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What is an Embedding? : A Problem for Category-theoretic Structuralism

Angere, Staffan LU (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)
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author
organization
publishing date
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unpublished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
2014-04-16 14:46:40
date last changed
2016-04-16 12:32:15
@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},
  keyword      = {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},
  year         = {2014},
}