $(X,Y)$ into the second are the same as elements of $(2^{Y+A})^X \\times (2^B)^Y \\cong 2^{X\\times(Y+A) + Y \\times B} \\cong 2^{X \\times A + Y \\times B + X \\times Y}$. These objects will themselves be non-isomorphic as long as at least one of $X$ and $Y$ is not the empty set; if both are, then clearly the functor still fails to distinguish objects up to isomorphism. (More technically, it does not reflect isomorphisms. [todo: explain or avoid this terminology])\nIntuitively, because objects of this category contain the data of two sets, the information cannot be captured by a single homset. This intuition is consistent with the fact that it can be captured with two: the generalized elements of shapes $(0,1)$ and $(1,0)$ together determine every object up to isomorphism.\n\n## Morphisms are functions on generalized elements ##\n If $x$ is an $A$-shaped element of $X$, and $f$ is a morphism from $X$ to $Y$, then $f(x) := f\\circ x$ is an $A$-shaped element of $Y$. The Yoneda lemma [todo: create Yoneda lemma page] states that every function on generalized elements which commutes with reparameterization, i.e. $f(xu) = f(x) u$, is actually given by a morphism in the category.', metaText: '', isTextLoaded: 'true', isSubscribedToDiscussion: 'false', isSubscribedToUser: 'false', isSubscribedAsMaintainer: 'false', discussionSubscriberCount: '2', maintainerCount: '2', 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: [ 'LukeSciarappa', 'DylanHendrickson' ], childIds: [], parentIds: [ 'math' ], commentIds: [], questionIds: [], tagIds: [], relatedIds: [], markIds: [], explanations: [], learnMore: [], requirements: [ { id: '6364', parentId: 'set_mathematics', childId: 'gen_elt', type: 'requirement', creatorId: 'LukeSciarappa', createdAt: '2016-08-31 04:24:24', level: '2', isStrong: 'true', everPublished: 'true' }, { id: '6365', parentId: 'function', childId: 'gen_elt', type: 'requirement', creatorId: 'LukeSciarappa', createdAt: '2016-08-31 04:24:36', level: '2', isStrong: 'true', everPublished: 'true' }, { id: '6366', parentId: 'category_mathematics', childId: 'gen_elt', type: 'requirement', creatorId: 'LukeSciarappa', createdAt: '2016-08-31 04:24:44', level: '1', isStrong: 'false', everPublished: 'true' } ], 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: '19887', pageId: 'gen_elt', userId: 'EricBruylant', edit: '0', type: 'newParent', createdAt: '2016-10-07 18:57:53', auxPageId: 'math', oldSettingsValue: '', newSettingsValue: '' }, { likeableId: '0', likeableType: 'changeLog', myLikeValue: '0', likeCount: '0', dislikeCount: '0', likeScore: '0', individualLikes: [], id: '19885', pageId: 'gen_elt', userId: 'DylanHendrickson', edit: '3', type: 'newEdit', createdAt: '2016-10-07 17:59:07', auxPageId: '', oldSettingsValue: '', newSettingsValue: '' }, { likeableId: '0', likeableType: 'changeLog', myLikeValue: '0', likeCount: '0', dislikeCount: '0', likeScore: '0', individualLikes: [], id: '19876', pageId: 'gen_elt', userId: 'LukeSciarappa', edit: '2', type: 'newEdit', createdAt: '2016-10-07 03:40:11', auxPageId: '', oldSettingsValue: '', newSettingsValue: '' }, { likeableId: '0', likeableType: 'changeLog', myLikeValue: '0', likeCount: '0', dislikeCount: '0', likeScore: '0', individualLikes: [], id: '19419', pageId: 'gen_elt', userId: 'LukeSciarappa', edit: '0', type: 'newRequirement', createdAt: '2016-08-31 04:42:07', auxPageId: 'set_mathematics', oldSettingsValue: '', newSettingsValue: '' }, { likeableId: '0', likeableType: 'changeLog', myLikeValue: '0', likeCount: '0', dislikeCount: '0', likeScore: '0', individualLikes: [], id: '19420', pageId: 'gen_elt', userId: 'LukeSciarappa', edit: '0', type: 'newRequirement', createdAt: '2016-08-31 04:42:07', auxPageId: 'function', oldSettingsValue: '', newSettingsValue: '' }, { likeableId: '0', likeableType: 'changeLog', myLikeValue: '0', likeCount: '0', dislikeCount: '0', likeScore: '0', individualLikes: [], id: '19421', pageId: 'gen_elt', userId: 'LukeSciarappa', edit: '0', type: 'newRequirement', createdAt: '2016-08-31 04:42:07', auxPageId: 'category_mathematics', oldSettingsValue: '', newSettingsValue: '' }, { likeableId: '3483', likeableType: 'changeLog', myLikeValue: '0', likeCount: '1', dislikeCount: '0', likeScore: '1', individualLikes: [], id: '19418', pageId: 'gen_elt', userId: 'LukeSciarappa', edit: '1', type: 'newEdit', createdAt: '2016-08-31 04:42:05', auxPageId: '', oldSettingsValue: '', newSettingsValue: '' } ], feedSubmissions: [], searchStrings: {}, hasChildren: 'false', hasParents: 'true', redAliases: {}, improvementTagIds: [], nonMetaTagIds: [], todos: [], slowDownMap: 'null', speedUpMap: 'null', arcPageIds: 'null', contentRequests: {} }