Homotopy Theory

user131753

4:20 AM

I was reading The Joy of Cats and in pp. 59-60, I find the shortly after giving the examples of concrete categories (for their definition of concrete categories see Definition 5.1(1)) they write (see Remark 5.3 (1)),

user131753

"Since faithful functors are injective on $\text{hom}$-sets, we usually assume...that $\operatorname{hom}_{\mathbf{A}}(A, B)$ is a **subset** of $\operatorname{hom}_{\mathbf{X}}(UA, UB)$ for each pair $(A, B)$ of $\mathbf{A}$-objects. This familiar convention allows one to express the property that "for $\mathbf{A}$-objects $A$ and and an $\mathbf{X}$-morphism $f:UA \to UB$ there exists a (necessarily unique) $\mathbf{A}$-morphism $A \to B$ with $\require{AMScd}U(A \to B) = \begin{CD}
UA @>{f}>> UB\end{CD}$" much more succinctly, by stating

user131753

(emphasis mine)

user131753

Can anyone explain to me some reason for taking $\operatorname{hom}_{\mathbf{A}}(A, B)$ to be a subset of $\operatorname{hom}_{\mathbf{X}}(UA, UB)$ when in fact they may not be? What is the rationale behind this convention?

user131753

To be sure in a footnote (after the last sentence) they do write, "Observe that analogues of this expression (i.e. $f:UA\to UX$ is an $\mathbf{A}$-morphism) are frequently used in “concrete situations”, e.g., in saying
that a certain function between vector spaces “is linear” or that a certain function between topological spaces “is continuous”, etc. " But I don't see how it fits with the convention.

David Roberts

5:50 AM

@DenisNardin "Understand"... Personally I think one should learn a bunch of category theory first.

I should say that this question is not about homotopy limits, but about the comma object in a (2,1)-category, hence very much simpler than homotopy pullbacks. Advising someone who is self-confessed "a little shaky" in category theory that what they are looking for in Hirschhorn is doing them no favours.

Praphulla Koushik

6:58 AM

@DavidRoberts Only category theory I know is from chapter 2 of Hilton and Stammbach A course in Homological algebra web.math.rochester.edu/people/faculty/doug/otherpapers//… I do not know much about (2,1)-categories and all that... I picked it from here and there and do not have solid background in that... I will learn...

Praphulla Koushik

7:10 AM

@DavidRoberts It was not me who said it has something to do with Homotopy limit.. So, I do not know what to say if you think this not about homotopy limits... Notion of $(2,1)$categories is not in Hilton/STammbach and not in Mac Lane's Working mathematician.. If you can point me to some reference, I will read it... Thanks,..

Denis Nardin

@DavidRoberts I'm not sure I understand what you mean. Are you disagreeing with my opinion that a stronger foundation in basic category theory would be beneficial? Also, it was not me that recommended Hirschhorn (I recommended MacLane :))

I agree that looking into the details of homotopy limits is probably not useful at this point, and I suspect that what they need is a more solid grounding in basic category theory

David Roberts

9:44 AM

@PraphullaKoushik (2,1)-categories are special 2-categories. Google is your friend. People here should have looked at your question and realised it was not homotopy theoretic in nature. Since everything is 2-categorical (or rather, (2,1)-categorical), homotopy is unnecessary, but one can use homotopy-theoretic ideas if one really only wants to work with the underlying 1-category as a model category etc etc.

@DenisNardin no, I totally agree with you that more category theory is necessary. (and the remark about the book recommendation was more aimed into the air cough Harry cough)

Denis Nardin

@DavidRoberts So the disagreement here might stem from the fact that I find (∞,1)-category theory significantly easier than 2-category theory (i.e. (2,2)-category theory) and more relevant for the topic at thand. But YMMV, and in any case one should get a strong grounding in 1-category theory first

Tom Bachmann

10:46 AM

AFAIK you can build (oo,n) and hence (oo,oo)-categories and also (n,n)-categories by a fairly straightforward construction internal to (oo,1)-categories. Hence in a sense (oo,1)-category theory already contains the higher versions. This cannot be said about (n,1) or (n,n)-category theory. Hence why it takes ingenuity to write down the axioms of bicategories, tricategories etc.

David Roberts

10:56 AM

@DenisNardin and (2,1)-categories are the easiest nontrivial case of (oo,1)-categories kerodon.net/tag/009P :-)

Transcript for