By K Heiner Kamps, Timothy Porter

Summary homotopy concept is predicated at the commentary that analogues of a lot of topological homotopy idea and straightforward homotopy conception exist in lots of different different types, comparable to areas over a hard and fast base, groupoids, chain complexes and module different types. learning specific types of homotopy constitution, resembling cylinders and course house structures permits not just a unified improvement of many examples of recognized homotopy theories, but in addition finds the interior operating of the classical spatial concept, truly indicating the logical interdependence of homes (in specific the life of convinced Kan fillers in linked cubical units) and effects (Puppe sequences, Vogt's lemma, Dold's Theorem on fibre homotopy equivalences, and homotopy coherence conception)

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**Sample text**

Let F = f∗ : A−comod → B −comod be the corresponding pushforward functor. Then F is surjective if and only if f is surjective. 7. Let C be a coalgebra. A nonzero element x ∈ C is called grouplike if Δ(x) = x ⊗ x. 13. 8. There is a bijection between grouplike elements of a coalgebra C and its one-dimensional subcoalgebras, given by x → kx. 9. Let X be a set. Then kX, the set of formal linear combinations of elements of X, is a coalgebra, with Δ(x) = x ⊗ x for x ∈ X. The grouplike elements of kX are precisely elements x ∈ X.

37) is a monoidal functor. 3. Show that actions of a group G on the category Vec viewed as an abelian category correspond to elements of H 2 (G, k× ), while any action of G on Vec viewed as a monoidal category is trivial. 8. The Mac Lane strictness theorem As we have seen above, it is simpler to work with monoidal categories in which the associativity and unit constrains are the identity maps. 1. A monoidal category C is strict if for all objects X, Y, Z in C one has equalities (X ⊗ Y ) ⊗ Z = X ⊗ (Y ⊗ Z) and X ⊗ 1 = X = 1 ⊗ X, and the associativity and unit constraints are the identity maps.

9. 1 is simpler. 6 implies that for a triple (C, ⊗, a) satisfying a pentagon axiom (which should perhaps be called a “semigroup category”, as it categoriﬁes the notion of a semigroup), being a monoidal category is a property and not a structure (similarly to how it is for semigroups and monoids). 4 In a sense, all this means that in constructions with monoidal categories, unit objects and isomorphisms always “go along for the ride”, and one need not worry about them especially seriously. For this reason, below we will typically take less care dealing with them than we have done in this section.