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1. • CommentRowNumber1.
   • CommentAuthorPaoloPerrone
   • CommentTimeSep 6th 2017
   • (edited Sep 6th 2017)
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   Author: PaoloPerrone
   Format: MarkdownItexHello. I've been reading the article ["Stuff, structure, property"](https://ncatlab.org/nlab/show/stuff%2C+structure%2C+property), and related entries. I love the idea, and the "homotopical flavour" to it. Here are anyway some things that are not clear to me: 1. What happens at the 0-level, i.e. decategorified? I would say that an injective function forgets properties, just like a fully faithful functor. However, a non-injective function seems to forget "stuff", rather than structure, in the sense that identifies "things" that were different before applying the function. 2. What happens at the 2-level? For example take the forgetful 2-functor from the 2-category of monoidal categories to Cat. I wouldn't say that what is forgotten is "stuff", but rather the monoidal "structure" (maybe "2-structure"?), i.e. something entirely analogous to a monoid structure, just "one level up". (And then one could say that "2-structure" can carry "1-structure", like a braiding, and "1-structure" carries properties, like symmetry, etcetera.) So, it looks to me that the lowest (nontrivial) level is always some kind of "property", but the highest level always looks like some kind of "stuff", and the things that change are in between (I would call them "structure", "2-structure", and so on, and properties "0-structure".) Am I looking at this wrong? Or what would be examples where it's clear that: * the monoidal structure of a category (or anything coming from a non-idempotent 2-monad) is rather "stuff", and that * what a non-injective function forgets is not "stuff" but rather "structure"? Can anyone answer? I hope that the question is clear, please tell me if I should explain more, and forgive my possible mistakes!

   Hello. I’ve been reading the article “Stuff, structure, property”, and related entries. I love the idea, and the “homotopical flavour” to it.

   Here are anyway some things that are not clear to me:

   1. What happens at the 0-level, i.e. decategorified? I would say that an injective function forgets properties, just like a fully faithful functor. However, a non-injective function seems to forget “stuff”, rather than structure, in the sense that identifies “things” that were different before applying the function.
   2. What happens at the 2-level? For example take the forgetful 2-functor from the 2-category of monoidal categories to Cat. I wouldn’t say that what is forgotten is “stuff”, but rather the monoidal “structure” (maybe “2-structure”?), i.e. something entirely analogous to a monoid structure, just “one level up”. (And then one could say that “2-structure” can carry “1-structure”, like a braiding, and “1-structure” carries properties, like symmetry, etcetera.)

   So, it looks to me that the lowest (nontrivial) level is always some kind of “property”, but the highest level always looks like some kind of “stuff”, and the things that change are in between (I would call them “structure”, “2-structure”, and so on, and properties “0-structure”.) Am I looking at this wrong? Or what would be examples where it’s clear that:

   • the monoidal structure of a category (or anything coming from a non-idempotent 2-monad) is rather “stuff”, and that
   • what a non-injective function forgets is not “stuff” but rather “structure”?

   Can anyone answer? I hope that the question is clear, please tell me if I should explain more, and forgive my possible mistakes!

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeSep 6th 2017
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   Author: Urs
   Format: MarkdownItexOn the level of functions of sets, there is not much to be seen regarding stuff, structure and property. Intuitively, the reason that functors can know so much about stuff, structure and property is because they see the web of homomorphisms between mathematical structures (aka their category), and because of the key insight of category theory: What some mathematical structure (object) really _is_ may be read off from just the web of homomorphisms that it participates in (aka the ambient category).

   On the level of functions of sets, there is not much to be seen regarding stuff, structure and property.

   Intuitively, the reason that functors can know so much about stuff, structure and property is because they see the web of homomorphisms between mathematical structures (aka their category), and because of the key insight of category theory: What some mathematical structure (object) really is may be read off from just the web of homomorphisms that it participates in (aka the ambient category).

