Zhen Lin

general recent conversations

$Mathematics$ Discussion on question by Rosélia Mah…

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Zhen Lin

Apr 20 07:59

"Legitimate" is a technical term in some circles. It means exactly what they say: the hom-sets are small. Or, more generally, the hom-objects are in $\mathcal{V}$, for whatever $\mathcal{V}$ they are using as the base of enrichment.

Zhen Lin

Apr 20 07:59

Thank you. I was not aware of this paper. This settles the matter for presheaves but the case of sheaves remains unclear to me. (Is a small sheaf a sheaf that happens to be a small presheaf? If so then it is not obvious that the category of small sheaves is cocomplete.)

Zhen Lin

Apr 20 07:59

@DmitriP. The category of small sheaves is cocomplete by definition (more or less) but it's not obvious to me that it is complete or has exponentials...

Zhen Lin

Apr 20 07:59

To be a convenient category of topological spaces the objects must, in the first place, be topological spaces. This is not so for the category of locales or any category of sheaves on the category locales.

$Mathematics$ Discussion on question by HomotopyStu…

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Zhen Lin

Apr 11 20:50

I am pretty sure the notion of homotopy limit before 1970 would not have been based on derived functors or categories with weak equivalences but rather explicit constructions with path spaces and so on.

Discussion on question by Vincent Gra…

Imported from a comment discussion on mathoverflow.net/questio...

Zhen Lin

May 4, 2023 22:53

That can be made into a precise claim which the number theorists probably have an answer for. You might want to see what they say first.

Zhen Lin

May 4, 2023 22:53

I’m not sure that counts as “random”…

$Mathematics$ Discussion on question by Seth: Meani…

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Zhen Lin

Apr 27, 2023 23:08

Category theory is not even a hundred years old yet. You can’t expect a new subject where even the practitioners can’t agree on terminology or definitions to be as well covered as undergraduate basics like linear algebra or group theory.

Zhen Lin

Apr 27, 2023 23:08

No, because you have not replaced $\textbf{Top}$ with its opposite.

Zhen Lin

Apr 27, 2023 23:08

It is in fact not necessary to know set theory in detail. I don’t think the issue is that you don’t understand “small”; I think the issue is that you don’t understand “has a colimit”.

Zhen Lin

Apr 27, 2023 23:08

The way you are using these words suggests to me you do not yet have a solid grasp of the basics. “The category of diagrams in $\textbf{Top}$ has a colimit.” is not a well formed assertion. Individual diagrams have colimits, categories of diagrams don’t.

$Mathematics$ Discussion between Enkidu and Zhen Lin

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Zhen Lin

Mar 28, 2023 07:07

It seems to me you don't actually understand the point I was trying to make. I regret upvoting this incomplete answer.

Zhen Lin

Mar 28, 2023 07:07

I said the associators don't matter. I did not say anything about associativity, which is precisely the non-trivial coherence axiom I am talking about! If you read carefully you would have seen this.

Zhen Lin

Mar 28, 2023 07:07

Associators are not relevant to my point. As I said, this is a non-trivial coherence condition even for strict monoidal categories. Try proving it yourself!

Zhen Lin

Mar 28, 2023 07:07

I don't agree that such a convention exists – but that is irrelevant. I am talking about a more subtle point which is often misunderstood in these "all diagrams commute" results. Choosing canonical bracketings is fine when you deal with formal variables, but when you start substituting expressions into the variables you may end up with non-canonical bracketings. To say nothing of all the confusions that can happen when you substitute concrete objects and partially evaluate expressions...

Zhen Lin

Mar 28, 2023 07:07

You do not always get a choice about bracketing. For example, if you substitute $m_1$ with $x \otimes y$. Then depending on whether you apply the isomorphism before or after substituting you get a priori different results!

Zhen Lin

Mar 28, 2023 07:07

No, even when both sides are strict monoidal categories (so that the pentagon axiom is trivial), for the traditional definition of monoidal functor, there is still a coherence axiom involving bracketing!

Zhen Lin

Mar 28, 2023 07:07

Perhaps it's worth mentioning that the isomorphism does not "depend" on the choice of bracketing – this is what you need the coherence axioms in the definition of monoidal functor for!

$Mathematics$ Discussion on question by Toney Leung…

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Zhen Lin

Sep 10, 2022 20:08

I think you are being misled by the notation. The object $A \times_B C$ depends on not only the objects $A, B, C$ but also the morphisms $A \to B$ and $C \to B$. I feel you may be trying to run before you can walk here and I suggest doing some basic exercises in category theory from a category theory texts rather than trying to pick it up from algebraic geometry texts.

Zhen Lin

Sep 10, 2022 20:08

It is a mystery to me what isomorphism you mean there. In the first place, what morphism $h_V \times_H G \to h_U$ do you have to even form a fibre product over $h_U$?

Zhen Lin

Sep 10, 2022 20:08

I don’t see how your argument proves representability of $G \to H$.

Zhen Lin

Sep 10, 2022 20:08

Well, it is false. Let $F$ be represented by $\emptyset$, let $H$ be any representable, and let $G$ be an algebraic space that is non-representable. Then you will find that $F \to G$ and $F \to H$ are representable for basically trivial reasons, but $G \to H$ cannot be representable because $G$ is not representable.

