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1. • CommentRowNumber1.
   • CommentAuthorAndrew Stacey
   • CommentTimeOct 2nd 2012
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   Author: Andrew Stacey
   Format: MarkdownItexI have a terminology question: suppose I have a cartesian closed category $\mathcal{S}$ and a functor $\mathfrak{F} \colon \mathcal{S} \to \mathcal{S}$. What do I call it when the induced set morphisms $\mathcal{S}(X,Y) \to \mathcal{S}(\mathfrak{F}(X), \mathfrak{F}(Y))$ lift to $\mathcal{S}$ morphisms between the internal homs? In my particular case there is only one choice for the lift, if that makes any difference (my expectation is that this means that any necessary coherences are trivially satisfied).

   I have a terminology question: suppose I have a cartesian closed category $\mathcal{S}$ and a functor $\mathfrak{F} \colon \mathcal{S} \to \mathcal{S}$. What do I call it when the induced set morphisms $\mathcal{S}(X,Y) \to \mathcal{S}(\mathfrak{F}(X), \mathfrak{F}(Y))$ lift to $\mathcal{S}$ morphisms between the internal homs?

   In my particular case there is only one choice for the lift, if that makes any difference (my expectation is that this means that any necessary coherences are trivially satisfied).

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeOct 2nd 2012
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   Author: Urs
   Format: MarkdownItex[[cartesian closed functor]]?

   cartesian closed functor?

3. • CommentRowNumber3.
   • CommentAuthorAndrew Stacey
   • CommentTimeOct 2nd 2012
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   Author: Andrew Stacey
   Format: MarkdownItexI looked at that and it seemed stronger than what I wanted. That has to preserve products and the morphism I'm talking about should be an isomorphism. It seemed more of an internal thing. I've now found the internal definition at [[functor]] which may be what I want (once I've unravelled all the definitions).

   I looked at that and it seemed stronger than what I wanted. That has to preserve products and the morphism I’m talking about should be an isomorphism. It seemed more of an internal thing. I’ve now found the internal definition at functor which may be what I want (once I’ve unravelled all the definitions).

4. • CommentRowNumber4.
   • CommentAuthorTodd_Trimble
   • CommentTimeOct 2nd 2012
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   Author: Todd_Trimble
   Format: MarkdownItexAndrew, the structure you need to lift to internal homs is called a [strength](http://ncatlab.org/nlab/show/tensorial+strength) (and functors equipped with strengths are called "strong"). You can also call such functors simply "$\mathcal{S}$-enriched".

   Andrew, the structure you need to lift to internal homs is called a strength (and functors equipped with strengths are called “strong”). You can also call such functors simply “$\mathcal{S}$-enriched”.

5. • CommentRowNumber5.
   • CommentAuthorTodd_Trimble
   • CommentTimeOct 2nd 2012
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   Author: Todd_Trimble
   Format: MarkdownItexBoy, that article [[tensorial strength]] could use some work.

   Boy, that article tensorial strength could use some work.

6. • CommentRowNumber6.
   • CommentAuthorAndrew Stacey
   • CommentTimeOct 2nd 2012
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   Author: Andrew Stacey
   Format: MarkdownItexRight, so the general principle that I can take from this is that if I'm dealing with a [[cartesian closed category]] and there's some concept for [[closed monoidal categories]] then I can apply it, using the cartesian product as the monoidal product. I'm so used to thinking of "monoidal" as "tensor product" (probably for symbolic reasons) that I forget that the cartesian product is a monoidal product.

   Right, so the general principle that I can take from this is that if I’m dealing with a cartesian closed category and there’s some concept for closed monoidal categories then I can apply it, using the cartesian product as the monoidal product. I’m so used to thinking of “monoidal” as “tensor product” (probably for symbolic reasons) that I forget that the cartesian product is a monoidal product.