Maps (left adjoint arrows) between Frobenius objects in a cartesian bicategory B are precisely comonoid homomorphisms and, for A Frobenius and any T in B, map(B)(T,A) is a groupoid.
We fix any bicategory $mathscr{A}$ together with a class of morphisms $mathbf{W}_{mathscr{A}}$, such that there is a bicategory of fractions $mathscr{A}[mathbf{W}_{mathscr{A}}^{-1}]$. Given another such pair $(mathscr{B},mathbf{W}_{mathscr{B}})$ and
any pseudofunctor $mathcal{F}:mathscr{A}rightarrowmathscr{B}$, we find necessary and sufficient conditions in order to have an induced equivalence of bicategories from $mathscr{A}[mathbf{W}_{mathscr{A}}^{-1}]$ to $mathscr{B}[mathbf{W}_{mathscr{B}}^{-1}]$. In particular, this gives necessary and sufficient conditions in order to have an equivalence from any bicategory of fractions $mathscr{A}[mathbf{W}_{mathscr{A}}^{-1}]$ to any given bicategory $mathscr{B}$.
We fix any bicategory $mathscr{A}$ together with a class of morphisms $mathbf{W}_{mathscr{A}}$, such that there is a bicategory of fractions $mathscr{A}[mathbf{W}_{mathscr{A}}^{-1}]$. Given another such pair $(mathscr{B},mathbf{W}_{mathscr{B}})$ and
any pseudofunctor $mathcal{F}:mathscr{A}rightarrowmathscr{B}$, we find necessary and sufficient conditions in order to have an induced pseudofunctor $mathcal{G}:mathscr{A}[mathbf{W}_{mathscr{A}}^{-1}]rightarrow mathscr{B}[mathbf{W}_{mathscr{B}}^{-1}]$. Moreover, we give a simple description of $mathcal{G}$ in the case when the class $mathbf{W}_{mathscr{B}}$ is right saturated.
We revisit the definition of Cartesian differential categories, showing that a slightly more general version is useful for a number of reasons. As one application, we show that these general differential categories are comonadic over Cartesian catego
ries, so that every Cartesian category has an associated cofree differential category. We also work out the corresponding results when the categories involved have restriction structure, and show that these categories are closed under splitting restriction idempotents.
Cartesian differential categories were introduced to provide an abstract axiomatization of categories of differentiable functions. The fundamental example is the category whose objects are Euclidean spaces and whose arrows are smooth maps. Tensor d
ifferential categories provide the framework for categorical models of differential linear logic. The coKleisli category of any tensor differential category is always a Cartesian differential category. Cartesian differential categories, besides arising in this manner as coKleisli categories, occur in many different and quite independent ways. Thus, it was not obvious how to pass from Cartesian differential categories back to tensor differential categories. This paper provides natural conditions under which the linear maps of a Cartesian differential category form a tensor differential category. This is a question of some practical importance as much of the machinery of modern differential geometry is based on models which implicitly allow such a passage, and thus the results and tools of the area tend to freely assume access to this structure. The purpose of this paper is to make precise the connection between the two types of differential categories. As a prelude to this, however, it is convenient to have available a general theory which relates the behaviour of linear maps in Cartesian categories to the structure of Seely categories. The latter were developed to provide the categorical semantics for (fragments of) linear logic which use a storage modality. The general theory of storage, which underlies the results mentioned above, is developed in the opening sections of the paper and is then applied to the case of differential categories.
This paper proves three different coherence theorems for symmetric monoidal bicategories. First, we show that in a free symmetric monoidal bicategory every diagram of 2-cells commutes. Second, we show that this implies that the free symmetric monoida
l bicategory on one object is equivalent, as a symmetric monoidal bicategory, to the discrete symmetric monoidal bicategory given by the disjoint union of the symmetric groups. Third, we show that every symmetric monoidal bicategory is equivalent to a strict one. We give two topological applications of these coherence results. First, we show that the classifying space of a symmetric monoidal bicategory can be equipped with an E_{infty} structure. Second, we show that the fundamental 2-groupoid of an E_n space, n geq 4, has a symmetric monoidal structure. These calculations also show that the fundamental 2-groupoid of an E_3 space has a sylleptic monoidal structure.