Zhen Lin

Sep 10, 2022 20:08

@ToneyLeung Why do you think it's true? I actually cannot think of any examples.

Zhen Lin

Sep 10, 2022 20:08

I would guess this is false. Have you tried looking for simple counterexamples? For instance, suppose $F$ the representable functor represented by $\emptyset$...

$Mathematics$ Discussion on question by Yuan Yang: …

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Zhen Lin

Feb 14, 2022 09:38

*any set-indexed family of sets

$Mathematics$ Discussion on question by giuseppe: C…

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Zhen Lin

Sep 17, 2020 16:19

The geometric realisation of a discrete simplicial set is homeomorphic to the set you start with. This is a straightforward calculation: geometric realisation preserves coproducts and the terminal object, and a discrete simplicial set is a coproduct of some set of copies of the terminal object.

Zhen Lin

Sep 17, 2020 16:19

This is one of the difficult points of working with bisimplicial sets, yes. You have to develop your own scheme for not confusing the roles of the two indices – they are not interchangeable once you privilege one of them by, say, introducing the projective model structure.

Zhen Lin

Sep 17, 2020 16:19

It’s degreewise discrete. Discrete simplicial sets are weakly equivalent if and only if they are isomorphic. So count the number of elements.

Zhen Lin

Sep 17, 2020 16:19

You are definitely confusing yourself. Yes, the simplicial simplicial set that is degreewise $\Delta^n$ is contractible. But there is also a simplicial simplicial set whose $m$-th level is the discrete simplicial set $\Delta^n_m$, and this is not contractible for $n > 0$. If Dugger says that the bisimplicial set $\Delta^n$ is not contractible then I would assume he means the latter.

Zhen Lin

Sep 17, 2020 16:19

I cannot read your mind so I cannot tell you where you have confused yourself. Maybe you have mistakenly assumed that your embedding $r$ preserves weak equivalences. I am not going to read the paper for you.

Zhen Lin

Sep 17, 2020 16:19

I don’t like repeating myself but it seems you didn’t process what I said. In a model structure where weak equivalences are degreewise only one of $\Delta^{n, 0}$ or $\Delta^{0, n}$ is contractible ($n > 0$). Which one depends on your indexing convention.

Zhen Lin

Sep 17, 2020 16:19

$\Delta^{n, 0}$ is not isomorphic to $\Delta^{0, n}$.

Zhen Lin

Sep 17, 2020 16:19

Let me be more direct. Either you have misinterpreted the notation or the author has confused the notation. There is a perfectly sensible scenario where $\Delta^n$ stands for a non-contractible bisimplicial set. I don’t know what convention you are using for the two indices but I guess this would be $\Delta^{0, n}$ if you say $\Delta^{n, 0}$ is contractible.

Zhen Lin

Sep 17, 2020 16:19

There are two obvious ways of embedding simplicial sets into bisimplical sets. One way preserves weak equivalencies. The other way embeds a simplicial set as a degreewise discrete bisimplicial set; this way does not preserve weak equivalencies. Both embeddings have their uses.

$Mathematics$ Discussion on question by canedidd: W…

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Zhen Lin

Sep 13, 2020 15:18

Yes, you get a morphism $\operatorname{Spec} k(x) \to X$, but if $k (x) \not\cong k$ then you don't get an element of $X (k)$.

Zhen Lin

Sep 13, 2020 15:18

@canedidd You seem to be assuming that the residue field of a point is automatically isomorphic to $k$. This is not true.

Homotopy Theory

A room for anyone interested in homotopy theory, or any nearby...

Zhen Lin

Apr 12, 2016 11:05

@SaalHardali There are many ways of defining open immersion. There's a sticks-and-stones way described in [Demazure and Gabriel]. It's also in my thesis, but only as a special case of more general ideas.

Zhen Lin

Apr 12, 2016 07:20

Well, you can find such a definition in [Demazure and Gabriel], but basically it's just a matter of realising how to define open immersion

Zhen Lin

Apr 12, 2016 07:08

What is "pure"?

Zhen Lin

Apr 12, 2016 07:06

to me, at least, it's not obvious

Zhen Lin

Apr 12, 2016 07:03

the algebra part is correct, but the problem is with globalisation

Zhen Lin

Apr 12, 2016 06:59

I'm not convinced it exists

Zhen Lin

Apr 12, 2016 06:41

@SaalHardali I don't think that's the right notion of "finite presentation" for this purpose

Zhen Lin

Mar 29, 2016 17:33

you can think of it as being an algebraic theory where you only impose new "axioms" without introducing new "operations"

Zhen Lin

Mar 29, 2016 17:32

every reflective subcategory is monadic

Zhen Lin

Mar 29, 2016 15:03

I suppose you could conclude that compact Hausdorff spaces are secretly algebraic

Zhen Lin

Mar 24, 2016 08:37

ensembles ou espaces

Zhen Lin

Mar 23, 2016 17:07

@SaalHardali not at all – the small object argument is nothing like that.

Zhen Lin

Mar 22, 2016 14:12

global sections is functorial, so yes

Zhen Lin

Mar 22, 2016 13:12

hence the join is also contained in the subsheaf, and therefore the subsheaf is representable