at *isofibration* I have added a pointer to Brown 70. And I replaced a paragraph saying that isofibrations are “evil” by this modified paragraph

added at *core* the remark that the core is right adjoint to the forgetful functor $Grpd \to Cat$.

The nLab page on Gray’s *formal category theory: adjointness for 2-categories*
reads:

The book was supposed to be the first part of a four volume work, but unfortunately later volumes/chapters never appeared.

Do drafts of these later volumes exist?

]]>Created Beck module, mentioned it (once) on the tangent category page.

]]>at *total category* I have added after the definition and after the first remark these two further remarks:

+– {: .num_remark}

Since the Yoneda embedding is a full and faithful functor, a total category $C$ induces an idempotent monad $Y \circ L$ on its category of presheaves, hence a modality. One says that $C$ is a totally distributive category if this modality is itself the right adjoint of an adjoint modality.

=–

+– {: .num_remark}

The $(L \dashv Y)$-adjunction of a total category is closely related to the
$(\mathcal{O} \dashv Spec)$-adjunction discussed at *Isbell duality* and at *function algebras on ∞-stacks*. In that context the $L Y$-modality deserves to be called the *affine modality*.

=–

]]>at internal hom the following discussion was sitting. I hereby move it from there to here

Here's some discussion on notation:

*Ronnie*: I have found it convenient in a number of categories to use the convention that if say the set of morphisms is $hom(x,y)$ then the internal hom when it exists is $HOM(x,y)$. In particular we have the exponential law for categories

Then one can get versions such as $CAT_a(y,z)$ if $y,z$ are objects over $a$.

Of course to use this the name of the category needs more than one letter. Also it obviates the use of those fonts which do not have upper and lower case, so I have tended to use mathsf, which does not work here!

How do people like this? Of course, panaceas do not exist.

*Toby*: I see, that fits with using $\CAT$ as the $2$-category of categories but $\Cat$ as the category of categories. (But I'm not sure if that's a good thing, since I never liked that convention much.) I only used ’Hom’ for the external hom here since Urs had already used ’hom’ for the internal hom.

Most of the time, I would actually use the same symbol for both, just as I use the same symbol for both a group and its underlying set. Every closed category is a concrete category (represented by $I$), and the underlying set of the internal hom is the external hom. So I would distinguish them only when looking at the theorems that relate them, much as I would bother parenthesising an expression like $a b c$ only when stating the associative law.

*Ronnie*: In the case of crossed complexes it would be possible to use $Crs_*(B,C)$ for the internal hom and then $Crs_0(B,C)$ is the actual set of morphisms, with $Crs_1(B,C)$ being the (left 1-) homotopies.

But if $G$ is a groupoid does $x \in G$ mean $x$ is an arrow or an object? The group example is special because a group has only one object.

If $G$ is a group I like to distinguish between the group $Aut(G)$ of automorphisms, and the crossed module $AUT(G)$, some people call it the *actor*, which is given by the inner automorphism map $G \to Aut(G)$, and this seems convenient. Similarly if $G$ is a groupoid we have a group $Aut(G)$ of automorphisms but also a group groupoid, and so crossed module, $AUT(G)$, which can be described as the maximal subgroup object of the monoid object $GPD(G,G)$ in the cartesian closed closed category of groupoids.

*Toby*: ’But if $G$ is a groupoid does $x \in G$ mean $x$ is an arrow or an object?’: I would take it to mean that $x$ is an object, but I also use $\mathbf{B}G$ for the pointed connected groupoid associated to a group $G$; I know that groupoid theorists descended from Brandt wouldn't like that. I would use $x \in \Arr(G)$, where $\Arr(G)$ is the arrow category (also a groupoid now) of $G$, if you want $x$ to be an arrow. (Actually I don't like to use $\in$ at all to introduce a variable, preferring the type theorist's colon. Then $x: G$ introduces $x$ as an object of the known groupoid $G$, $f: x \to y$ introduces $f$ as a morphism between the known objects $x$ and $y$, and $f: x \to y: G$ introduces all three variables. This generalises consistently to higher morphisms, and of course it invites a new notation for a hom-set: $x \to y$.)

continued in next comment…

]]>created an entry *[[modal type theory]]*; tried to collect pointers I could find to articles which discuss the interpretation of modalities in terms of (co)monads. I was expecting to find much less, but there are a whole lot of articles discussing this. Also cross-linked with *[[monad (in computer science)]]*.

