We provide an elementary proof of a bicategorical pasting theorem that does not rely on Powers 2-categorical pasting theorem, the bicategorical coherence theorem, or the local characterization of a biequivalence.
The theory of shadows is an axiomatic, bicategorical framework that generalizes topological Hochschild homology (THH) and satisfies analogous important properties, such as Morita invariance. Using Bermans extension of THH to bicategories, we prove th
at there is an equivalence between functors out of THH of a bicategory and shadows on that bicategory. As an application we provide a new, conceptual proof that shadows are Morita invariant.
We prove a bicategorical analogue of Quillens Theorem A. As an application, we deduce the well-known result that a pseudofunctor is a biequivalence if and only if it is essentially surjective on objects, essentially full on 1-cells, and fully faithful on 2-cells.
We formulate and prove a twofold generalisation of Lies second theorem that integrates homomorphisms between formal group laws to homomorphisms between Lie groups. Firstly we generalise classical Lie theory by replacing groups with categories. Second
ly we include categories whose underlying spaces are not smooth manifolds. The main intended application is when we replace the category of smooth manifolds with a well-adapted model of synthetic differential geometry. In addition we provide an axiomatic system that specifies the abstract structures that are required to prove Lies second theorem. As a part of this abstract structure we define the notion of enriched mono-coreflective subcategory which makes precise the notion of a subcategory of local models.
A classification theorem for three different sorts of Maltsev categories is proven. The theorem provides a classification for Maltsev category, naturally Maltsev category, and weakly Maltsev category in terms of classifying classes of spans. The clas
s of all spans characterizes naturally Maltsev categories. The class of relations (i.e. jointly monomorphic spans) characterizes Maltsev categories. The class of strong relations (i.e. jointly strongly monomorphic spans) characterizes weakly Maltsev categories. The result is based on the uniqueness of internal categorical structures such as internal category and internal groupoid (Lawvere condition). The uniqueness of these structures is viewed as a property on their underlying reflexive graphs, restricted to the classifying spans. The class of classifying spans is combined, via a new compatibility condition, with split squares. This is analogous to orthogonality between spans and cospans. The result is a general classifying scheme which covers the main characterizations for Maltsev like categories. The class of positive relations has recently been shown to characterize Goursat categories and hence it is a new example that fits in this general scheme.
This paper presents the proof of the coherence theorem for Ann-categories whose set of axioms and original basic properties were given in [9]. Let $$A=(A,{Ah},c,(0,g,d),a,(1,l,r),{Lh},{Rh})$$ be an Ann-category. The coherence theorem states that in t
he category $ A$, any morphism built from the above isomorphisms and the identification by composition and the two operations $tx$, $ts$ only depends on its source and its target. The first coherence theorems were built for monoidal and symmetric monoidal categories by Mac Lane [7]. After that, as shown in the References, there are many results relating to the coherence problem for certain classes of categories. For Ann-categories, applying Hoang Xuan Sinhs ideas used for Gr-categories in [2], the proof of the coherence theorem is constructed by faithfully ``embedding each arbitrary Ann-category into a quite strict Ann-category. Here, a {it quite strict} Ann-categogy is an Ann-category whose all constraints are strict, except for the commutativity and left distributivity ones. This paper is the work continuing from [9]. If there is no explanation, the terminologies and notations in this paper mean as in [9].