The n-Category Café

Hmmm… just as the things themselves, 3 is just that $f$ has a section and a retraction; 1 is that $f$ has an equal section and retraction; since $f$ having both a section and a retraction already implies there is a canonical homotopy between them, 1 therefore means a section and a retraction and an extra homotopy between them; 2 is a 1 with … a contraction of something? Off the top of my head I can’t tell what, just because the datum is in the wrong space for my intuition; though of course mapping back by the inverse equivalence should fix that for me. But this is what you were calling “adjoint equivalence” before, yes? 4/5 of course … “has contractible fibers” is really quite straightforward. 6 sounds dangerous, just because I can’t tell whether you mean merely existing or what. It seems to me it might be interpretable as a component bijection, though that’s surely not what you intend here.

Anyways, I therefore prognosticate

• 3 shall be “isomorphism” because it says there is an inverse, and really that’s all it says;
• 1 shall be “redundant isomorphism” because of the redundant homotopy;
• 2 therefore shall be a “rectified redundant isomorphism”.
• I think I’d call 4/5 a “weak homotopy equivalence”, because where I come from it ‘immediately’ implies the same property for the map of loops at any basepoint of $A$. Not a very good reason, nor a very good name, but since you asked.

Or at least, that’s a quick slap-dash and idiosyncratic assignment of names which just as likely won’t help anyone.

Is not (1) what was once upon a time called homtopy equivalence?

I am not sure to understand all the subtleties, but here is what comes to my mind:

1. the data of $g$, $\sigma$ and $\rho$ is called a ‘quasi-inverse’ to $f$.

2. the data of $g$, $\sigma$ and $\rho$ is called an ‘adjoint inverse’ to $f$.

3. the data of $g$ and $\rho$ is an ‘homotopy section’ of $f$ (or ‘quasi-section’), while the data of $h$ and $\sigma$ is an ‘homotopy retraction’ of $f$ (or ‘quasi-retraction’). Therefore, the data of $g$, $h$, $\sigma$ and $\rho$ is an ‘homotopy section together with an homotopy retraction’ of $f$.

4. such an $f$ a ‘weak homotopy equivalence’ (or ‘weak equivalence’, for short).

5. such an $f$ is a ‘bijection’ (or is ‘bijective’).

6. such an $f$ is both an ‘injection’ and a ‘surjection’ (or is both ‘injective’ and ‘surjective’).

After Steve’s Awodey’s excellent talk in the MIT topology seminar a few weeks ago, a number of folks in the Boston area have been remarking on the analogy between the univalence axiom and the completeness condition that arises in many models for (oo,n)-categories.

Speaking very loosely, the completeness condition says that the canonical map from the space of identities to the space of equivalences is a weak homotopy equivalence. With this analogy in mind, I think it would be excellent if the corresponding thing in homotopy type theory were also called equivalence. I think that’s definition (2), right?

The social sciences create a lot of new words or labels. Sometimes it seems that every researcher will come up with a new name for a concept whenever it deviates ever so slightly from existing concepts. At first this seems pompous, but I’ve come to appreciate that this lends an exactness to their discussions, in absence of equations that make small second and third order terms evident. The crowning achievement of exact naming can however be found in chemistry, where the structure of a molecule can be inferred directly from its full name. It seems that the task at hand here lies somewhere in between.

However, I’m also wary of names that attempts to be helpful by painting an apt analogy. As a physics student and non-native english speaker, I got halfway through my PhD before I was made aware that “field” and “fieldlines” was originally supposed to conjour up an image of the furrows a plough makes in a farmers’ field. (When do others figure this out?)

That said, we’ve got fibres and we’ve got a structure. Intuitively some of the functions treat what you put into them nicely, while others crumple it and spit out something poorly behaved. The mapping itself could be represented by several (intransitive) verbs which evoke well-tried routes and smoothness or an uglier distortion, which is to be eschewed.

Examples where the analogy of mapping and function is a journey:
Shuttling - a going back and forth regularly over an often short route by a vehicle
Plying - to make a practice of rowing or sailing over or on , to apply oneself steadily
Steering, meandering, traipsing and trudging are other connotation-heavy words that to my knowledge no mathematician has called dibs on yet.

Examples where the analogy is an (in principle) reversible morphing of an object:
Pleat/pleating - (intransitive verb) to nicely arrange in pleats / folds.
Writhing - to twist into coils or folds, to twist so as to distort.
Crooking - to bend, curl or flex, negative connotations with “crook”. Notching, pinching and gashing could be equally negative.

There also appears to be a need for nouns, but nouns for structures that are conceptually tightly interwoven with a verb or the process of creating the noun-objects. A promising approach which can be tied in with some of the above verbs is to consider the world of textiles. “Twill fabrics” have a front and a back side and are created by the process of “twilling”, unlike plain weaves where the two sides are alike. The weaves are ofcourse “woven”.

Playing with these concepts evoke many-to-one mappings and the combination of two objects into one, as with the interwoven warp and weft threads. If one wish to associate varying degrees of intricacy, simplicity or crudeness to the resulting output, one can evoke tacticle sensations by naming it after fabrics. For instance:
“The mapping g:B->A results in a brocade, which is rich in mathematical structure. Let G be a flannel, upon which a tartan T is imposed. Let A be a linen space and J a jute matrix. (Not to be confused with Hessian matrix)”