added pointer to Lemma 3.5 in Felix’s thesis

]]>Outlined the notion of a blockchain and the key notion of ’proof of work’ that underlies it, along with some criticisms.

]]>I split off an entry regular monomorphism in an (infinity,1)-category from regular monomorphism. But not much there yet.

]]>Added an example from coinductive calculus (generalized binomial theorem).

]]>while bringing some more structure into the section-outline at *comma category* I noticed the following old discussion there, which hereby I am moving from there to here:

[begin forwarded discussion]

+–{.query} It's a very natural notation, as it generalises the notation $(x,y)$ (or $[x,y]$ as is now more common) for a hom-set. But personally, I like $(f \rightarrow g)$ (or $(f \searrow g)$ if you want to differentiate from a cocomma category, but that seems an unlikely confusion), as it is a category of arrows from $f$ to $g$. —Toby Bartels

Mike: Perhaps. I never write $(x,y)$ for a hom-set, only $A(x,y)$ or $hom_A(x,y)$ where $A$ is the category involved, and this is also the common practice in nearly all mathematics I have read. I have seen $[x,y]$ for an internal-hom object in a closed monoidal category, and for a hom-set in a homotopy category, but not for a hom-set in an arbitrary category.

I would be okay with calling the comma category (or more generally the comma object) $E(f,g)$ or $hom_E(f,g)$ *if* you are considering it as a discrete fibration from $A$ to $B$. But if you are considering it as a *category* in its own right, I think that such notation is confusing. I don’t mind the arrow notations, but I prefer $(f/g)$ as less visually distracting, and evidently a generalization of the common notation $C/x$ for a slice category.

*Toby*: Well, I never stick ‘$E$’ in there unless necessary to avoid ambiguity. I agree that the slice-generalising notation is also good. I'll use it too, but I edited the text to not denigrate the hom-set generalising notation so much.

*Mike*: The main reason I don’t like unadorned $(f,g)$ for either comma objects or hom-sets is that it’s already such an overloaded notation. My first thought when I see $(f,g)$ in a category is that we have $f:X\to A$ and $g:X\to B$ and we’re talking about the pair $(f,g):X\to A\times B$ — surely also a natural generalization of the *very* well-established notation for ordered pairs.

*Toby*: The notation $(f/g/h)$ for a double comma object makes me like $(f \to g \to h)$ even more!

*Mike*: I’d rather avoid using $\to$ in the name of an object; talking about projections $p:(f\to g)\to A$ looks a good deal more confusing to me than $p:(f/g)\to A$.

*Toby*: I can handle that, but after thinking about it more, I've realised that the arrow doesn't really work. If $f, g: A \to B$, then $f \to g$ ought to be the set of transformations between them. (Or $f \Rightarrow g$, but you can't keep that decoration up.)

Mike: Let me summarize this discussion so far, and try to get some other people into it. So far the only argument I have heard in favor of the notation $(f,g)$ is that it generalizes a notation for hom-sets. In my experience that notation for hom-sets is rare-to-nonexistent, nor do I like it as a notation for hom-sets: for one thing it doesn’t indicate the category in question, and for another it looks like an ordered pair. The notation $(f,g)$ for a comma category also looks like an ordered pair, which it isn’t. I also don’t think that a comma category is very much like a hom-set; it happens to be a hom-set when the domains of $f$ and $g$ are the point, but in general it seems to me that a more natural notion of hom-set between functors is a set of natural transformations. It’s really the *fibers* of the comma category, considered as a fibration from $C$ to $D$, that are hom-sets. Finally, I don’t think the notation $(f,g)$ scales well to double comma objects; we could write $(f,g,h)$ but it is now even less like a hom-set.

Urs: to be frank, I used it without thinking much about it. Which of the other two is your favorite? By the way, Kashiwara-Schapira use $M[C\stackrel{f}{\to} E \stackrel{g}{\leftarrow} D]$. Maybe $comma[C\stackrel{f}{\to} E \stackrel{g}{\leftarrow} D]$? Lengthy, but at least unambiguous. Or maybe ${}_f {E^I}_g$?

Zoran Skoda: $(f/g)$ or $(f\downarrow g)$ are the only two standard notations nowdays, I think the original $(f,g)$ which was done for typographical reasons in archaic period is abandonded by the LaTeX era. $(f/g)$ is more popular among practical mathematicians, and special cases, like when $g = id_D$) and $(f\downarrow g)$ among category experts…other possibilities for notation should be avoided I think.

Urs: sounds good. I’ll try to stick to $(f/g)$ then.

Mike: There are many category theorists who write $(f/g)$, including (in my experience) most Australians. I prefer $(f/g)$ myself, although I occasionally write $(f\downarrow g)$ if I’m talking to someone who I worry might be confused by $(f/g)$.

Urs: recently in a talk when an over-category appeared as $C/a$ somebody in the audience asked: “What’s that quotient?”. But $(C/a)$ already looks different. And of course the proper $(Id_C/const_a)$ even more so.

Anyway, that just to say: i like $(f/g)$, find it less cumbersome than $(f\downarrow g)$ and apologize for having written $(f,g)$ so often.

*Toby*: I find $(f \downarrow g)$ more self explanatory, but $(f/g)$ is cool. $(f,g)$ was reasonable, but we now have better options.

