Timeline for Naturally occurring examples of badly behaved categories
Current License: CC BY-SA 4.0
12 events
| when toggle format | what | by | license | comment | |
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| Apr 25, 2021 at 3:05 | comment | added | Dmitri Pavlov | @AndréHenriques: I looked in the original Ghez–Lima–Roberts paper, and they only talk about infinite direct sums of families {A_i}_{i∈I}, defined as an object A with maps ι_i:A_i→A and π_i:A→A_i satisfying π_i ι_i = id and ∑_i ι_i π_ι = id, which is different from how I remembered it. Concerning (co)limits, I rechecked the adjunction identities, and they only hold if one allows unbounded maps, which is not allowed for a W*-category, so Hilb only has infinite direct sums, but not infinite (co)products in this sense. | |
| Apr 24, 2021 at 22:40 | comment | added | André Henriques | @DmitriPavlov I don't understand you. Which "diagonal functor"? (Are you maybe just talking about the existence of binary (co)products in Hilb?) | |
| Apr 24, 2021 at 20:38 | comment | added | Dmitri Pavlov | @AndréHenriques: It means that the diagonal functor has a left adjoint (as a 1-morphism in the indicated bicategory of W*-categories). | |
| Apr 24, 2021 at 20:35 | comment | added | André Henriques | @DmitriPavlov Can you please explain what you mean by "it becomes (co)complete"? What does it mean, for you, for a W*category to be (co)complete? | |
| Apr 24, 2021 at 19:35 | comment | added | Dmitri Pavlov | This is an artifact of considering Hilb to be an ordinary category, i.e., an object of the bicategory of categories, functors, and natural transformations. Once we consider Hilb with its appropriate structure as an object of the bicategory of W*-categories, W*-functors, and bounded natural transformations, it becomes complete, cocomplete, and the tensor product becomes cocontinuous in each argument (and thus is uniquely characterized by the fact that C⊗C≅C). | |
| Apr 21, 2021 at 18:04 | comment | added | Tim Campion | This is interesting. So Hilb has finite colimits and $\aleph_1$-filtered colimits, which leaves a sort of "gap in its colimit spectrum", where it specifically lacks filtered colimits which are not $\aleph_1$-filtered. I wonder if other "categories of objects with metric-like completeness conditions" follow this pattern. | |
| Apr 21, 2021 at 14:17 | comment | added | André Henriques | @MartinBrandenburg Yes and yes. Finite limits and colimits exist in Hilb. (Note however that the functor which sends a Hilbert space to its underlying vector does not preserve coequalisers.) The Hilbert space tensor product is very nice, but I don't know how to characterise is in category-theory language. | |
| Apr 20, 2021 at 21:04 | comment | added | Martin Brandenburg | We have finite limits in $\mathbf{Hilb}$, right? And $\mathbf{Hilb}$ has a nice symmetric monoidal structure (Hilbert space tensor product). | |
| Apr 19, 2021 at 20:47 | comment | added | André Henriques | @TimCampion Restricting to maps of norm $\le 1$ is a good idea for Banach spaces, but doesn't do much good when working with Hilbert spaces (binary coproducts exist in the former case, but not in the latter case). | |
| Apr 18, 2021 at 20:51 | comment | added | Tim Campion | Banach spaces and bounded linear maps is similarly bad (I think one has $\aleph_1$-filtered colimits at least.) Of course, both examples are improved by restricting to maps of norm $\leq 1$. | |
| S Apr 18, 2021 at 20:29 | history | answered | André Henriques | CC BY-SA 4.0 | |
| S Apr 18, 2021 at 20:29 | history | made wiki | Post Made Community Wiki by André Henriques |