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In principle, many different abstractions of the set-theoretic notion of point / element are available in the framework of categories, but they are not equally effective and, what is more interesting to me, there doesn't seem to exist one of them subsuming all the others (hence to be preferred in every possible occurrence). Since this is a key issue in my current research on structures, I would appreciate much if

you could provide me with references to existing literature about various interpretations of the notion of point in category theory and, if possible, a (short) historical overview of its evolution

(I don't think it's really necessary to stress that its refers to the notion of point in category theory, and not to the theory of categories - in fact, I'm not really stressing it). Those which I'm already aware of are listed below.


Say that $\bf C$ is a category and $A$ an object of $\bf C$. Then:

  1. If $\bf C$ has a terminal object $\top$, a point of $A$ is any arrow $a$ to $A$ such that $\mathrm{src}(a) \cong \top$ (I use ${\rm src}(\cdot)$ for the source map of $\bf C$). This notion is fine, for instance, if you work with $\bf Set$ (sets + functions), $\bf Top$ (topological spaces + continuous functions), $\bf Pre$ (preordered sets + monotonic functions), etc, but not equally effective with algebraic categories such as $\bf Grp$ (groups + group homomorphisms), $\bf Rg$ (semirings + semiring homomorphisms) and ${\bf Mod}_R$ (left modules over a ring $R$ + homomorphisms of left modules over $R$), basically due to the the fact that, for each of these, terminals are zero objects [it doesn't even work well with $\bf Sgrp$ (semigroups + semigroup homomorphisms) but for other reasons]. A point in the sense of the present definition is usually called a global element
  2. A pont of $A$ is any morphism to $A$. In this case, one commonly speaks of generalized elements. I cannot profit from this notion, and I won't spend one more word on it.
  3. If $\bf C$ is a closed category and $I$ is one of its units (unique up to iso), a point of $A$ is any morphism $a$ to $A$ with ${\rm src}(a) \cong I$. I have no clue how such points are usually referred to. This generalizes Definition 1 in some relevant cases (notably including closed monoidal categories), but $\bf Top$ is not closed monoidal, and let me guess that it is not even closed (I've not checked).
  4. If $\mathcal{G}_1(\mathbf{C})$ is the class of the $1$-generators of $\bf C$, a point of $A$ is any arrow $a$ to $A$ with $\mathrm{src}(a) \in \mathcal G_1(\mathbf{C})$. I don't know how this kind of points are called in the literature.

I would be especially interested in variants, generalizations or alternatives to the previous definitions involving monomorphisms (as for the first one in the list) rather than arbitrary arrows. Thank you in advance for any help.

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Funny, the generalised elements are the ones which are most profitable with respect to the possibility of interpreting internal logic... –  Zhen Lin May 27 '12 at 18:47
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There are many notions of "points" because there are many different contexts where such a notion makes sense and there is no single interpretation that makes sense in all contexts. This question is far too broad if you're asking for all such contexts or a general classification thereof. Perhaps you could specialize the context somewhat? As is, even the foremost experts are likely to miss some and, even then, few would have the patience to answer such a question... –  François G. Dorais May 27 '12 at 19:28
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I am surprised that you say you cannot profit from the notion of generalized elements at all. I think of it as a very natural and very convenient way to think of points in categories (althought, yes, it ouf course depends on the situation). One reason why the notion of generalized elements might deserve its name is seeing that it constructs a natural connection between some notions from category and their set-theoretic counterpart. To take one of the most obvious ones: a morphism is a monomorphism if and only if it acts injectively on the collection of generalized elements. –  Niemi May 27 '12 at 19:40
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Also, these generalized elements are extremely important in functorial algebraic geometry. –  Martin Brandenburg May 27 '12 at 19:49
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Thanks you all. In one way or another, your comments have unhinged a little door in my mind. And yes, I'm definitely reconsidering the option of generalized elements (though not in its full generality). In any event, I'm still interested in a historical overview of the notion itself of point in category theory as well as in a list, as complete as possible, of its most significant embodiments. I see that it may be an overwhelming request but asking is not a sin. –  Salvo Tringali May 28 '12 at 15:14
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5. There is a variation of the generalized elements, called members in Mac Lane's book CWM. Namely, two morphisms $x,y$ with codomain $A$ are identified when there are epis $u,v$ such that $xu=yv$. Clearly this is symmetric and reflexive; in an abelian category it is also transitive. Mac Lane uses the notation $x \in_m A$. Then some properties of members are established, which are used to prove diagram lemmas in arbitrary abelian categories. In my opinion, this is far more efficient and enlightening than proving them via Freyd-Mitchell. And for me this is a perfect translation of the "element calculus" in abstract abelian category.

For other categories, you won't have a general answer what the best "element calculus" might be. It really depends on the context. I strongly agree with François G. Dorais' comment above. A better question would be: Given the problem xyz, how can I define elements in context abc in order to solve xyz?

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