论文标题
双函数的双函数定理和严格的张量产品，用于双重功能的双重类别
Bifunctor Theorem and strictification tensor product for double categories with lax double functors
论文作者
论文摘要
我们介绍了$ dbl^{st} _ {lx} $的内HOM的候选人，严格的双类别类别和LAX Double Double Foundors的类别，并将Lax Double Double Foundor表征到获得LAX Double Quasi punctor中。后者由一对宽松的双函子和四个类似于分配定律的宽松双函子组成。我们将此特征扩展到2类同构$ q \ x \ lax_ {hop}（\ aa \ times \ bb，\ cc）\ iso \ iSo \ lax_ {hop}（\ aa，\ aa，\ llbracket \ bb，\ bb，\ bb，\ cc \ rrbrabracket）$。我们表明，我们在$ dbl^{st} _ {lx} $中而不是灰色单型产品中，我们获得了一种限制Lax Double Quasi-functors的产品。我们证明了一个双肢定理，通过该定理，某些类型的松弛双准函数会产生笛卡尔产品上的Lax Double Foundor，将其扩展到2个函数$ q \ x \ lax_ {hop}^{ns}（\ aa \ times \ bb，\ cc）\ to \ lax_ {hop}（\ aa \ aa \ times \ times \ bb，\ cc）$，并显示如何限制biequivalence。研究（联合国）咖喱2个功能。我们证明，来自Trivial Double类别的Lax Double Foundor是Codomain Double类别中的单元，并证明我们的上述2函数以$ q \ x \ x \ x \ lax_ {hop}（*\ times*，\ dd）\ to \ lax_ \ to \ lax_ {hop}（hop}（*，\ dd）$恢复$ \ comp（\ hh（\ dd））：\ mnd \ mnd（\ hh（\ dd））\ to \ mnd（\ hh（\ dd））$ \ comp $ \ comp $ the Street引入的$ \ comp $，在$ \ hh（\ dd）$是$ \ hh（\ dd）的情况
We introduce a candidate for the inner hom for $Dbl^{st}_{lx}$, the category of strict double categories and lax double functors, and characterize a lax double functor into it obtaining a lax double quasi-functor. The latter consists of a pair of lax double functors with four 2-cells resembling distributive laws. We extend this characterization to a 2-category isomorphism $q\x\Lax_{hop}(\Aa\times\Bb,\Cc) \iso \Lax_{hop}(\Aa, \llbracket\Bb,\Cc\rrbracket)$. We show that instead of a Gray monoidal product in $Dbl^{st}_{lx}$ we obtain a product that in a sense strictifies lax double quasi-functors. We prove a bifunctor theorem by which certain type of lax double quasi-functors give rise to lax double functors on the Cartesian product, extend it to a 2-functor $q\x\Lax_{hop}^{ns}(\Aa\times\Bb,\Cc)\to\Lax_{hop}(\Aa\times\Bb,\Cc)$ and show how it restricts to a biequivalence. The (un)currying 2-functors are studied. We prove that a lax double functor from the trivial double category is a monad in the codomain double category, and show that our above 2-functor in the form $q\x\Lax_{hop}(*\times *,\Dd)\to\Lax_{hop}(*,\Dd)$ recovers the specification $\Comp(\HH(\Dd)):\Mnd\Mnd(\HH(\Dd))\to\Mnd(\HH(\Dd))$ of the natural transformation $\Comp$ introduced by Street, where $\HH(\Dd)$ is the horizontal 2-category of $\Dd$.
