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1. • CommentRowNumber1.
   • CommentAuthorTodd_Trimble
   • CommentTimeDec 2nd 2013
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   Author: Todd_Trimble
   Format: MarkdownItexThe definition of [[concrete category]] includes the condition that $U: C \to Set$ is representable, but then says that many authors do not include that condition. The characterization theorem in the article also excludes that condition. I'm not all that convinced that the term should default to the notion that includes a representation $U \cong \hom(c, -)$. In so much of the literature, for example in the work of Freyd and of Isbell and in the observation that the homotopy category is not concrete, this condition is not included. Also we have the result that if $C$ is concretizable, then so is $C^{op}$, but this doesn't work if we include representability. I might propose that we reserve "concrete" for the weaker notion, and use "representably concrete" for the stronger notion. I think it would make for less awkwardness.

   The definition of concrete category includes the condition that $U: C \to Set$ is representable, but then says that many authors do not include that condition. The characterization theorem in the article also excludes that condition.

   I’m not all that convinced that the term should default to the notion that includes a representation $U \cong \hom(c, -)$. In so much of the literature, for example in the work of Freyd and of Isbell and in the observation that the homotopy category is not concrete, this condition is not included. Also we have the result that if $C$ is concretizable, then so is $C^{op}$, but this doesn’t work if we include representability.

   I might propose that we reserve “concrete” for the weaker notion, and use “representably concrete” for the stronger notion. I think it would make for less awkwardness.

2. • CommentRowNumber2.
   • CommentAuthorMike Shulman
   • CommentTimeDec 2nd 2013
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   Author: Mike Shulman
   Format: MarkdownItexI agree.

   I agree.

3. • CommentRowNumber3.
   • CommentAuthorTodd_Trimble
   • CommentTimeDec 2nd 2013
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   Author: Todd_Trimble
   Format: MarkdownItexThanks, Mike. I went ahead and made some changes (and additions and corrections) to [[concrete category]], but please speak up if there are any objections.

   Thanks, Mike.

   I went ahead and made some changes (and additions and corrections) to concrete category, but please speak up if there are any objections.

4. • CommentRowNumber4.
   • CommentAuthorMike Shulman
   • CommentTimeDec 3rd 2013
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   Author: Mike Shulman
   Format: MarkdownItexLooks good to me, thanks!

   Looks good to me, thanks!

5. • CommentRowNumber5.
   • CommentAuthorUrs
   • CommentTimeMar 31st 2016
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   Author: Urs
   Format: MarkdownItexIt looks like the discussion at _[[concrete category]]_ was wanting to point to this here, I have added pointer now: * [[Peter Freyd]], _Homotopy is not concrete_, in _The Steenrod Algebra and its Applications_, Springer Lecture Notes in Mathematics Vol. 168, Springer-Verlag, 1970, Republished in: Reprints in Theory and Applications of Categories, No. 6 (2004) pp 1-10 ([web](http://www.tac.mta.ca/tac/reprints/articles/6/tr6abs.html))

   It looks like the discussion at concrete category was wanting to point to this here, I have added pointer now:

   • Peter Freyd, Homotopy is not concrete, in The Steenrod Algebra and its Applications, Springer Lecture Notes in Mathematics Vol. 168, Springer-Verlag, 1970, Republished in: Reprints in Theory and Applications of Categories, No. 6 (2004) pp 1-10 (web)