Difference between revisions of "Universal property"
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{{WikiEntry|key=Universal property|qCode=1417809}}, also known as [[Universal construction]], is a crucial concept pervasive in the literature of [[Category Theory]]. This concept, or the thesis of [[Category Theory]], offered a very different approach to organize knowledge. According to [[John Peloquin|Peloquin]]<ref name="PeloquinUniversalProperty">{{:Video/Category Theory Part 3 of 3: Universal Properties}}</ref>: | {{WikiEntry|key=Universal property|qCode=1417809}}, also known as [[Universal construction]], is a crucial concept pervasive in the literature of [[Category Theory]]. This concept, or the thesis of [[Category Theory]], offered a very different approach to organize knowledge. According to [[John Peloquin|Peloquin]]<ref name="PeloquinUniversalProperty">{{:Video/Category Theory Part 3 of 3: Universal Properties}}</ref>: | ||
A universal property characterizes an object in a category, in an essentially [[unique]] way, in terms of its relation to other objects through arrows. | A universal property characterizes an object in a category, in an essentially [[unique]] way, in terms of its relation to other objects through arrows. | ||
The statement shown above used a critical term, [[unique]]. It implies that all things, all phenomenon, if they were to be characterized uniquely, the unique way can be determined by the relation of the object to all other things<ref>See [[Yoneda lemma]]</ref>. For staters, the uniqueness of relations may be verified using [[product]], [[co-product]], [[exponential]], and other kind of arithmetic constructs. | The statement shown above used a critical term, [[unique]]. It implies that all things, all phenomenon, if they were to be characterized uniquely, the unique way can be determined by the relation of the object to all other things<ref>See [[Yoneda lemma]]</ref>. For staters, the uniqueness of relations may be verified using [[product]], [[co-product]], [[exponential]], and other kind of arithmetic constructs. Handwiki has a detailed entry on the subject of [https://handwiki.org/wiki/Universal_property|Universal property]. | ||
This implication portrait a single, generalized relational mechanisms to define properties of individual things. All things can be characterized through relations with other things. This relationally-based approach means that you don't need to study anything else, unique properties of every individual can be universally represented using nothing but relations. All properties are just variants or descendants of this relational [[universal construct]]. [[John Peloquin|Peloquin]]<ref extends="PeloquinUniversalProperty">[https://youtu.be/bPvWEN8UGuo?t=66 ...'avoid repeating the same proofs over and over again...']</ref>: | This implication portrait a single, generalized relational mechanisms to define properties of individual things. All things can be characterized through relations with other things. This relationally-based approach means that you don't need to study anything else, unique properties of every individual can be universally represented using nothing but relations. All properties are just variants or descendants of this relational [[universal construct]]. [[John Peloquin|Peloquin]]<ref extends="PeloquinUniversalProperty">[https://youtu.be/bPvWEN8UGuo?t=66 ...'avoid repeating the same proofs over and over again...']</ref>: | ||
Revision as of 03:08, 24 March 2022
Universal property(Q1417809), also known as Universal construction, is a crucial concept pervasive in the literature of Category Theory. This concept, or the thesis of Category Theory, offered a very different approach to organize knowledge. According to Peloquin[1]:
A universal property characterizes an object in a category, in an essentially unique way, in terms of its relation to other objects through arrows.
The statement shown above used a critical term, unique. It implies that all things, all phenomenon, if they were to be characterized uniquely, the unique way can be determined by the relation of the object to all other things[2]. For staters, the uniqueness of relations may be verified using product, co-product, exponential, and other kind of arithmetic constructs. Handwiki has a detailed entry on the subject of property.
This implication portrait a single, generalized relational mechanisms to define properties of individual things. All things can be characterized through relations with other things. This relationally-based approach means that you don't need to study anything else, unique properties of every individual can be universally represented using nothing but relations. All properties are just variants or descendants of this relational universal construct. PeloquinCite error: Invalid <ref> tag; invalid names, e.g. too many:
It allows us to avoid repeating the same proofs over and over again in different categories.
By not repeating oneself, it means we can apply this proven theory to generalize many kinds of proven results. This kind generalization, can be thought of as a rigorous belief mechanism, a universal construct.
Using the phrase:universal construct, Saint John Henry Newman who argued for all knowledge, including theology must be taught in a university, may have thought about the following logical argument, too. He stated:
If all things are descendants from a certain universal construct, religions must be derived from such construct.
The notion of utilizing universal construct to bridge understanding between different domain knowledge has been explicitly articulated by Olivia Caramello. She has several videos on this subject by Olivia Caramello[3][4][5].
There are a few videos[1][6][7] on Universal Properties.
References
- ↑ 1.0 1.1 Peloquin, John (Jun 15, 2020). Category Theory Part 3 of 3: Universal Properties. local page: blargoner.
- ↑ See Yoneda lemma
- ↑ Caramello, Olivia (May 24, 2021). Unification and morphogenesis : a topos-theoretic perspective. local page: Institut des Hautes Études Scientifiques.
- ↑ Caramello, Olivia (Dec 1, 2018). The idea of 'bridge' and its unifying role. local page: TEDxLakeComo.
- ↑ Meyerson, Michael (2002). Political numeracy : mathematical perspectives on our chaotic constitution. local page: Norton Publisher. ISBN 0393323722.
- ↑ Tubbenhauer, Daniel (Jan 12, 2022). What are...universal properties?. local page: VisualMath.
- ↑ Forsberg, Fredrik Nordvall (Jan 12, 2022). Universal properties. local page: Fredrik Nordvall Forsberg.