homotopy theory, (∞,1)-category theory, homotopy type theory
flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed…
models: topological, simplicial, localic, …
see also algebraic topology
Introductions
Definitions
Paths and cylinders
Homotopy groups
Basic facts
Theorems
superalgebra and (synthetic ) supergeometry
This page collects material related to the book
Die Wissenschaft der extensive Grössen oder die Ausdehnungslehre
Erster Teil, die lineale Ausdehnungslehre,
1844
which introduced for the first time basic concepts of what today is known as linear algebra (including affine spaces as torsors over vector spaces) and introduced in addition an exterior product (§37, §55) on vectors, forming what today is known as exterior algebra or Grassmann algebra, hence in fact superalgebra.
Grassmann advertizes his work (p. xxv) as being the theory of extensive quantity. The modern way of speaking about this is that the elements of the exterior algebra he considered are differential forms on Euclidean space.
Discussion of the book includes
and the similar text
which says at one point that full appreciation of the Ausdehnungslehre requires concepts of category theory
> The modern conceptual apparatus, involving levels of structure, categories of morphisms preserving given structure, forgetful reduct functors between categories, the adjoints to such functors, etc., seems to be necessary for ordinary mortals to be able to find their way through the riches of Grassmann’s geometry.
The first part of the introduction of the Ausdehnungslehre is concerned with philosophy, about which
> Grassmann insists that his reason for including it is an attempt to provide an orientation to help the student form for himself the proper estimation of the relation between general and particular at every stage of the learning process (Lawvere 95).
The second part of the introduction, titled Survey of the general theory of forms considers key concepts of algebra. For instance it considers the associativity law and states its coherence law (§3). Grassmann writes that he uses the term “form” in place of “quantity” (German: “Grösse”) (Introduction A.3, §2). It is “forms” that his algebraic operations are defined on, and which are produced by these.
> The last half of that introduction is essentially one of the first expositions of the rudimentary principles of what today might be called universal algebra. The content of the first half, after considerable study of the compact formulations, appears to be a simple and clear natural scientist’s version of the basic principles of dialectical materialism, as applied to the formal sciences. (Lawvere 95)
Curiously, while Grassmann complains (on p. xv) about the “unclarity and arbitrariness” of Hegel’s school of philosophy (German idealism, predominant in Germany at Grassmann’s time), the introduction of the Ausdehnungslehre has much the same sound as Hegel, notably it discusses “categories” such as being, becoming (p. xxii), particulars (p.xx) and the dialectic of opposites such as discrete $\dashv$ continuous (p.xxii) and, notably, of intensive and extensive quantity (p. xxiv-xxv), which Grassmann advertizes as the very topic of his mathematical theory. That of course is the difference to Hegel, that unambiguous mathematical formalization of these otherwise vague concepts is provided (according to Lawvere 95 Grassmannn’s formalization of the pair being and becoming is via points and vectors in an affine space), and in this sense Grassmann is clearly a forerunner of Lawvere’s various proposals for formalizing Hegel’s objective logic in categorical logic/topos theory (as discussed at Science of Logic).