What is an Embedding? : A Problem for Category-theoretic Structuralism
(2014) In Preprint without journal information- Abstract
- This paper concerns the proper definition of embeddings in purely category-theoretical terms. It is argued that plain category theory cannot capture what, in the general case, constitutes an embedding of one structure in another. We discuss three available solutions to this problem: variants of monics, concrete categories, and allegories. The first and last of these are found to be unable to solve the problem, and the second to be philosophically unsatisfactory. Instead, we introduce a theory of forms and relators, which, like allegory theory, attempts to abstract from relation algebras in the way that categories abstract from monoids, but which does not have the shortcomings we have identified in allegories. We show that the theory in... (More)
- This paper concerns the proper definition of embeddings in purely category-theoretical terms. It is argued that plain category theory cannot capture what, in the general case, constitutes an embedding of one structure in another. We discuss three available solutions to this problem: variants of monics, concrete categories, and allegories. The first and last of these are found to be unable to solve the problem, and the second to be philosophically unsatisfactory. Instead, we introduce a theory of forms and relators, which, like allegory theory, attempts to abstract from relation algebras in the way that categories abstract from monoids, but which does not have the shortcomings we have identified in allegories. We show that the theory in question does indeed solve the problem of defining embeddings. (Less)
Please use this url to cite or link to this publication:
https://lup.lub.lu.se/record/4393825
- author
- Angere, Staffan LU
- organization
- publishing date
- 2014
- type
- Contribution to journal
- publication status
- unpublished
- subject
- keywords
- structuralism, embedding, category theory
- in
- Preprint without journal information
- publisher
- Manne Siegbahn Institute
- ISSN
- 0348-7911
- language
- English
- LU publication?
- yes
- id
- 94f8ddcd-be1d-496f-b0c2-1d64cff9886a (old id 4393825)
- date added to LUP
- 2016-04-04 14:26:15
- date last changed
- 2018-11-21 21:20:18
@article{94f8ddcd-be1d-496f-b0c2-1d64cff9886a, abstract = {{This paper concerns the proper definition of embeddings in purely category-theoretical terms. It is argued that plain category theory cannot capture what, in the general case, constitutes an embedding of one structure in another. We discuss three available solutions to this problem: variants of monics, concrete categories, and allegories. The first and last of these are found to be unable to solve the problem, and the second to be philosophically unsatisfactory. Instead, we introduce a theory of forms and relators, which, like allegory theory, attempts to abstract from relation algebras in the way that categories abstract from monoids, but which does not have the shortcomings we have identified in allegories. We show that the theory in question does indeed solve the problem of defining embeddings.}}, author = {{Angere, Staffan}}, issn = {{0348-7911}}, keywords = {{structuralism; embedding; category theory}}, language = {{eng}}, publisher = {{Manne Siegbahn Institute}}, series = {{Preprint without journal information}}, title = {{What is an Embedding? : A Problem for Category-theoretic Structuralism}}, url = {{https://lup.lub.lu.se/search/files/6360188/4393826.pdf}}, year = {{2014}}, }