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1. • CommentRowNumber1.
   • CommentAuthorAndrew Stacey
   • CommentTimeAug 24th 2012
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   Author: Andrew Stacey
   Format: MarkdownItexWe can think of vector spaces completely algebraically and then define "vector space objects" in arbitrary categories. However, if that category already has a copy of the real line, it is often preferred to insist that the multiplication maps (which in the algebraic presentation are unary) fit together to a morphism in the category $\mathbb{R} \times V \to V$. Viz [[topological vector space]]. Is there a standard name for this?

   We can think of vector spaces completely algebraically and then define “vector space objects” in arbitrary categories. However, if that category already has a copy of the real line, it is often preferred to insist that the multiplication maps (which in the algebraic presentation are unary) fit together to a morphism in the category $\mathbb{R} \times V \to V$. Viz topological vector space. Is there a standard name for this?

2. • CommentRowNumber2.
   • CommentAuthorZhen Lin
   • CommentTimeAug 24th 2012
   • (edited Aug 24th 2012)
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   Author: Zhen Lin
   Format: MarkdownItexSurely an internal vector space should be an internal field (whatever that is!) that acts on an internal abelian group? (This corresponds to the two-sorted axiomatisation of vector spaces.) Otherwise we would only be able to capture vector spaces over discrete fields...

   Surely an internal vector space should be an internal field (whatever that is!) that acts on an internal abelian group? (This corresponds to the two-sorted axiomatisation of vector spaces.) Otherwise we would only be able to capture vector spaces over discrete fields…

3. • CommentRowNumber3.
   • CommentAuthorTodd_Trimble
   • CommentTimeAug 24th 2012
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   Author: Todd_Trimble
   Format: MarkdownItexYou could just say e.g., "module over the internal ring $\mathbb{R}$ in $Top$".

   You could just say e.g., “module over the internal ring $\mathbb{R}$ in $Top$”.

4. • CommentRowNumber4.
   • CommentAuthorMike Shulman
   • CommentTimeAug 25th 2012
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   Author: Mike Shulman
   Format: MarkdownItex+1 for Todd's answer. It's usually better not to get into trying to decide what kind of "internal field" you mean, whereas internal rings are well-behaved and completely algebraic.

   +1 for Todd’s answer. It’s usually better not to get into trying to decide what kind of “internal field” you mean, whereas internal rings are well-behaved and completely algebraic.

5. • CommentRowNumber5.
   • CommentAuthorDavidRoberts
   • CommentTimeAug 25th 2012
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   Author: DavidRoberts
   Format: MarkdownItexModule objects would be the way to go, as the vector space axioms don't use the multiplicative group.

   Module objects would be the way to go, as the vector space axioms don’t use the multiplicative group.

6. • CommentRowNumber6.
   • CommentAuthorTobyBartels
   • CommentTimeAug 25th 2012
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   Author: TobyBartels
   Format: MarkdownItexAndrew's ‘completely algebraic’ definition of a real vector space uses an algebraic theory with one unary operation for each real number. In other words, this uses *external* real numbers. But to get (say) a topological vector space, then (as everybody else is saying) you must use *internal* real numbers. It's not at all clear to me that the term ‘real vector space object’ wouldn't mean the latter by default.

   Andrew’s ‘completely algebraic’ definition of a real vector space uses an algebraic theory with one unary operation for each real number. In other words, this uses external real numbers. But to get (say) a topological vector space, then (as everybody else is saying) you must use internal real numbers.

   It’s not at all clear to me that the term ‘real vector space object’ wouldn’t mean the latter by default.

7. • CommentRowNumber7.
   • CommentAuthorMike Shulman
   • CommentTimeAug 25th 2012
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   Author: Mike Shulman
   Format: MarkdownItexBut are the real numbers with their usual topology the "internal real numbers" in $Top$, for any general meaning of "internal real numbers"? I know they are the Dedekind real numbers object in the [[topological topos]]...

   But are the real numbers with their usual topology the “internal real numbers” in $Top$, for any general meaning of “internal real numbers”? I know they are the Dedekind real numbers object in the topological topos…

8. • CommentRowNumber8.
   • CommentAuthorTodd_Trimble
   • CommentTimeAug 25th 2012
   • (edited Aug 25th 2012)
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   Author: Todd_Trimble
   Format: MarkdownItexSo as not to get sidetracked by discussions of what "internal real numbers" are in some category or other, it might be better to amend my #2 to say something like, "Consider $\mathbb{R}$, the usual field of real numbers (in $Set$, if you want to be pedantic!), as a ring object or internal ring in $Top$. Then a topological vector space over $\mathbb{R}$ is by definition an internal module over $\mathbb{R}$." Obviously $\mathbb{R}$ here can be replaced by other things like [[local field]]s.

   So as not to get sidetracked by discussions of what “internal real numbers” are in some category or other, it might be better to amend my #2 to say something like, “Consider $\mathbb{R}$, the usual field of real numbers (in $Set$, if you want to be pedantic!), as a ring object or internal ring in $Top$. Then a topological vector space over $\mathbb{R}$ is by definition an internal module over $\mathbb{R}$.” Obviously $\mathbb{R}$ here can be replaced by other things like local fields.

9. • CommentRowNumber9.
   • CommentAuthorTobyBartels
   • CommentTimeAug 25th 2012
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   Author: TobyBartels
   Format: MarkdownItexBy ‘internal real numbers’ I meant something vaguer than a [[real numbers object]]; although it's interesting that we get what we want as the RNO in Johnstone's topos!

   By ‘internal real numbers’ I meant something vaguer than a real numbers object; although it’s interesting that we get what we want as the RNO in Johnstone’s topos!