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High Quality Content by WIKIPEDIA articles! In mathematics, a scheme is an important concept connecting the fields of algebraic geometry, commutative algebra and number theory. Schemes were introduced by Alexander Grothendieck so as to broaden the notion of algebraic variety; some consider schemes to be the basic object of study of modern algebraic geometry. Technically, a scheme is a topological space together with commutative rings for all its open sets, which arises from "glueing together" spectra (spaces of prime ideals) of commutative rings. ... Read more

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Category theory is a general mathematical theory of structures and of structures of structures. It occupied a central position in contemporary mathematics as well as computer science. This book describes the history of category theory whereby illuminating its symbiotic relationship to algebraic topology, homological algebra, algebraic geometry and mathematical logic and elaboratively develops the connections with the epistemological significance.

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Review of Krömer's Tool and Object
The main goal of the book under review is to provide a systematic and profound analysis of several important historical and epistemological aspects of category theory. The central theme of the book is to give an analysis of the remarkable fact that category theory gained position in daily mathematics as a useful and legitimate conceptual innovation, in spite of the difficulties of its set-theoretical foundations and the challenge it caused to this formerly well-established mathematical foundation and to some epistemological positions.
The philosophical stance to category theory developed here is inspired by the pragmatism of Peirce and by Wittgenstein's criticisms of reductionism, which represents a highly interesting alternative to more traditional approaches in philosophy of mathematics like logicism, intuitionism, formalism, realism, fictionalism, etc. In this vein, the author's philosophical position focusses on the "use" of concepts, instead of formal syntax and semantics, and on the thesis that that philosophical justification of mathematical reasoning is an accurate description of the way mathematicians work with categories.
I missed, in the context of a philosophy of category theory, more detailed discussions on some category-theorists philosophical positions, like Lawvere's "dialectical" philosophy of mathematics, different versions of structuralism and different "topos foundations" (for instance, those of Lambek, Bell, Mac Lane) and in this sense the book is more a history than a philosophy of category theory. Some passages are obscured rather than clarified by the philosophical tone, and a methodological fault is that the author sometimes regards spontaneous declarations of some mathematicians as well-elaborated philosophical conceptions or official historical explanations.
Nonetheless, this work is a serious attempt to discuss the history and a philosophy of category theory, and historians of mathematics, philosophers of mathematics, and also "working" mathematicians can profit to a large extent from Krömer's analysis.
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