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15 - Monics and epics

from Part Two - Doing Category Theory

Published online by Cambridge University Press:  13 October 2022

Eugenia Cheng
Affiliation:
School of the Art Institute of Chicago
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Summary

Another example of doing things categorically, that is, moving away from elements and expressing structure in terms of morphisms in a category. We examine the asymmetry in the definition of a function, leading to the definitions of injective and surjective functions. We examine examples and non-examples from life, which include people stepping on your foot, and people experiencing homelessness; mathematical examples and non-examples include functions from the integers to the integers that add 1 or multiply by 2, and also the empty function. We define monics and epics as categorical versions of injective and surjective functions, and show that in Set they actually correspond to injections and surjections. We show that isomorphisms are necessarily monic and epic, but that a monic and epic morphism is not necessarily an isomorphism in other categories. We examine this in the category of monoids. We conclude by mentioning some further topics: density, the use of monics to define subobjects, and the issues of generalizing monics and epics to higher dimensions.

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The Joy of Abstraction
An Exploration of Math, Category Theory, and Life
, pp. 186 - 205
Publisher: Cambridge University Press
Print publication year: 2022

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  • Monics and epics
  • Eugenia Cheng, School of the Art Institute of Chicago
  • Book: The Joy of Abstraction
  • Online publication: 13 October 2022
  • Chapter DOI: https://doi.org/10.1017/9781108769389.019
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  • Monics and epics
  • Eugenia Cheng, School of the Art Institute of Chicago
  • Book: The Joy of Abstraction
  • Online publication: 13 October 2022
  • Chapter DOI: https://doi.org/10.1017/9781108769389.019
Available formats
×

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To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

  • Monics and epics
  • Eugenia Cheng, School of the Art Institute of Chicago
  • Book: The Joy of Abstraction
  • Online publication: 13 October 2022
  • Chapter DOI: https://doi.org/10.1017/9781108769389.019
Available formats
×