An Introduction to Topology and Homotopy by Allan J. Sieradski

By Allan J. Sieradski

The remedy of the topic of this article isn't really encyclopedic, nor used to be it designed to be compatible as a reference handbook for specialists. fairly, it introduces the subjects slowly of their old demeanour, in order that scholars aren't beaten through the last word achievements of a number of generations of mathematicians. cautious readers will see how topologists have steadily subtle and prolonged the paintings in their predecessors and the way such a lot reliable principles achieve past what their originators expected. To motivate the improvement of topological instinct, the textual content is abundantly illustrated. Examples, too a number of to be thoroughly coated in semesters of lectures, make this article appropriate for self sufficient research and make allowance teachers the liberty to choose what they're going to emphasize. the 1st 8 chapters are compatible for a one-semester path often topology. the total textual content is appropriate for a year-long undergraduate or graduate point curse, and gives a powerful beginning for a next algebraic topology path dedicated to the better homotopy teams, homology, and cohomology.

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The source of ao is reduced (no tensorands of I) and completely left-parenthesized (resp. reduced and completely left-parenthesized and free from images of monoidal products under F in the lax case, and free from products both of whose factors are images under F in the oplax and strong cases). 2. The target of an is reduced and completely right-parenthesized (resp. reduced and completely right-parenthesized and free from products both of whose factors are images under F in the lax case, and free from images of monoidal products under F in the oplax and strong cases).

Within a neighborhood bounding the new arc away from the triangles of later moves, if the non-regularity is removed by moves not involving the new arc, or 2. so t h a t the convex hull of the triangle of the move removing the non-regularity and the image of the nearest-neighbor projection of its starting arc across the thin triangle(s) does not intersect the remainder of the link, if the non-regularity is removed by a move involving the new arc. In either case, we replace the sequence of moves with a sequence in which 24 Functorial Knot Theory the move introducing the non-regularity is replaced by the move(s) across the large triangle(s).

Consider the one-object, onemap monoidal category 1^ with object * = I and the obvious monoidal category structure. Then the assignment * H-> A and Id* H-> Id A , with structure maps F : F ( * ) F(*) —> F(* F(I) = 1 : K —>• A is a lax monoidal functor from 1_ to K — v . s . Conversely, every lax monoidal functor from 1 to K — v . s . is of this form. It is an amusing exercise, left to the reader, to verify both of the statements in the last example. 48 Functorial Knot Theory We will be particularly interested in examples of strong monoidal functors whose source is one of our categories of tangles.

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