Geometry of the category of complexes and algebraic K-theory

Homotopy fiber sequences induced by 2-functors

“…that carries a p-simplex χ of W S 1 , as in (21), to the p-simplex χ ′ of W S 2 , as in (22), in which all data x ′ s, u ′ s α ′ s and a ′ s are the same as those in χ, and the morphisms f j m : a m → u j m * a m−1 are respectively given by:…”

“…From this point of view the use of 2-categories and their classifying spaces in homotopy theory goes back to Whitehead [40] and Mac Lane-Whitehead [28] since 2-groupoids with only one object (= strict 2-groups, in the terminology of Baez [2]) are the same as crossed modules (this observation is attributed to Verdier in [6]). But, beyond homotopy theory, the use of classifying spaces of 2-categories has also shown its relevance in several other mathematical contexts such as in algebraic K-theory [21], conformal field theory [39], or in the study of geometric structures on low-dimensional manifolds [38].…”

Homotopy colimits of 2-functors

“…Similarly, if D, E : C op → 2Cat are 2-functors and Γ : D ⇒ E is a 2-transformation, for any objects c ∈ C and y ∈ E c , there is a commutative square (25)…”

“…XII]. Also, for C a 2-category and D : C → Cat ⊆ 2Cat a 2-functor into the category of categories, functors, and natural transformations, one recovers the homotopy colimit of a 2-diagram of categories by Hinich and Schechtman in [25].…”

Classifying Spaces for Monoidal Categories Through Geometric Nerves

“…Id are those arising from the associativity and unit constraints of M ⊗ (see [8,9]). In particular, the space at level 1 of the spectrum associated to a symmetric monoidal category (M ⊗ , c) is the space ||M ⊗ ||.…”