ArtsAutosBooksBusinessEducationEntertainmentFamilyFashionFoodGamesGenderHealthHolidaysHomeHubPagesPersonal FinancePetsPoliticsReligionSportsTechnologyTravel

Conflict Theory Development

Updated on June 15, 2012

A Category Theory of Conflict

In what follows, we will assume that each word being treated as an object will be an object in some category C, and each arrow below, to be a morphism in the same category C. For those who lack a knowledge of category theory, the following link to a series of short video lectures on category theory by theCatsters is an excellent resource to learn some basic category theory quickly:

To motivate the use of category theory as the mathematical language in which to develop our mathematical theory of conflict, we shall use the following simple example of a conflict:

Let P be the predicate "The cat is _____"

Let h: alive-> P be the arrow that takes the word red and maps it into the predicate P to produce the statement "The cat is alive." Like wise, let the arrow h':dead->P be the arrow that takes the word dead and maps it into the predicate P to produce the statement "The cat is dead."

Notice, that like in our previous examples in the previous hub (see, the claims "The cat is alive" and "The cat is dead" conflict if we assert they cannot both be true at the same time: i.e. if we assert that the same cat cannot be both dead and alive at the same time.

Let q: alive->dead be the arrow that takes to word "alive" to the word "dead": i.e.

Since h'oq(alive): alive->P and h:alive->P are morphisms in our underlying catogory, we might naturally ask whether h=h'oq. The following computation shows h is not equal to h'oq.

h'oq(alive): alive->P is an arrow from alive to P with h'oq(alive)=h'(q(alive))=h'(dead)="The cat is dead."

On the other hand, h(alive): alive->P is an arrow from alive to P with h(alive)="The cat is alive." Since "The cat is alive." is not equal to "The cat is dead.", we see that h'oq(alive) is not equal to h(alive), that is, h'oq ≠ h.

So our conflict has reduced to a diagram in a category C that does not commute. This suggests defining a generalized conflict in a category C to simply be a non-commutative diagram in C.

Definition 1: Let C be a category. A generalized conflict in C is a non-commutative diagram in C.

For those with some knowledge in category theory, a necessary and sufficient condition for the following diagram below to commute would have been for q to be a morphism in the slice category C/P. In our examples from the previous hub, and the example above, the notion of having two distinct items being forced into a space too small to hold both of those items was essential for the conflicts of the form given in examples 1-3 of the following hub:

The diagram above encodes the notion of having two distinct items mapped into a single space, and the non-commutativity of the same diagram encodes the notion of the space being too small to fit those two distinct items.

This motivates the following definition of a strict S-Conflict in a category C.

Definition 2: Let C be a category, P and object of C, and consider the slice category C/P. A strict S-Conflict in C is a morphism f: x->x' in C such that f is not a morphism in C/P. That is, f is a morphism in C such that the following diagram fails to commute:


There are two cannonical solutions to this conflict corresponding to the two cannonical solutions listed in the two cannonical solutions listed in the link above are to remove the distinction between two distinguished items x and x', or create two distinct spaces S and S' and put each distinct item, x and x', into a distinct space large enough to hold them: i.e. put x in S, and x' into S', say.

The first solution (or rather a generalization of the first solution) is simply to find a morphism q':alive -> dead in the slice category of C/P. Such a morphism q', by the definition of a morphism in C/P, would make the diagram above commute thus resolving the conflict.

The second cannonical solution is to find an endofunctor G:C->C and a morphism q':alive -> dead in C such that q' is a morphism in the category Coalg(G) of coalgebras of G: the reasoning for this is that if C is a category, then a predicate on C may be thought of as a functor G:C->C and for a morphism q:x->G(x), and q':x'->G(x') (or in our case,

f:alive ->G(alive), g:dead->G(dead)) a morphism from from g:x->x' making q'og=Ggoq would resolve our conflict. Intuitively, the space is defined to be the functor G, and the existance of a G-coalgebra morphism from q:x->G(x) to q':x'->G(x') corresponds to saying the space G is large enough to hold both x and x', thereby, preventing a conflict. A dual notion of a strict S-conflict, a strict S-Co-conflict, an interpretation of the meaning of a strict S-Co-conflict, and the interpretation of the two cannonical co-solutions to the conflict will all be elaborated on in a future hub.

