Some while ago we discussed using the English terms “little topos” and “big topos” when writing in English instead of the French “petit” and “gros”. I don’t recall any objections being raised, so I took the initiative to move petit topos to big and little topos as I did some expansion of it. (The floor is still open for objections, of course.)

]]>Is there a term to denote collectively all the constructions of “theories” of some sort, without giving rise to ambiguities?

What I exactly mean is a collective term for monads, operads, clubs, Lawvere theories, and in general all categorical constructions that may have “algebras”, or which describe formal operations in some (possibly generalized) way. In my mind they all encode some sort of “theory”, but the word “theory” seems to be reserved for “algebraic theory”/”Lawvere theory”.

Which term would you use instead?

(I hope it’s clear enough what I’m asking.)

]]>Is there a word to describe the process of making something categorical? “Categoricalization”?

By this I don’t mean a categorification, which is instead something “one level up”, I mean rather something like, for example, going to set theory to the category of sets and functions.

]]>As far as I can tell, the difference between a monadic functor and a strictly monadic functor boils down to this: a strictly monadic functor $U : \mathcal{D} \to \mathcal{C}$ has the property that, for any object $D$ in $\mathcal{D}$ and any isomorphism $f : C \to U D$ in $\mathcal{C}$, there is a *unique* object $\tilde{C}$ and an (automatically unique) isomorphism $\tilde{f} : \tilde{C} \to D$ such that $U \tilde{f} = f$. Conversely, any monadic functor with this property is strictly monadic. This is very reminiscent of the definition of isofibration, but a monadic isofibration need not be strictly monadic. What’s a good phrase to describe functors like these for which isomorphisms lift uniquely?

- “Creates isomorphisms” on its own might be construed as “conservative”.
- “Isomorphisms lift uniquely” suggests something a little bit stronger than what I’m going for – the unique functor from a group to the terminal category admits unique solutions to the object part of the lifting problem (for obvious reasons) but not the isomorphism part.
- Maybe “strong/strict isofibration”…?

Certain people who occasionally write about such things (Richard Garner, Peter Lumsdaine, Mike Shulman?, myself) have conspired to start using the terminology “algebraic weak factorization system” (or with British spelling) in place of the original natural weak factorization system. I’d like to indicate this on the nLab entry. (In fact, I’d love to change the name and instead indicate the old terminology, but perhaps you might object.) What is the best way to handle this?

]]>The word “effective” is being used for many different things in the same context on ncatlab, and using “coeffective” can cut down on this overuse by at least half. For example,

http://ncatlab.org/nlab/show/congruence

applies the word “effective” to certain epimorphisms, as well as to certain congruences, which is confusing for two reasons: in the context of a diagram $R\rightrightarrows U \to X$,

1) neither usage implies the other, and

2) we have no word for when the diagram is “effective on both sides”.

I suggest the following revision:

A pair of parallel morphisms $R\rightrightarrows U$ (necessarily a congruence) is called

or*effective*if $R$ is the kernel pair of its coequalizer (and these operations all exist).*an effective pair*A morphism $U\to X$ (necessarily epi) is called

if it is the coequalizer of its kernel pair (and these things all exist).*coeffective*

Then for a diagram $R\rightrightarrows U \to X$, TFAE:

1) X is the coequalizer and R is effective,

2) R is the kernel pair and X is coeffective.

So if either holds, following the usual use of “bi” in category theory, we can call the diagram “bieffective”.

I’d also recommend the phrase “h is a coeffective epimorphism” over “effective epimorphism”, because it can be shortened to “h is coeffective” with no confusion; coeffective implies epi.

Example usages:

“Gluing diagrams” are usually bieffective diagrams: X can be a scheme, U the disjoint union of a Zariski cover, and R the disjoint union of the intersections.

In the category of algebraic spaces, etale congruences are always effective, and etale surjections are always coeffective, so bieffective diagrams abound, and their bieffectiveness is what allows us to think of them intuitively as “gluing diagrams”.

In GIT, the geometric quotient of a free group action is bieffective (geometric means $G\times U\to U\times_X U$ is schematically surjective, and free means it’s an immersion, hence an isomorphism).

What do people think about this?

]]>Apologies for the slew of paper related questions, but this one was bugging me too.

Given a pretopology, or more generally, a coverage $J$ on a category, and the class of arrows ($J$-epi) of arrows that admit local sections relative to $J$. This class is interesting, but I’m interested in the subclass of arrows of which all pullbacks exist and which is stable under pullback (hence forms another pretopology). I denoted this $J_{sing}$ in my paper, because it is, if you like, the singletonification of $J$. This is clearly a Bad Name (TM), but I can’t think of a good name. ’The class of pullback-stable $J$-epimorphisms’ is also too much of a mouthful. It’s a sort of saturation of $J$, but isn’t saturated as I define the notion (and I have good reason to keep the definition of saturation as is).

Any ideas?

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