<p>There isn't one. There is no collection of prerequisites which necessarily forms a market. Markets are not algebraic varieties (<a href="https://en.wikipedia.org/wiki/Algebraic_variety" rel="nofollow noreferrer">WP</a>, <a href="https://ncatlab.org/nlab/show/algebraic+variety" rel="nofollow noreferrer">nLab</a>). Markets do not have any universal properties (<a href="https://en.wikipedia.org/wiki/Universal_property" rel="nofollow noreferrer">WP</a>, <a href="https://ncatlab.org/nlab/show/universal+construction" rel="nofollow noreferrer">nLab</a>). Markets do not have any conserved quantities (<a href="https://en.wikipedia.org/wiki/Conservation_law" rel="nofollow noreferrer">WP</a>, <a href="https://ncatlab.org/nlab/show/Noether%27s%20theorem" rel="nofollow noreferrer">nLab</a>). There is no unified concept of homomorphism or structure-preserving mapping between markets, and so there is no most-general category of markets.</p>

CC BY-SA 4.0
2024-05-07T19:31:53.710Z
2024-05-07T19:31:53.710Z
Corbin
<p>In the history of economic theories, various formalisms of &quot;a market&quot; or &quot;the marketplace&quot; have been used to explain various facets of economic behavior, starting roughly with <a href="https://web.archive.org/web/20150608125658/http://www.princeton.edu/mudd/news/faq/topics/Non-Cooperative_Games_Nash.pdf" rel="nofollow noreferrer">Nash 1950</a>. In recent years, there has been interest in generalizing these formalisms using the language of category theory, a branch of mathematics which formalizes formalisms (see <a href="https://math.stackexchange.com/q/246715/425071">this Math.SE question, 2012</a>); however, there's not yet a unified explanation of how these formalisms fit into the standard language of economics.</p>
<p><strong>What's the most general category theory which can be applied to the economist's usage of a market?</strong></p>
<p>I have done some research on this topic. For example, a Goel market (<a href="https://sites.lsa.umich.edu/mje/wp-content/uploads/sites/909/2023/04/An_Algebraic_Presentation_of_Market_Structure.pdf" rel="nofollow noreferrer">Goel 2023</a>) is an object in a subcategory of rings (<a href="https://en.wikipedia.org/wiki/Ring_theory" rel="nofollow noreferrer">WP</a>, <a href="https://ncatlab.org/nlab/show/ring" rel="nofollow noreferrer">nLab</a>); that is, a market is like a number, and a system of markets is like a number system. A Markowitz market (<a href="https://arxiv.org/abs/1611.07741" rel="nofollow noreferrer">Armstrong 2016</a>) is a vector space (<a href="https://en.wikipedia.org/wiki/Vector_space" rel="nofollow noreferrer">WP</a>, <a href="https://ncatlab.org/nlab/show/vector%20space" rel="nofollow noreferrer">nLab</a>) over the real numbers, and a system of markets is like a diagram of linear maps. But it would be difficult to unify them without serious changes to their algebraic signatures, as both authors acknowledge.</p>

CC BY-SA 4.0
2023-10-16T22:47:35.517Z
Sorry if this has been asked before. I'd like to cite something for an upcoming nLab page, and this isn't a question for Math.SE, because mathematicians don't have a particular sentiment for any algebraic signatures; we're happy to work with whatever economists suggest is workable a a real-world model. Feel free to retag and ask for improvements.
Corbin
CC BY-SA 4.0
2023-10-16T22:54:45.840Z
Market was described formally way before Nash, for example in Marshalls Principles
1muflon1
CC BY-SA 4.0
2023-10-17T07:31:36.160Z
Nash 1950 is not about markets at all.
Michael Greinecker
CC BY-SA 4.0
2023-10-17T20:04:34.227Z
problem is that in different subfields markets will be mathematically modeled differently. Market is just a place that supply and demand meets. Such place can be mathematically modeled in various equally valid ways. In macroeconomics a market might be described just as a simple vector of market price and quantity, in mechanism design you will have elaborate functions defining marketplace. This makes your question difficult to answer.
csilvia
CC BY-SA 4.0
2023-10-18T15:59:20.593Z
@csilvia: That would be directly at odds with the typical economics lecture. I learned micro- and macroeconomic principles in high school, and "a market" was a very specific sort of diagram with very specific associated data. If economics is mathematical, then these diagrams should admit formalized representations.
Corbin
CC BY-SA 4.0
2023-10-18T22:58:48.713Z
@Corbin those diagrams you refer to are supply and demand diagrams, am I correct? Those are one way how to represent market. They are often used in microeconomics. However, in input output tables used in some branches of macroeconomics that all gets compressed into one entry into price vector and one entry into the quantity matrix. Also just because some concept is formal does not mean that it cannot be mathematically modeled in different ways. The same earth can be in some branches of astronomy modeled as a point in another as a sphere
csilvia
<mathematical-economics>
What is the most general category-theoretic description of a market?
Corbin
# What is the most general category-theoretic description of a market?