A lightning quick stubby note:

abstract general, concrete general and concrete particular

Is the last one right?

]]>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.

=–

]]>Todd,

when you see this here and have a minute, would you mind having a look at *monoidal category* to see if you can remove the query-box discussion there and maybe replace it by some crisp statement?

Thanks!

]]>created Cisinski model structure

]]>added to *polynomial functor* the evident but previously missing remark why it is called a “polynomial”, here.

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

I have given *Grothendieck construction for model categories* its own entry, in order to have a place for recording references. In particular I added pointer to the original references (Roig 94, Stanculescu 12)

(There used to be two places in the entry *Grothendieck construction* where an attempt was made to list the literature on the model category version, but they didn’t coincide and were both inclomplete. So I have replaced them with pointers to the new entry.)

This question on math.stackexchange piqued my interest. It asks what the monads on a given monoid are. That is, if we treat a monoid as a category with one object we can ask what the monads on that category are. I would have thought that this question would have had an elegant answer, because monads and monoids are both so fundamental. But in fact I can’t find a nice characterisation.

Writing down the definitions we find that a monad on a monoid $X$ is equivalent to “an endomorphism $\theta\colon X\to X$ together with two elements $m,h\in X$ such that:

$\forall x\in X$, $\theta(x)m = m\theta(\theta(x))$,

$\forall x\in X$, $\theta(x)h = hx$,

$m^2 = m\theta(m)$,

$m\theta(h) = mh = 1$.”

Now, in the nice cases when $X$ is a group or commutative, one can prove that $mh=hm=1$ and that $\theta$ is just the inner automorphism given by conjugation by $h$. But in the general case I’m not able to prove that $\theta$ can still only be an inner automorphism. So does anyone know what kind of structure this is?

]]>I have added the characterization of Quillen equivalences in the case that the right adjoint creates weak equivalences, here.

]]>The entry *unit of an adjunction* had a big chunk of mixed itex+svg code at the beginning to display an adjunction. On my machine though the output of that code was ill typeset. So I have removed the code and replaced it by plain iTex encoding of an adjunction.

(Just in case anyone deeply cares about the svg that was there. It’s still in the history. If it is preferred by anyone, it needs to be fixed first.)

]]>quickly added at [[accessible category]] parts of the MO discussion here. Since Mike participated there, I am hoping he could add more, if necessary.

]]>We have several entries that used to mention *Lawvere’s fixed point theorem* without linking to it. I have now created a brief entry with citations and linked to it from relevant entries.

created an entry *category of being*, for completeness.

A general remark: people often write that, unfortunately or not, Eilenberg-MacLane’s term “category” is not that of, say, Kant. But in fact if read this way here, following Lawvere, then the former is a good formalization of the latter, after all.

]]>I did the following to dependent product:

rearranged the section outline somewhat

added the statement that in type theory $\prod_{x \in X} P_x$ may be written $\forall x : X , P x$;

added a Properties-section

added statement and proof of the relation of dependent product to spaces of sections

added to *coherent category* a brief section *Subobjects, slices and internal logic*.

added to *perverse sheaf* a paragraph on the issue with and the origin of the terminology, here.

the entry *modular tensor category* was lacking (among many things that it is still lacking) some pointers to literature that reviews the relation to QFT. I have added a handful, maybe the best one is this here:

- Eric Rowell,
*From quantum groups to Unitary modular tensor categories*, Contemporary Mathematics 2005 (arXiv:math/0503226)

created [[Theta category]] (made [[disk category]] a redirect).

In the "Idea" section I had the idea that the quickest way to define is as the full subcategory of strict n-catgeories on n-computads. Is that right? If not, something along these lines must be right.

]]>