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2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry book bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics complex complex-geometry computable-mathematics computer-science constructive cosmology definitions deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration foundation foundations functional-analysis functor gauge-theory gebra geometric-quantization geometry graph graphs gravity grothendieck group group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology nlab noncommutative noncommutative-geometry number-theory object of operads operator operator-algebra order-theory pages pasting philosophy physics pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes set set-theory sheaf simplicial space spin-geometry stable-homotopy-theory stack string string-theory superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory tqft type type-theory universal variational-calculus

1. • CommentRowNumber1.
   • CommentAuthorMike Shulman
   • CommentTimeJun 6th 2010
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   Author: Mike Shulman
   Format: MarkdownItexCreated [[paracategory]] and [[Kleene equality]].

   Created paracategory and Kleene equality.

2. • CommentRowNumber2.
   • CommentAuthorTobyBartels
   • CommentTimeJun 6th 2010
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   Author: TobyBartels
   Format: MarkdownItexThanks for [[Kleene equality]]; I have added some examples to it. (Teaching junior-college algebra makes one think about this sort of thing, at least if one has a mind like mine.)

   Thanks for Kleene equality; I have added some examples to it. (Teaching junior-college algebra makes one think about this sort of thing, at least if one has a mind like mine.)

3. • CommentRowNumber3.
   • CommentAuthorMike Shulman
   • CommentTimeJun 6th 2010
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   Author: Mike Shulman
   Format: MarkdownItexIs there a word for an equality which means "if the LHS is defined, then so is the RHS and they are equal"?

   Is there a word for an equality which means “if the LHS is defined, then so is the RHS and they are equal”?

4. • CommentRowNumber4.
   • CommentAuthorTodd_Trimble
   • CommentTimeJun 6th 2010
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   Author: Todd_Trimble
   Format: TextFreyd calls it "directed equality" on page 3 of Categories, Allegories.

   Freyd calls it "directed equality" on page 3 of Categories, Allegories.
5. • CommentRowNumber5.
   • CommentAuthorTobyBartels
   • CommentTimeJun 6th 2010
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   Author: TobyBartels
   Format: MarkdownItexDirected equality is particularly important for rewriting rules. So if you ever see $x^2/x$ and are confident that it arises legitimately (that is, that it must be defined in the context at hand, or it would never have been written down), then you are allowed to simplify it to $x$. But if you write down $x^2/x$ yourself, without first checking that it is defined, then you do not know that it is equivalent to $x$ (but only to $x, x \ne 0$).

   Directed equality is particularly important for rewriting rules. So if you ever see $x^2/x$ and are confident that it arises legitimately (that is, that it must be defined in the context at hand, or it would never have been written down), then you are allowed to simplify it to $x$. But if you write down $x^2/x$ yourself, without first checking that it is defined, then you do not know that it is equivalent to $x$ (but only to $x, x \ne 0$).

6. • CommentRowNumber6.
   • CommentAuthorMike Shulman
   • CommentTimeJun 6th 2010
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   Author: Mike Shulman
   Format: MarkdownItex"Directed equality" isn't quite ideally evocative for me, although I see the intent, but calling it a "rewrite rule" is pretty good.

   “Directed equality” isn’t quite ideally evocative for me, although I see the intent, but calling it a “rewrite rule” is pretty good.

7. • CommentRowNumber7.
   • CommentAuthorTobyBartels
   • CommentTimeJun 7th 2010
   • (edited Jun 7th 2010)
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   Author: TobyBartels
   Format: MarkdownItexBut the proposition that $A = B$ if $A$ is defined is different from the rewrite rule that $A$ can be replaced by $B$. Rewrite rules are usually rules for simplifying expressions, or at least for getting them into a more useful form, and one is selective about them. Given a collection of terms and a collection of rewrite rules between them, there is a derived notion of equality (two terms are equal iff they can be rewritten to the same result), which is not the same thing; and even this (which is symmetric after all) doesn't match the *directed* equality. For example, nobody would propose $7 \rightsquigarrow 3 + 4$ as a rewrite rule for elementary arithmetic, even though adding it to the usual rewrite rules would preserve the derived notion of equality and $7 \gt\!\!\!= 3 + 4$ (there should be no space at all between ‘$\gt$’ and ‘$=$’, to make a single symbol for directed equality) is certainly true; that's because it's not a useful step to take.

   But the proposition that $A = B$ if $A$ is defined is different from the rewrite rule that $A$ can be replaced by $B$. Rewrite rules are usually rules for simplifying expressions, or at least for getting them into a more useful form, and one is selective about them. Given a collection of terms and a collection of rewrite rules between them, there is a derived notion of equality (two terms are equal iff they can be rewritten to the same result), which is not the same thing; and even this (which is symmetric after all) doesn’t match the directed equality.

   For example, nobody would propose $7 \rightsquigarrow 3 + 4$ as a rewrite rule for elementary arithmetic, even though adding it to the usual rewrite rules would preserve the derived notion of equality and $7 \gt\!\!\!= 3 + 4$ (there should be no space at all between ‘$\gt$’ and ‘$=$’, to make a single symbol for directed equality) is certainly true; that’s because it’s not a useful step to take.

8. • CommentRowNumber8.
   • CommentAuthorTobyBartels
   • CommentTimeJun 7th 2010
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   Author: TobyBartels
   Format: MarkdownItexI added a note about models which also alludes to directed equality.

   I added a note about models which also alludes to directed equality.

9. • CommentRowNumber9.
   • CommentAuthorMike Shulman
   • CommentTimeJun 7th 2010
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   Author: Mike Shulman
   Format: MarkdownItexOk, fair enough.

   Ok, fair enough.