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2-categories 2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry bundles calculus categories category category-theory chern-weil-theory cohesion cohesive-homotopy-theory cohesive-homotopy-type-theory cohomology colimits combinatorics complex-geometry computable-mathematics computer-science constructive constructive-mathematics cosmology definitions deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry differential-topology digraphs duality elliptic-cohomology enriched fibration finite foundations functional-analysis functor gauge-theory gebra geometric-quantization geometry goodwillie-calculus graph graphs gravity grothendieck 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 infinity integration integration-theory k-theory lie lie-theory limit limits linear linear-algebra locale localization logic manifolds mathematics measure-theory modal-logic model model-category-theory monads monoidal monoidal-category-theory morphism motives motivic-cohomology multicategories nonassociative noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory 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 string-theory subobject superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory type type-theory universal variational-calculus

1. • CommentRowNumber1.
   • CommentAuthorTodd_Trimble
   • CommentTimeSep 9th 2012
   • (edited Sep 9th 2012)
   • PermaLink
   Author: Todd_Trimble
   Format: MarkdownItexI added to the "abstract nonsense" section in [[free monoid]] a helpful general observation on how to construct free monoids. "Adjoint functor theorem" is overkill for free monoids over $Set$.

   I added to the “abstract nonsense” section in free monoid a helpful general observation on how to construct free monoids. “Adjoint functor theorem” is overkill for free monoids over $Set$.

2. • CommentRowNumber2.
   • CommentAuthorRodMcGuire
   • CommentTimeSep 9th 2012
   • PermaLink
   Author: RodMcGuire
   Format: MarkdownItexummm, shouldn't >Then a left adjoint to the forgetful functor $Mon(C) \to C$ exists, taking an object $c$ to $$\sum_{n \geq 0} c^{\otimes n},$$ which thereby becomes the free monoid on $C$. really be >Then a left adjoint to the forgetful functor $Mon(C) \to C$ exists, taking $C$ to $$\sum_{n \geq 0} C^{\otimes n},$$ which thereby becomes the free monoid on $C$. an object $c$ of $C$ is not involved.

   ummm, shouldn’t

   Then a left adjoint to the forgetful functor $Mon(C) \to C$ exists, taking an object $c$ to

   $$\sum_{n \geq 0} c^{\otimes n},$$

   which thereby becomes the free monoid on $C$.

   really be

   Then a left adjoint to the forgetful functor $Mon(C) \to C$ exists, taking $C$ to

   $$\sum_{n \geq 0} C^{\otimes n},$$

   which thereby becomes the free monoid on $C$.

   an object $c$ of $C$ is not involved.

3. • CommentRowNumber3.
   • CommentAuthorTodd_Trimble
   • CommentTimeSep 9th 2012
   • (edited Sep 9th 2012)
   • PermaLink
   Author: Todd_Trimble
   Format: MarkdownItexummm... no. There was one typo in what I wrote: that should have been a lower-case $c$ before the period. I'll go fix that. (Edit: done.)

   ummm… no. There was one typo in what I wrote: that should have been a lower-case $c$ before the period. I’ll go fix that. (Edit: done.)

4. • CommentRowNumber4.
   • CommentAuthorTobyBartels
   • CommentTimeSep 9th 2012
   • PermaLink
   Author: TobyBartels
   Format: MarkdownItexThe adjoint functor theorem is useful to do the proofs. (None of the constructions currently come with proofs that they are what we claim they are.)

   The adjoint functor theorem is useful to do the proofs. (None of the constructions currently come with proofs that they are what we claim they are.)

5. • CommentRowNumber5.
   • CommentAuthorTodd_Trimble
   • CommentTimeSep 10th 2012
   • PermaLink
   Author: Todd_Trimble
   Format: MarkdownItexShould I include a proof of the theorem I quoted? (Hm, not sure I *really* want to put myself out there, but I'll ask anyway.)

   Should I include a proof of the theorem I quoted? (Hm, not sure I really want to put myself out there, but I’ll ask anyway.)

6. • CommentRowNumber6.
   • CommentAuthorTobyBartels
   • CommentTimeSep 10th 2012
   • PermaLink
   Author: TobyBartels
   Format: MarkdownItexI don't think that it's necessary now. It might be better to leave the proofs for somebody who doesn't find the result obvious and wants to write down what they think of. (That's usually what I do … not that I always find proofs obvious when I leave them out if I'm quoting the results from elsewhere.)

   I don’t think that it’s necessary now. It might be better to leave the proofs for somebody who doesn’t find the result obvious and wants to write down what they think of. (That’s usually what I do … not that I always find proofs obvious when I leave them out if I’m quoting the results from elsewhere.)