=–

]]>I’m having trouble understanding the relationship between what is described in the page generalized algebraic theory and what is in the referenced article by Cartmell.

For example, the nLab page seems to contemplate three levels of symbols—“supersort”, “sort”, and “operation”—while if I understand correctly, Cartmell’s GATs only have symbols at two levels, for sorts and operations. Also, I would only expect a GAT’s sort symbols to be applied to terms, not types as the nLab page seems to contemplate. (The nLab page speaks of “derived operations in the theory of sorts” rather than types, but I believe the same concept is intended.) My intuition is that the world of GATs in Cartmell’s sense more or less corresponds to a certain sublanguage of LF (rather than $F_\omega$), so there shouldn’t be anything like a symbol that is applied to arguments that are types and yields a type.

]]>Following discussion in some other threads, I thought one should make it explicit and so I created an entry

Currently this contains some (hopefully) evident remarks of what “dependent linear type theory” reasonably should be at least, namely a hyperdoctrine with values in linear type theories.

The entry keeps saying “should”. I’d ask readers to please either point to previous proposals for what “linear dependent type theory” is/should be, or criticise or else further expand/refine what hopefully are the obvious definitions.

This is hopefully uncontroversial and should be regarded an obvious triviality. But it seems it might be one of those hidden trivialities which deserve to be highlighted a bit more. I am getting the impression that there is a big story hiding here.

Thanks for whatever input you might have.

]]>I want to be able to point to *category of V-enriched categories*, so I created an entry, so far just with a brief Idea-paragraph.

started an entry *global equivariant stable homotopy theory* with an Idea-section and some references.

I have also created a brief entry for the unstable version: *global equivariant homotopy theory*.

For anyone who wants to edit and wondering where to add what, let me just highlight that there is the following collection of existing entries (some of them with genuine content, some mostly stubby)

homotopy theory | stable homotopy theory |
---|---|

equivariant homotopy theory | equivariant stable homotopy theory |

global equivariant homotopy theory | global equivariant stable homotopy theory |

added references to *essentially algebraic theory*. Also equipped the text with a few more hyperlinks.

I thought it worthwhile to add the view of monoids as strict monoidal categories. This seems missing here.

dorchard

]]>Unfortunately, I need to discuss with you another terminological problem. I am lightly doing a circle of entries related to combinatorial aspects of representation theory. I stumbled accross permutation representation entry. It says that the permutation representation is the representation in category $Set$. Well, nice but not that standard among representation theorists themselves. Over there one takes such a thing – representation by permutations of a *finite* group $G$ on a set $X$, and looks what happens in the vector space of functions into a field $K$. As we know, for a group element $g$ the definition is, $(g f)(x) = f(g^{-1} x)$, for $f: X\to K$ is the way to induce a representation on the function space $K^X$. The latter representation is called the **permutation representation** in the standard representation theory books like in

- Claudio Procesi,
*Lie groups, an approach through invariants and representations*, Universitext, Springer 2006, gBooks

I know what to do approximately, we should probably keep both notions in the entry (and be careful when refering to this page – do we mean representation by permutations, what is current content or permutation representation in the rep. theory on vector spaces sense). But maybe people (Todd?) have some experience with this terminology.

Edit: new (related) entries for Claudio Procesi and Arun Ram.

]]>added a subsection “Properties – As algebraic K-theory over the field with one element” (here)

]]>added pointer to the slides Licata 18

]]>added a brief historical comment to *Higgs field* and added the historical references

stub for *confinement*, but nothing much there yet. Just wanted to record the last references there somewhere.

I have scratched the rough Idea-section that I had here previously, and started afresh, now with mentioning of the K-theoretic McKay correspondence. Also added references. But it’s still just a stub entry

]]>In the article connection on a cubical set the definition of a connection uses two types of maps, denoted Γ^+_i and Γ^-_i. Roughly, the former corresponds to a map of cubes that takes the minimum of some coordinates, whereas the latter takes the maximum of some coordinates.

However, in the book by Brown-Higgins-Sivera in Definition 13.1.3, page 446, only the maps Γ^-_i are used. There they are denoted simply by Γ_i. The paper by Maltsiniotis about the strict test category of cubes with connection also uses the same definition.

Which definition is correct? What is the reference for nLab’s definition and why does it deviate from the definition of Brown-Higgins-Sivera?

]]>added pointer to

- Steve Awodey, Nicola Gambino, Kristina Sojakova,
*Homotopy-initial algebras in type theory*(arXiv:1504.05531)

to *initial algebra over an endofunctor*, *higher inductive type* and *W-type*

started stubs for *K-theory of a permutative category* and *K-theory of a bipermutative category* .

New (tagged philosophy) entry structuralism, very stubby so far.

]]>added to *equivariant K-theory* comments on the relation to the operator K-theory of crossed product algebras and to the ordinary K-theory of homotopy quotient spaces (Borel constructions). Also added a bunch of references.

(Also finally added references to Green and Julg at *Green-Julg theorem*).

This all deserves to be prettified further, but I have to quit now.

]]>Finally split two-sided fibration off of Grothendieck fibration. Thanks to Emily Riehl for adding the definitions here.

]]>I added to transchromatic character

- Nathaniel Stapleton,
*An Introduction to HKR Character Theory*(pdf)

and

- Takeshi Torii,
*HKR characters, p-divisible groups and the generalized Chern character*, (pdf)

just for the record

]]>