{
localUrl: '../page/product_category_theory.html',
arbitalUrl: 'https://arbital.com/p/product_category_theory',
rawJsonUrl: '../raw/4mj.json',
likeableId: '2777',
likeableType: 'page',
myLikeValue: '0',
likeCount: '1',
dislikeCount: '0',
likeScore: '1',
individualLikes: [
'EricBruylant'
],
pageId: 'product_category_theory',
edit: '11',
editSummary: '',
prevEdit: '10',
currentEdit: '11',
wasPublished: 'true',
type: 'wiki',
title: 'Product (Category Theory)',
clickbait: 'How a product is characterized rather than how it's constructed',
textLength: '875',
alias: 'product_category_theory',
externalUrl: '',
sortChildrenBy: 'likes',
hasVote: 'false',
voteType: '',
votesAnonymous: 'false',
editCreatorId: 'NateSoares',
editCreatedAt: '2016-06-24 03:55:45',
pageCreatorId: 'MarkChimes',
pageCreatedAt: '2016-06-20 14:33:36',
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: '726',
text: 'This simultaneously captures the concept of a product of [-3jz sets], [-3rb posets], [-3gd groups], [-topological_space topological spaces] etc. In addition, like any [-universal_property universal construction], this characterization does not differentiate between [-4f4 isomorphic] versions of the product, thus allowing one to abstract away from an arbitrary, [-specific_construction_category_theory specific construction].\n\n##Definition##\nGiven a pair of objects $X$ and $Y$ in a category $\\mathbb{C}$, the **product** of $X$ and $Y$ is an object $P$ *along with a pair of morphisms* $f: P \\rightarrow X$ and $g: P \\rightarrow Y$ satisfying the following [-universal_property universal] condition:\n\nGiven any other object $W$ and morphisms $u: W \\rightarrow X$ and $v:W \\rightarrow Y$ there is a *unique* morphism $h: W \\rightarrow P$ such that $fh = u$ and $gh = v$.\n\n\n\n',
metaText: '',
isTextLoaded: 'true',
isSubscribedToDiscussion: 'false',
isSubscribedToUser: 'false',
isSubscribedAsMaintainer: 'false',
discussionSubscriberCount: '2',
maintainerCount: '2',
userSubscriberCount: '0',
lastVisit: '',
hasDraft: 'false',
votes: [],
voteSummary: [
'0',
'0',
'0',
'0',
'0',
'0',
'0',
'0',
'0',
'0'
],
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: {
Summary: 'This simultaneously captures the concept of a product of [-3jz sets], [-3rb posets], [-3gd groups], [-topological_space topological spaces] etc. In addition, like any [-universal_property universal construction], this characterization does not differentiate between [-4f4 isomorphic] versions of the product, thus allowing one to abstract away from an arbitrary, [-specific_construction_category_theory specific construction].'
