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  1. Hi,
    If x is an object of a category C, one usually says that x is if finite presentation (or compact) if for any direct filtered system (y_i) in C, the canonical map

    f : colim_i Hom(x,y_i) -> Hom(x, colim y_i)

    is bijective. One usually says x is of finite type if this holds only for direct filtered systems of monomophisms.

    However, I am interested in objects x for which f is injective (without additional condition on the direct filtered system). It seems to me that, in the category of (right) modules over a ring, such objects are exactly finitely generated modules. My question is, has this been considered, and is there a name for this property of x?

    Thanks in advance

    Alain Bruguières
    • CommentRowNumber2.
    • CommentAuthorMike Shulman
    • CommentTimeApr 21st 2017

    I’ve usually heard “finitely generated” used to refer to the property you call “finite type”.

  2. Well I thought it was called "finite type". And consequently, I was planning to call the other property (injectivity of f) "finitely generated". But if the name is already taken, I should probably exchange the terms. Fact is, I need both notions (the one with f bijective and monic filtered systems and the one with f injective). And in the category of modules, they coincide...
    • CommentRowNumber4.
    • CommentAuthorMike Shulman
    • CommentTimeApr 25th 2017

    A little googling suggests that “finite type” may be more commonly used for ring homomorphisms, to distinguish finite generation as an algebra from finite generation as a module. If they mean essentially the same thing for modules, then it doesn’t seem like a good idea to me to arbitrarily decide to generalize them differently to the abstract context.

    This might be a good question for mathoverflow, or the categories mailing list?

    • CommentRowNumber5.
    • CommentAuthorDavidRoberts
    • CommentTimeApr 25th 2017
    • (edited Apr 25th 2017)

    A ring map of finite type means the codomain is finitely generated as an algebra over the domain, though…