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In the Historical notes section of the Wikipedia article on category theory, it is mentioned that in 1942-1945, Samuel Eilenberg and Saunders Mac Lane, in the course of their work in algebraic topology, established the notion of category and functor in order to study natural transformations.

The article goes on to say that Stanislaw Ulam and some others have claimed that category-like ideas had existed since at least the late 1930s, and that category theory was, in a sense, related to the work of Emmy Noether during even earlier years.

I have not formally studied category theory, just done some reading on it, so it didn't occur to me that there are many trivial and early examples of functors. In light of this, I'll attempt a question that is perhaps easier to answer (if longer).

What exactly were these pre-algebraic topology notions of category theory developed in the 1930s? Also, even though particular classes of mathematical objects have existed for longer periods of time, presumably they weren't actually viewed as categories until later on. What is the earliest category mentioned in the literature (e.g. the category of groups or the category of topological spaces)?

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    $\begingroup$ wouldn't be set theory? $\endgroup$ – janmarqz Feb 26 '14 at 3:30
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    $\begingroup$ I think set theory is very distinct from category theory. I would say that the essence of category theory is its refusal to look at the internal structure of its objects and to characterize every property in terms of arrows. For example, in set theory, to define a product, one starts talking about ordered pairs of elements, and it was historically important that the ordered pair itself could be identified as a set-theoretic object. Category theory dispenses with the ordered pairs completely, and defines a product as an object that possesses a pair of projection arrows with certain properties. $\endgroup$ – MJD Feb 26 '14 at 4:07
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    $\begingroup$ The question "what is the earliest functorial construction" is too hard to answer and also a little bit boring since functors are everywhere (also hundreds of years ago). And your question also doesn't exclude trivial examples such as the functor which maps a set $X$ to $X$. A better question would be what these category-like ideas were exactly around the 30s. $\endgroup$ – Martin Brandenburg Feb 26 '14 at 10:21
  • $\begingroup$ @MartinBrandenburg: Would it make the question significantly less trivial/more interesting to require that the functor not be an endofunctor? $\endgroup$ – Alex Petzke Feb 27 '14 at 4:38
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    $\begingroup$ Isn't it the case that algebraic topology is the source of the earliest nontrivial functors? The functor that maps a topological space to its homotopy group, for example. $\endgroup$ – MJD Feb 27 '14 at 14:56

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