},
creatorIds: [
'MarkChimes',
'NateSoares'
],
childIds: [
'product_universal_property'
],
parentIds: [
'category_theory'
],
commentIds: [
'4mp',
'4my',
'4n0'
],
questionIds: [],
tagIds: [
'stub_meta_tag'
],
relatedIds: [],
markIds: [],
explanations: [],
learnMore: [
{
id: '4310',
parentId: 'product_category_theory',
childId: 'product_category_theory',
type: 'subject',
creatorId: 'MarkChimes',
createdAt: '2016-06-20 14:16:50',
level: '1',
isStrong: 'false',
everPublished: 'true'
}
],
requirements: [
{
id: '4308',
parentId: 'morphism',
childId: 'product_category_theory',
type: 'requirement',
creatorId: 'MarkChimes',
createdAt: '2016-06-20 14:16:29',
level: '1',
isStrong: 'false',
everPublished: 'true'
},
{
id: '4309',
parentId: 'category_mathematics',
childId: 'product_category_theory',
type: 'requirement',
creatorId: 'MarkChimes',
createdAt: '2016-06-20 14:16:47',
level: '1',
isStrong: 'false',
everPublished: 'true'
}
],
subjects: [
{
id: '4310',
parentId: 'product_category_theory',
childId: 'product_category_theory',
type: 'subject',
creatorId: 'MarkChimes',
createdAt: '2016-06-20 14:16:50',
level: '1',
isStrong: 'false',
everPublished: 'true'
}
],
lenses: [
{
id: '123',
pageId: 'product_category_theory',
lensId: 'product_universal_property',
lensIndex: '0',
lensName: 'Universal property',
lensSubtitle: '',
createdBy: '267',
createdAt: '2016-08-25 15:48:14',
updatedBy: '267',
updatedAt: '2016-08-25 15:48:21'
}
],
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: '19629',
pageId: 'product_category_theory',
userId: 'AlexeiAndreev',
edit: '0',
type: 'deleteUsedAsTag',
createdAt: '2016-09-15 21:23:25',
auxPageId: 'product_universal_property',
oldSettingsValue: '',
newSettingsValue: ''
},
{
likeableId: '0',
likeableType: 'changeLog',
myLikeValue: '0',
likeCount: '0',
dislikeCount: '0',
likeScore: '0',
individualLikes: [],
id: '19136',
pageId: 'product_category_theory',
userId: 'PatrickStevens',
edit: '0',
type: 'newChild',
createdAt: '2016-08-25 09:46:53',
auxPageId: 'product_universal_property',
oldSettingsValue: '',
newSettingsValue: ''
},
{
likeableId: '0',
likeableType: 'changeLog',
myLikeValue: '0',
likeCount: '0',
dislikeCount: '0',
likeScore: '0',
individualLikes: [],
id: '14539',
pageId: 'product_category_theory',
userId: 'NateSoares',
edit: '11',
type: 'newEdit',
createdAt: '2016-06-24 03:55:45',
auxPageId: '',
oldSettingsValue: '',
newSettingsValue: ''
},
{
likeableId: '0',
likeableType: 'changeLog',
myLikeValue: '0',
likeCount: '0',
dislikeCount: '0',
likeScore: '0',
individualLikes: [],
id: '14247',
pageId: 'product_category_theory',
userId: 'MarkChimes',
edit: '10',
type: 'newEdit',
createdAt: '2016-06-21 04:45:20',
auxPageId: '',
oldSettingsValue: '',
newSettingsValue: ''
},
{
likeableId: '0',
likeableType: 'changeLog',
myLikeValue: '0',
likeCount: '0',
dislikeCount: '0',
likeScore: '0',
individualLikes: [],
id: '14246',
pageId: 'product_category_theory',
userId: 'MarkChimes',
edit: '9',
type: 'newEdit',
createdAt: '2016-06-21 04:44:47',
auxPageId: '',
oldSettingsValue: '',
newSettingsValue: ''
},
{
likeableId: '0',
likeableType: 'changeLog',
myLikeValue: '0',
likeCount: '0',
dislikeCount: '0',
likeScore: '0',
individualLikes: [],
id: '14110',
pageId: 'product_category_theory',
userId: 'MarkChimes',
edit: '7',
type: 'newEdit',
createdAt: '2016-06-20 15:32:49',
auxPageId: '',
oldSettingsValue: '',
