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What is the monoidal equivalent of a locally cartesian closed category?

If a closed monoidal category is the monoidal equivalent of a Cartesian closed category, is there an analogous equivalent for locally cartesian closed categories? Is there a standard terminology or ...
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Suppose that $C$ is a complete closed monoidal category and $I$ is any small category. Then the functor category $Fun(I,C)$ is again a closed monoidal category with the pointwise tensor product $F \... 1answer 384 views Cartesian closed category Let$\bf{C}$be a category with finite products. (1) An object$X$of$\bf{C}$is called cartesian if the functor$(-)\times X$has a right adjoint. (2) A morphism$s:X\rightarrow B$... 1answer 205 views Product in the free CCC over a CCC When you start with a CCC$C$, take the underlying graph of$C$via the forgetful$U : Cat \to Graph$, and then construct the free CCC over$U(C)$via$Free : Graph \to Cat$: what's the relationship ... 3answers 656 views When is the projective model structure cartesian? When is the internal hom invariant? If M is a sufficiently nice model category and D is a small category then there are two natural model structures we can impose on the functor category$Fun(D,M)$where the weak equivalences are the ... 2answers 318 views A (too?) simple notion of “closed multicategory” Suppose I define a multicategory$M=(Ob(M),Hom_M)$to be simply closed if for every sequence$S=(b_1,\ldots,b_n;x)$of$n+1$objects in$M$, we provide an object$Exp(S)\in Ob(M)$, and for every ... 1answer 274 views Example of a non-closed cocomplete symmetric monoidal category Background By a cocomplete symmetric monoidal category$C$I mean a symmetric monoidal category whose underlying category is cocomplete and such that$- \otimes X : C \to C$is cocontinuous for all$...
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Take a Cartesian (or monoidal) closed category; define Reader monad for a given object $E$ as $X \mapsto X^E$; and take a strong monad $M$ (strong means preserves product or tensor product). Now the ...
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Exponentiable objects in a category, valued in a larger, containing category

Recall that when dealing with topological spaces one usually likes dealing with a subcategory of $Top$ which is convenient, one facet of which is that it is cartesian closed. However to get to a ...
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Do Categorical Quotients Preserve Covering Maps?

Before asking a question, please let me write down settings. SETTINGS: Let $C$ be a category with fiber products and $B$ be a closed subcategory of $C$ (i.e. $B$ contains any isomorphism of $C$, and ...
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