3. • CommentRowNumber3.
   • CommentAuthorDavid_Corfield
   • CommentTimeSep 6th 2017
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   Author: David_Corfield
   Format: MarkdownItex>* what a non-injective function forgets is not "stuff" but rather "structure"? If you're treating this as a set (0-groupoid) fibred over another set, you shouldn't consider the whole fibre, but rather an element of a fibre. In the original case of, say, the underlying functor $Grp$ to $Set$, saying that this forgets structure is not saying the fibre over a particular set forgets structure, but that a particular element, i.e., a group, forgets its structure. Consider the absolute value function $\mathbb{Z} \to \mathbb{N}$. Both $+3$ and $-3$ are mapped to $3$ and in the process both forget their sign.

   • what a non-injective function forgets is not “stuff” but rather “structure”?

   If you’re treating this as a set (0-groupoid) fibred over another set, you shouldn’t consider the whole fibre, but rather an element of a fibre. In the original case of, say, the underlying functor $Grp$ to $Set$, saying that this forgets structure is not saying the fibre over a particular set forgets structure, but that a particular element, i.e., a group, forgets its structure.

   Consider the absolute value function $\mathbb{Z} \to \mathbb{N}$. Both $+3$ and $-3$ are mapped to $3$ and in the process both forget their sign.

4. • CommentRowNumber4.
   • CommentAuthorMike Shulman
   • CommentTimeSep 6th 2017
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   Author: Mike Shulman
   Format: MarkdownItexRight, David answered your first question: an integer is a natural number equipped with extra structure, namely a sign if it's nonzero. For the second question, at higher dimensions it's not obvious what the best naming conventions are. I can see an argument for "property, 1-structure, 2-structure, stuff"), if you're thinking of "stuff" as the "most" that can be forgotten. But if you regard $n$-categories as particular $(n+1)$-categories with only identity $(n+1)$-morphisms, then it's clear that it has to be "property, structure, stuff, 2-stuff" for 2-categories in order to specialize correctly to 1-categories regarded as degenerate 2-categories. Intuitively, "structure" is composed of *elements* of sets, whereas "stuff" is composed of sets themselves (or more generally objects of categories), and 2-stuff is composed of categories (objects of 2-categories), and so on. The monoidal structure of a category consists of, among other things, a functor $C\times C\to C$, which assigns an object of the category $C$ to each pair of objects; thus it is composed of objects of categories in the same way that the multiplication of a group is composed of elements of sets. An example of forgetting 2-stuff is the forgetful functor $Cat\times Cat\to Cat$, where what is forgotten is a whole *category*, not just an object of one.

   Right, David answered your first question: an integer is a natural number equipped with extra structure, namely a sign if it’s nonzero.

   For the second question, at higher dimensions it’s not obvious what the best naming conventions are. I can see an argument for “property, 1-structure, 2-structure, stuff”), if you’re thinking of “stuff” as the “most” that can be forgotten. But if you regard $n$-categories as particular $(n+1)$-categories with only identity $(n+1)$-morphisms, then it’s clear that it has to be “property, structure, stuff, 2-stuff” for 2-categories in order to specialize correctly to 1-categories regarded as degenerate 2-categories. Intuitively, “structure” is composed of elements of sets, whereas “stuff” is composed of sets themselves (or more generally objects of categories), and 2-stuff is composed of categories (objects of 2-categories), and so on. The monoidal structure of a category consists of, among other things, a functor $C\times C\to C$, which assigns an object of the category $C$ to each pair of objects; thus it is composed of objects of categories in the same way that the multiplication of a group is composed of elements of sets. An example of forgetting 2-stuff is the forgetful functor $Cat\times Cat\to Cat$, where what is forgotten is a whole category, not just an object of one.

5. • CommentRowNumber5.
   • CommentAuthorPaoloPerrone
   • CommentTimeSep 6th 2017
   • (edited Sep 6th 2017)
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   Author: PaoloPerrone
   Format: MarkdownItex> Consider the absolute value function ℤ→ℕ. Both +3 and -3 are mapped to 3 and in the process both forget their sign. That's precisely the example I needed! Thank you so much. > Intuitively, “structure” is composed of elements of sets, whereas “stuff” is composed of sets themselves (or more generally objects of categories), and 2-stuff is composed of categories (objects of 2-categories), and so on. I see. So I guess I was seeing as "stuff" what instead should be "n-stuff" (for n-categories). I think that for my own intuition, the most suggestive word is "2-structure", and then "n+1-structure" for n-stuff. But that's just personal taste, as long as one can translate between the two (as in "higher stack vs. higher sheaf"). Great, thanks!