newSettingsValue: ''
},
{
likeableId: '0',
likeableType: 'changeLog',
myLikeValue: '0',
likeCount: '0',
dislikeCount: '0',
likeScore: '0',
individualLikes: [],
id: '14109',
pageId: 'product_category_theory',
userId: 'MarkChimes',
edit: '0',
type: 'deleteTag',
createdAt: '2016-06-20 15:31:23',
auxPageId: 'work_in_progress_meta_tag',
oldSettingsValue: '',
newSettingsValue: ''
},
{
likeableId: '0',
likeableType: 'changeLog',
myLikeValue: '0',
likeCount: '0',
dislikeCount: '0',
likeScore: '0',
individualLikes: [],
id: '14104',
pageId: 'product_category_theory',
userId: 'MarkChimes',
edit: '0',
type: 'newRequirement',
createdAt: '2016-06-20 14:33:39',
auxPageId: 'morphism',
oldSettingsValue: '',
newSettingsValue: ''
},
{
likeableId: '0',
likeableType: 'changeLog',
myLikeValue: '0',
likeCount: '0',
dislikeCount: '0',
likeScore: '0',
individualLikes: [],
id: '14105',
pageId: 'product_category_theory',
userId: 'MarkChimes',
edit: '0',
type: 'newRequirement',
createdAt: '2016-06-20 14:33:39',
auxPageId: 'category_mathematics',
oldSettingsValue: '',
newSettingsValue: ''
},
{
likeableId: '0',
likeableType: 'changeLog',
myLikeValue: '0',
likeCount: '0',
dislikeCount: '0',
likeScore: '0',
individualLikes: [],
id: '14106',
pageId: 'product_category_theory',
userId: 'MarkChimes',
edit: '0',
type: 'newTeacher',
createdAt: '2016-06-20 14:33:39',
auxPageId: 'product_category_theory',
oldSettingsValue: '',
newSettingsValue: ''
},
{
likeableId: '0',
likeableType: 'changeLog',
myLikeValue: '0',
likeCount: '0',
dislikeCount: '0',
likeScore: '0',
individualLikes: [],
id: '14107',
pageId: 'product_category_theory',
userId: 'MarkChimes',
edit: '0',
type: 'newSubject',
createdAt: '2016-06-20 14:33:39',
auxPageId: 'product_category_theory',
oldSettingsValue: '',
newSettingsValue: ''
},
{
likeableId: '0',
likeableType: 'changeLog',
myLikeValue: '0',
likeCount: '0',
dislikeCount: '0',
likeScore: '0',
individualLikes: [],
id: '14101',
pageId: 'product_category_theory',
userId: 'MarkChimes',
edit: '0',
type: 'newParent',
createdAt: '2016-06-20 14:33:38',
auxPageId: 'category_theory',
oldSettingsValue: '',
newSettingsValue: ''
},
{
likeableId: '0',
likeableType: 'changeLog',
myLikeValue: '0',
likeCount: '0',
dislikeCount: '0',
likeScore: '0',
individualLikes: [],
id: '14102',
pageId: 'product_category_theory',
userId: 'MarkChimes',
edit: '0',
type: 'newTag',
createdAt: '2016-06-20 14:33:38',
auxPageId: 'work_in_progress_meta_tag',
oldSettingsValue: '',
newSettingsValue: ''
},
{
likeableId: '0',
likeableType: 'changeLog',
myLikeValue: '0',
likeCount: '0',
dislikeCount: '0',
likeScore: '0',
individualLikes: [],
id: '14103',
pageId: 'product_category_theory',
userId: 'MarkChimes',
edit: '0',
type: 'newTag',
createdAt: '2016-06-20 14:33:38',
auxPageId: 'stub_meta_tag',
oldSettingsValue: '',
newSettingsValue: ''
},
{
likeableId: '0',
likeableType: 'changeLog',
myLikeValue: '0',
likeCount: '0',
dislikeCount: '0',
likeScore: '0',
individualLikes: [],
id: '14099',
pageId: 'product_category_theory',
userId: 'MarkChimes',
edit: '6',
type: 'newEdit',
createdAt: '2016-06-20 14:33:36',
auxPageId: '',
oldSettingsValue: '',
newSettingsValue: ''
}
],
feedSubmissions: [],
searchStrings: {},
hasChildren: 'true',
hasParents: 'true',
redAliases: {},
improvementTagIds: [],
nonMetaTagIds: [],
todos: [],
slowDownMap: 'null',
speedUpMap: 'null',
arcPageIds: 'null',
contentRequests: {}
}