(Asked by: username_0 on 2023-10-16)

In the history of economic theories, various formalisms of "a market" or "the marketplace" have been used to explain various facets of economic behavior, starting roughly with [Nash 1950](https://web.archive.org/web/20150608125658/http://www.princeton.edu/mudd/news/faq/topics/Non-Cooperative_Games_Nash.pdf). In recent years, there has been interest in generalizing these formalisms using the language of category theory, a branch of mathematics which formalizes formalisms (see [this Math.SE question, 2012](https://math.stackexchange.com/q/246715/425071)); however, there's not yet a unified explanation of how these formalisms fit into the standard language of economics.

**What's the most general category theory which can be applied to the economist's usage of a market?**

I have done some research on this topic. For example, a Goel market ([Goel 2023](https://sites.lsa.umich.edu/mje/wp-content/uploads/sites/909/2023/04/An_Algebraic_Presentation_of_Market_Structure.pdf)) is an object in a subcategory of rings ([WP](https://en.wikipedia.org/wiki/Ring_theory), [nLab](https://ncatlab.org/nlab/show/ring)); that is, a market is like a number, and a system of markets is like a number system. A Markowitz market ([Armstrong 2016](https://arxiv.org/abs/1611.07741)) is a vector space ([WP](https://en.wikipedia.org/wiki/Vector_space), [nLab](https://ncatlab.org/nlab/show/vector%20space)) over the real numbers, and a system of markets is like a diagram of linear maps. But it would be difficult to unify them without serious changes to their algebraic signatures, as both authors acknowledge.

**Tags:** mathematical-economics
> **Comments:**
> **username_0**: Sorry if this has been asked before. I'd like to cite something for an upcoming nLab page, and this isn't a question for Math.SE, because mathematicians don't have a particular sentiment for any algebraic signatures; we're happy to work with whatever economists suggest is workable a a real-world model. Feel free to retag and ask for improvements.  
> **username_1**: Market was described formally way before Nash, for example in Marshalls Principles  
> **username_2**: Nash 1950 is not about markets at all.  
> **username_3**: problem is that in different subfields markets will be mathematically modeled differently. Market is just a place that supply and demand meets. Such place can be mathematically modeled in various equally valid ways. In macroeconomics a market might be described just as a simple vector of market price and quantity, in mechanism design you will have elaborate functions defining marketplace. This makes your question difficult to answer.  
> **username_0**: @csilvia: That would be directly at odds with the typical economics lecture. I learned micro- and macroeconomic principles in high school, and "a market" was a very specific sort of diagram with very specific associated data. If economics is mathematical, then these diagrams should admit formalized representations.  
> **username_3**: @Corbin those diagrams you refer to are supply and demand diagrams, am I correct? Those are one way how to represent market. They are often used in microeconomics. However, in input output tables used in some branches of macroeconomics that all gets compressed into one entry into price vector and one entry into the quantity matrix. Also just because some concept is formal does not mean that it cannot be mathematically modeled in different ways. The same earth can be in some branches of astronomy modeled as a point in another as a sphere  

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## 1 answers

### Answer by username_0 on 2024-05-07. Score: 0 | Accepted answer

There isn't one. There is no collection of prerequisites which necessarily forms a market. Markets are not algebraic varieties ([WP](https://en.wikipedia.org/wiki/Algebraic_variety), [nLab](https://ncatlab.org/nlab/show/algebraic+variety)). Markets do not have any universal properties ([WP](https://en.wikipedia.org/wiki/Universal_property), [nLab](https://ncatlab.org/nlab/show/universal+construction)). Markets do not have any conserved quantities ([WP](https://en.wikipedia.org/wiki/Conservation_law), [nLab](https://ncatlab.org/nlab/show/Noether%27s%20theorem)). There is no unified concept of homomorphism or structure-preserving mapping between markets, and so there is no most-general category of markets.


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