For those who lack a background in category theory, the following series of short video lectures by theCatsters is an excellent resource to learning basic category theory quickly:


    0 of 8192 characters used
    Post Comment

    No comments yet.


    This website uses cookies

    As a user in the EEA, your approval is needed on a few things. To provide a better website experience, uses cookies (and other similar technologies) and may collect, process, and share personal data. Please choose which areas of our service you consent to our doing so.

    For more information on managing or withdrawing consents and how we handle data, visit our Privacy Policy at:

    Show Details
    HubPages Device IDThis is used to identify particular browsers or devices when the access the service, and is used for security reasons.
    LoginThis is necessary to sign in to the HubPages Service.
    Google RecaptchaThis is used to prevent bots and spam. (Privacy Policy)
    AkismetThis is used to detect comment spam. (Privacy Policy)
    HubPages Google AnalyticsThis is used to provide data on traffic to our website, all personally identifyable data is anonymized. (Privacy Policy)
    HubPages Traffic PixelThis is used to collect data on traffic to articles and other pages on our site. Unless you are signed in to a HubPages account, all personally identifiable information is anonymized.
    Amazon Web ServicesThis is a cloud services platform that we used to host our service. (Privacy Policy)
    CloudflareThis is a cloud CDN service that we use to efficiently deliver files required for our service to operate such as javascript, cascading style sheets, images, and videos. (Privacy Policy)
    Google Hosted LibrariesJavascript software libraries such as jQuery are loaded at endpoints on the or domains, for performance and efficiency reasons. (Privacy Policy)
    Google Custom SearchThis is feature allows you to search the site. (Privacy Policy)
    Google MapsSome articles have Google Maps embedded in them. (Privacy Policy)
    Google ChartsThis is used to display charts and graphs on articles and the author center. (Privacy Policy)
    Google AdSense Host APIThis service allows you to sign up for or associate a Google AdSense account with HubPages, so that you can earn money from ads on your articles. No data is shared unless you engage with this feature. (Privacy Policy)
    Google YouTubeSome articles have YouTube videos embedded in them. (Privacy Policy)
    VimeoSome articles have Vimeo videos embedded in them. (Privacy Policy)
    PaypalThis is used for a registered author who enrolls in the HubPages Earnings program and requests to be paid via PayPal. No data is shared with Paypal unless you engage with this feature. (Privacy Policy)
    Facebook LoginYou can use this to streamline signing up for, or signing in to your Hubpages account. No data is shared with Facebook unless you engage with this feature. (Privacy Policy)
    MavenThis supports the Maven widget and search functionality. (Privacy Policy)
    Google AdSenseThis is an ad network. (Privacy Policy)
    Google DoubleClickGoogle provides ad serving technology and runs an ad network. (Privacy Policy)
    Index ExchangeThis is an ad network. (Privacy Policy)
    SovrnThis is an ad network. (Privacy Policy)
    Facebook AdsThis is an ad network. (Privacy Policy)
    Amazon Unified Ad MarketplaceThis is an ad network. (Privacy Policy)
    AppNexusThis is an ad network. (Privacy Policy)
    OpenxThis is an ad network. (Privacy Policy)
    Rubicon ProjectThis is an ad network. (Privacy Policy)
    TripleLiftThis is an ad network. (Privacy Policy)
    Say MediaWe partner with Say Media to deliver ad campaigns on our sites. (Privacy Policy)
    Remarketing PixelsWe may use remarketing pixels from advertising networks such as Google AdWords, Bing Ads, and Facebook in order to advertise the HubPages Service to people that have visited our sites.
    Conversion Tracking PixelsWe may use conversion tracking pixels from advertising networks such as Google AdWords, Bing Ads, and Facebook in order to identify when an advertisement has successfully resulted in the desired action, such as signing up for the HubPages Service or publishing an article on the HubPages Service.
    Author Google AnalyticsThis is used to provide traffic data and reports to the authors of articles on the HubPages Service. (Privacy Policy)
    ComscoreComScore is a media measurement and analytics company providing marketing data and analytics to enterprises, media and advertising agencies, and publishers. Non-consent will result in ComScore only processing obfuscated personal data. (Privacy Policy)
    Amazon Tracking PixelSome articles display amazon products as part of the Amazon Affiliate program, this pixel provides traffic statistics for those products (Privacy Policy)