Abstract
Category theory is a framework for the investigation of mathematical form and structure in their most general manifestations. Central to it is the concept of structure-preserving map, or transformation. While the importance of this notion was long recognized in geometry (witness, for example, Klein’s Erlanger Programm of 1872), its pervasiveness in mathematics did not really begin to be appreciated until the rise of abstract algebra in the 1920s and 30s, where, in the form of homomorphism, the idea had been central from the beginning. So emerged the view that the essence of a mathematical structure lies not in its internal constitution as a set-theoretical construct, but rather in the nature of its relationship with other structures of the same kind through the network of transformations between them.
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Notes
- 1.
The kernel of Klein’s Erlanger Programm was the characterization of geometry as the study of those properties of figures that remain invariant under a particular group of transformations, See, e.g. E.T. Bell (1945), pp. 443–9.
- 2.
See, e.g., Kline (1972), Ch. 49
- 3.
Eilenberg , S. and Mac Lane , S., General Theory of Natural Equivalences, Transactions of the American Mathematical Society 58, 1945, pp. 231–294.
- 4.
It was the attempt to create a tractable theory of homology groups that first led Eilenberg and Mac Lane to formulate the idea of a functor , and then of a category.
- 5.
At the same time, functors embody a version of the Principle of Continuity in that composites and identities are preserved.
- 6.
See Johnstone (1983).
- 7.
Here we write x ∧ y for {x, y}.
- 8.
It should be mentioned, however, that the concept of a point can be defined for locales and locales arising as topologies characterized in terms of that notion. See Johnstone (1982), Ch. 2.
- 9.
Mac Lane and Moerdijk (1992), p. 64. Grothendieck describes a sheaf on a space as a “ruler that can be used for taking measurements on it.” (1986, Promenade 13).
- 10.
A presheaf on X may be seen as a functor of a certain kind. Any partially ordered set (P, ≤) can be regarded as a category in which the members of P constitute the objects and, for each p, q ∈ P, there is given exactly one map p → q just when p ≤ q. In particular the family O(X) of open sets of a topological space X, partially ordered by inclusion, may be regarded as a category. The same family, partially ordered by reverse inclusion, yields the “opposite” category O(X)op. Functors from the latter to Set are just the presheaves on X.
- 11.
Grothendieck sees the category of sheaves on a space as a “superstructure of measurement”, which may be “taken to incorporate all that is most essential about that space.” (1986 , Promenade 13).
- 12.
To be precise, C op has the same objects as C; and in C op the maps from an object A to an object B correspond precisely to the maps B → A in C, with composites and identity maps defined analogously.
- 13.
A functor F: C → D is an embedding if, for any objects A, B of C, any map FA → FB in D is of the form Ff for a unique f: A → B in C.
- 14.
Here we view a map f: D → C as an “inclusion” of D in C.
- 15.
If F and G are two functors C op → Set , F is called a subfunctor of G if FX ⊆ GX for any object X of C.
- 16.
Strictly speaking, in defining a site one should specify that the underling category C be small, that is, its collections of objects and maps should both be sets rather than proper classes in the sense of Gödel-Bernays set theory.
- 17.
For a detailed account, see Mac Lane and Moerdijk (1992), Ch. III.
- 18.
See, e.g., Mac Lane and Moerdijk (1992), p. 87.
- 19.
The term is a back-formation from the word “topology” to its original Greek source “topos”, “place”.
- 20.
Grothendieck (1986), Promenade 13. He goes on:
As is often the case in mathematics, we’ve succeeded … in expressing a certain idea—that of a “space ” in this instance—in terms of another one—that of a “category”. Each time the discovery of such a translation from one notion (representing one kind of situation) to another (which corresponds to a different situation) enriches our understanding of both notions, owing to the unanticipated confluence of specific intuitions which relate first to one and then to the other. Thus we see that a situation said to have a “topological” character (embodied in some given space) has been translated into a situation whose character is “algebraic” (embodied in the category); or, if you wish, “continuity” (as present in the space) finds itself “translated” or “expressed by a categorical structure of an algebraic character, which until then had been understood in terms of something “discrete” or “discontinuous”.
- 21.
More generally, elements or points of an object A of a category are maps from the terminal object to A.
- 22.
See Chap. 9.
- 23.
See Bell (1988).
- 24.
For these see, e.g., Bell (1998), Ch. 8.
- 25.
Of course, many important toposes such as the topos of sets have an internal logic that is classical. These are exceptions, however.
- 26.
By intuitionistic set theory IST we mean the theory in intuitionistic first-order logic whose axioms are the “usual” axioms of Zermelo set theory (without the axiom of choice ), namely: Extensionality, Pairing, Union, Power set, Infinity and Separation. For an exposition of IST, see Bell (2014).
- 27.
In fact, this is the case for the topos Shv (ℝ) of sheaves on the usual (classical) topological space ℝ of real numbers . For an account see Johnstone (2002), §D4.7.
- 28.
Others are mentioned in Johnstone, op. cit.
- 29.
Op. cit., Thm. 4.7.11.
- 30.
This is the only one of De Morgan’s laws which does not generally hold in constructive logic . It fails, in particular, in Shv(ℝ).
- 31.
As shown by Dana Scott in his paper Extending the topological interpretation to intuitionistic analysis II, pp. 235–255 of Buffalo Conference in Proof Theory and Intuitionism , North-Holland, 1970. Scott’s result was extended by Martin Hyland in Continuity and spatial toposes, in Fourman, Mulvey and Scott (1979), pp. 440–465. In Mac Lane and Moerdijk (1992), Ch. VI, §9 a topos is constructed in which all real-valued functions defined on the whole real line are continuous. See Appendix C for further discussion.
- 32.
See Appendix A.
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Bell, J.L. (2019). Category/Topos Theory. In: The Continuous, the Discrete and the Infinitesimal in Philosophy and Mathematics. The Western Ontario Series in Philosophy of Science, vol 82. Springer, Cham. https://doi.org/10.1007/978-3-030-18707-1_7
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