论文标题
通过跨度和矩阵中的内在化和富集
Internalization and enrichment via spans and matrices in a tricategory
论文作者
论文摘要
我们介绍了tricateGory $ \ bicat_3 $ bicateGiores，伪函数，伪自然变换和修改的tricategory $ \ bicat_3 $中的$ \ m $和$§$内部类别的类别。他们的横向三角形是矩阵和跨度$ v $的三角形。内部和富集的结构都是1类中相应构造的三角形。遵循\ cite {fgk}，我们在三曲酸中内部的类别中引入了单元及其垂直形态。当矩阵的内部类别$ \ m $和跨越1分的三件三角$ v $等效时，我们证明了同等的条件，并且在这种情况下，在这种情况下，它们的（严格）和垂直的monad形态的相应类别也相当于。我们证明，后一种类别对于分别在$ v $中富含和离散的类别的类别是同构的。作为我们三角构结构的副产品，我们从\ cite {fem}中恢复了一些结果。截断为1类别，我们从\ cite {cfp}和\ cite {ehr}恢复结果，对富集和离散的内部1类类别的等效性。
We introduce categories $\M$ and $§$ internal in the tricategory $\Bicat_3$ of bicategories, pseudofunctors, pseudonatural transformations and modifications, for matrices and spans in a 1-strict tricategory $V$. Their horizontal tricategories are the tricategories of matrices and spans in $V$. Both the internal and the enriched constructions are tricategorifications of the corresponding constructions in 1-categories. Following \cite{FGK} we introduce monads and their vertical morphisms in categories internal in tricategories. We prove an equivalent condition for when the internal categories for matrices $\M$ and spans $§$ in a 1-strict tricategory $V$ are equivalent, and deduce that in that case their corresponding categories of (strict) monads and vertical monad morphisms are equivalent, too. We prove that the latter categories are isomorphic to those of categories enriched and discretely internal in $V$, respectively. As a byproduct of our tricategorical constructions we recover some results from \cite{Fem}. Truncating to 1-categories we recover results from \cite{CFP} and \cite{Ehr} on the equivalence of enriched and discretely internal 1-categories.