   Consider the absolute value function ℤ→ℕ. Both +3 and -3 are mapped to 3 and in the process both forget their sign.

   That’s precisely the example I needed! Thank you so much.

   Intuitively, “structure” is composed of elements of sets, whereas “stuff” is composed of sets themselves (or more generally objects of categories), and 2-stuff is composed of categories (objects of 2-categories), and so on.

   I see. So I guess I was seeing as “stuff” what instead should be “n-stuff” (for n-categories). I think that for my own intuition, the most suggestive word is “2-structure”, and then “n+1-structure” for n-stuff. But that’s just personal taste, as long as one can translate between the two (as in “higher stack vs. higher sheaf”). Great, thanks!

6. • CommentRowNumber6.
   • CommentAuthorOscar_Cunningham
   • CommentTimeSep 18th 2018
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   Author: Oscar_Cunningham
   Format: MarkdownItexI have another question about stuff. A typical example of an eso and full functor is the one from the category of topological spaces and continuous maps to the category of topological spaces and equivalence classes of continuous maps under homotopy. So it should be possible to think of this functor as forgetting stuff. What's the stuff in this case? By Yoneda, two maps are homotopy equivalent if they agree on homotopy-equivalent-classes-of-maps-into-the-domain (or even just maps of spheres, by Whitehead's theorem). So we could think of the functor as forgetting the points of the space and only remembering these classes. But that seems very artificial. Is there a nice way to define "elements of a topological space" so that we can get the homotopy equivalence classes by restricting to a subset of them? Of course there might not be a nice answer, but it would be good if there was since the homotopy category is a classic example of a quotient category.

   I have another question about stuff.

   A typical example of an eso and full functor is the one from the category of topological spaces and continuous maps to the category of topological spaces and equivalence classes of continuous maps under homotopy. So it should be possible to think of this functor as forgetting stuff. What’s the stuff in this case?

   By Yoneda, two maps are homotopy equivalent if they agree on homotopy-equivalent-classes-of-maps-into-the-domain (or even just maps of spheres, by Whitehead’s theorem). So we could think of the functor as forgetting the points of the space and only remembering these classes. But that seems very artificial.

   Is there a nice way to define “elements of a topological space” so that we can get the homotopy equivalence classes by restricting to a subset of them?

   Of course there might not be a nice answer, but it would be good if there was since the homotopy category is a classic example of a quotient category.

7. • CommentRowNumber7.
   • CommentAuthorMike Shulman
   • CommentTimeSep 18th 2018
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   Author: Mike Shulman
   Format: MarkdownItexI would probably call the stuff that it forgets something like "a topological presentation of a given homotopy type".

   I would probably call the stuff that it forgets something like “a topological presentation of a given homotopy type”.

8. • CommentRowNumber8.
   • CommentAuthorUrs
   • CommentTimeSep 20th 2018
   • (edited Sep 20th 2018)
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   Author: Urs
   Format: MarkdownItexBy the way, in which generality do localization functors happen to be full functors? Should be true as soon as there is some minimum of calculus of fractions available, certainly for the functors on homotopy categories induced by localization of model categories. There must be some general discussion about localization and fullness, somewhere?

   By the way, in which generality do localization functors happen to be full functors?

   Should be true as soon as there is some minimum of calculus of fractions available, certainly for the functors on homotopy categories induced by localization of model categories.

   There must be some general discussion about localization and fullness, somewhere?

9. • CommentRowNumber9.
   • CommentAuthorUrs
   • CommentTimeSep 20th 2018
   • (edited Sep 20th 2018)
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   Author: Urs
   Format: MarkdownItex[ hm, the "should be true" part was wrong... ]

   [ hm, the “should be true” part was wrong… ]