I have expanded Lawvere-Tierney topology, also reorganized it in the process

]]>following Zoran’s suggestion I added to the beginning of the Idea-section at monad a few sentences on the general idea, leading then over to the Idea with respect to algebraic theories that used to be the only idea given there.

Also added a brief stub-subsection on monads in arbitrary 2-categories. This entry deserves a bit more atention.

]]>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!

]]>started a stub for *ambidextrous adjunction*, but not much there yet

I added to *cylinder object* a pointer to a reference that goes through the trouble of spelling out the precise proof that for $X$ a CW-complex, then the standard cyclinder $X \times I$ is again a cell complex (and the inclusion $X \sqcup X \to X\times I$ a relative cell complex).

What would be a text that features a *graphics* which illustrates the simple idea of the proof, visualizing the induction step where we have the cylinder over $X_n$, then the cells of $X_{n+1}$ glued in at top and bottom, then the further $(n+1)$-cells glued into all the resulting hollow cylinders? (I’d like to grab such graphics to put it in the entry, too lazy to do it myself. )

created *traced monoidal category* with a bare minimum

I would have sworn that we already had an entry on that, but it seems we didn’t. If I somehow missed it , let me know and we need to fix things then.

]]>the entry *[[group algebra]]* had been full of notation mismatch and also of typos. I have reworked it now.

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.)

]]>Hi, I am new here so please excuse any formatting errors.

I am looking for a reference on what I do not have a term for so I will call it subobject philosophy. What I mean is that the “right” definition of subobject often depends on the category. Broadly, it is generally agreed that subobjects should be *at most* monomorphisms (up to slice isomorphism) and *at least* regular monomorphisms. However, I have gotten this information purely from a friend and I wish to fact-check it. Is there any article about this or similar on the nLab or elsewhere? I do see mention of these concepts on their relevant pages, but no article pointedly about the “philosophy” of how one chooses which to use, or a compilation of the list of criteria currently in use across various categories.

Thank you all for your time.

]]>I have added to *monoidal model category* statement and proof (here) of the basic statement:

Let $(\mathcal{C}, \otimes)$ be a monoidal model category. Then 1) the left derived functor of the tensor product exsists and makes the homotopy category into a monoidal category $(Ho(\mathcal{C}), \otimes^L, \gamma(I))$. If in in addition $(\mathcal{C}, \otimes)$ satisfies the monoid axiom, then 2) the localization functor $\gamma\colon \mathcal{C}\to Ho(\mathcal{C})$ carries the structure of a lax monoidal functor

$\gamma \;\colon\; (\mathcal{C}, \otimes, I) \longrightarrow (Ho(\mathcal{C}), \otimes^L , \gamma(I)) \,.$The first part is immediate and is what all authors mention. But this is useful in practice typically only with the second part.

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

I fixed the first sentence at *doctrine*. It used to say

A

doctrine, as the word was originally used by Jon Beck, is a categorification of a “theory”.

I have changed it to

The concept of

doctrine, as the word was originally used by Jon Beck, is a categorification of the concept of “theory”.

If you see what I mean.

Then I added to the References-section this:

]]>The word “doctrine” itself is entirely due to Jon Beck and signifies something which is like a theory, except appropriate to be interpreted in the category of categories, rather than, for example, in the category of sets; of course, an important example of a doctrine is a 2-monad, and among 2-monads there are key examples whose category of “algebras” is actually a category of theories in the set-interpretable sense. Among such “theories of theories”, there is a special kind whose study I proposed in that paper. This kind has come to be known as “Kock-Zoeberlein” doctrine in honor of those who first worked out some of the basic properties and ramifications, but the recognition of its probable importance had emerged from those discussions with Jon.

finally a stub for *Segal condition*. Just for completeness (and to have a sensible place to put the references about Segal conditions in terms of sheaf conditions).

Am trying to get some historical citations straight at *linear type theory*, maybe somebody can help me:

what are the original sources of the idea that linear logic/type theory should generally be that of symmetric monoidal categories (“multiplicative intuitionistic LL”)?

In order of appearance, I am aware of

de Paiva 89 gives one particular example of a non-star-autonomous SMC that deserves to be said to interpret “linear logic” and clearly identifies the general perspective.

Bierman 95 discusses semantics in general SMCs more generally

Barber 97 reviews this and explores a bit more.

What (other) articles would be central to cite for this idea/perspective?

I am aware of more recent reviews such as

but I am looking for the correct “original sources”.

]]>added reference to dendroidal version of Dold-Kan correspondence

]]>I have finally split off *dependent sum* from *dependent product*. And added a few more paragraphs.

edited [[reflective subcategory]] and expanded a bit the beginning

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

split off *strict initial object* from *initial object* (in order to be able to point to it directly from within proofs elsewhere)

edited dualizable object a little, added a brief paragraph on dualizable objects in symmetric monoidal $(\infty,n)$-categories

]]>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.

=–

]]>started a minimum at *writer comonad*

added to *partial function* a new section *Definition – General abstract* with a brief paragraph on how partial functions form the Kleisli category of the maybe-monad.

added to [[Set]] the statement that is the terminal topos.

]]>started a minimum at *continuation monad*, but not really good yet