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    • CommentRowNumber1.
    • CommentAuthorPeter Heinig
    • CommentTimeJun 19th 2017
    • (edited Jun 19th 2017)

    More or less a vague reference request: do you recommend some articles or books which (somehow) treat categories in the style of “combinatorial group theory”, in particular, developing notation for “relators” and results on “finitely presented” categories (both in the sense of combinatorial group theory).

    For obvious reasons, every book on combinatorial group theory, and many articles within semigroup theory are something in that direction, but the question is rather asking for a more category-theoretic treatment.

    Something of a moderate extension of combinatorial group theory to (some classes of) categories?

    • CommentRowNumber2.
    • CommentAuthorMike Shulman
    • CommentTimeJun 19th 2017

    I don’t really know what “combinatorial group theory” entails. One can present objects by generators and relations in any category of algebras for a monad, while locally finitely presentable categories are “finitely presented” in a different sense. What sort of definitions/theorems are you looking to generalize?

    • CommentRowNumber3.
    • CommentAuthorDavidRoberts
    • CommentTimeJun 19th 2017

    @Mike presumably he means something like considering a finite quiver with finitely many commuting diagrams imposed.

    • CommentRowNumber4.
    • CommentAuthorTim_Porter
    • CommentTimeJun 20th 2017

    There is a lot of that sort of thing looked at in rewriting theory. From the direction you are considering, Peter, look first at Mitchell’s Rings with several Objects paper(Advances in Mathematics 8 (1972), 1–161). Questions of identities among relations for presentations of small categories and also for theories have been looked at by Lafont, Malbos, Guiraud and others. (see for example: Yves Guiraud and Philippe Malbos, Higher-dimensional categories with finite derivation type, Theory and Applications of Categories 22 (2009), no. 18, 420–478.). Other keywords to search on are polygraph or computad.

    • CommentRowNumber5.
    • CommentAuthorThomas Holder
    • CommentTimeJun 20th 2017
    • (edited Jun 20th 2017)

    In the 1980s Burstall and Rydeheard implemented parts of category theory on a computer under the heading “computational category theory” using finite presentations. There was (and probably still is) a website of that name with some software as well as a copy of their Prentice-Hall (1988) book of the same name. Bob Walters and students also worked on this, for which you could consult chapter VII of Walters “Categories and computer science”, CUP 1991.

    • CommentRowNumber6.
    • CommentAuthorPeter Heinig
    • CommentTimeJun 20th 2017

    Thanks for the pointers. Will look into them and at most then ask a more precise question on this topic.