## December 20, 2012

### Names For Equivalences

#### Posted by Mike Shulman

Homotopy type theory has a problem: we need names for a bunch of slightly different kinds of “equivalences”. Until now, we’ve been muddling along with some fairly ad hoc choices. Can you help think of better names?

Hardly any knowledge of type theory is necessary in order to help. Just read on…

In the world of type theory, we have things called types, which are kind of like sets and kind of like groupoids and kind of like spaces. We also have functions or maps between types, and homotopies between functions, and higher homotopies and so on. Moreover, we can also make logical statements about types and their elements, as if they were sets or spaces, and these logical statements can be interpreted in a canonical way as specifying certain data.

Let $A$ and $B$ be types and $f:A\to B$ a function, and consider the following additional structures.

1. A function $g:B\to A$ and homotopies $\rho : f g \sim 1_B$ and $\sigma : g f \sim 1_A$.

2. A function $g:B\to A$, homotopies $\rho : f g \sim 1_B$ and $\sigma : g f \sim 1_A$, and a higher homotopy $f(\sigma) \sim \rho_f$.

3. Two functions $g:B\to A$ and $h:B\to A$, and homotopies $\rho : f g \sim 1_B$ and $\sigma : h f \sim 1_A$.

4. The assertion that every homotopy fiber of $f$ is a contractible space (= type).

5. The assertion that every fiber of $f$ is a singleton. A type $S$ is called a singleton if “there exists an $s_0$ in $S$, and for all $s$ in $S$ we have $s=s_0$”.

6. The assertions that $f$ is both surjective (“for every $b$ in $B$, there exists an $a$ in $A$ such that $f(a)=b$”) and injective (“if $f(a) = f(a')$, then $a=a'$”).

The question is, what names would you give to each of these notions? Before you answer, however, some remarks are in order.

Firstly, (4) and (5) are actually identical. But I’ve stated them separately in case the different phrasings suggest different names to your mind.

Secondly, all of these notions are “logically equivalent” in the sense that given the data in any one of them, you can construct the data in any other. However, not all of them are on an equal footing homotopically. In particular, (1) and (6) are poorly behaved, and to be eschewed. More precisely, if the data specified in any (hence all) of these notions exists, then each collection of data (2), (3), (4), and (5) is itself a contractible space, whereas this is not true of (1) and (6).

For this reason, I would like to avoid applying convenient and familiar words to (1) and (6), because we don’t use them very much, so it would be a waste of a good word. On the other hand, we don’t want to choose names in a way that will be unnecessarily confusing to newcomers, either.

Thirdly, the seeming asymmetry in (2) is an illusion: it would be equivalent to ask for a higher homotopy $g(\rho) \sim \sigma_g$ instead. (But if we asked for both, then we would get something ill-behaved like (1) and (6), unless we also asked for one tertiary homotopy.)

There are words for some of these notions that we’ve been using in homotopy type theory for the past year or two. However, they were chosen in a sort of ad hoc way, and right now we’re overhauling the HoTT Coq library, so we have a chance to try to improve the common terminology. Hence, I’m not going to mention yet what the words are that we have been using. If you happen to know, try to forget, and look at the definitions afresh.

Posted at December 20, 2012 12:23 AM UTC

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### Re: Names For Equivalences

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.

Posted by: Jesse McKeown on December 20, 2012 2:43 AM | Permalink | Reply to this

### Re: Names For Equivalences

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

Posted by: jim stasheff on December 20, 2012 12:58 PM | Permalink | Reply to this

### Re: Names For Equivalences

Would you say that a homotopy equivalence is a map equipped with data as in (1), or such that there merely exists data as in (1)? While classical homotopy theory may not be very aware of the distinction, I think the latter is a more faithful description of how homotopy equivalences are treated. For instance, the space of homotopy equivalences from $X$ to $Y$ is defined as a subspace of $Y^X$, not as a subspace of $Y^X \times X^Y \times Y^{[0,1]} \times X^{[0,1]}$. The latter definition wouldn’t give the right homotopy type — which is the same problem with (1) in type theory.

We can make all six types equivalent in type theory if we are willing to “squash” all types into mere properties. But as I’ve phrased them all above, though, they are meant to be interpreted in an unsquashed manner, although (2) through (5) are nevertheless automatically “mere properties” in the homotopical/type-theoretic sense (i.e. contractible when inhabited).

Posted by: Mike Shulman on December 20, 2012 3:17 PM | Permalink | Reply to this

### Re: Names For Equivalences

yes, such that there exists…
similar problem wiht H-map of H-spaces

Posted by: jim stasheff on December 21, 2012 1:12 PM | Permalink | Reply to this

### Re: Names For Equivalences

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’).

Posted by: Denis-Charles Cisinski on December 20, 2012 9:35 PM | Permalink | Reply to this

### Re: Names For Equivalences

I very much like “quasi-inverse” for (1). It’s wieldy(?) enough for frequent use, while still conveying the fact that this shouldn’t be thought of as the fundamental notion.

Posted by: Peter LeFanu Lumsdaine on December 21, 2012 2:48 PM | Permalink | Reply to this

### Re: Names For Equivalences

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?

Posted by: Emily Riehl on December 20, 2012 10:46 PM | Permalink | Reply to this

### Re: Names For Equivalences

the analogy between the univalence axiom and the completeness condition that arises in many models for $(\infty,n)$-categories.

Indeed! It’s more than an analogy, in fact. But I don’t think I’d noticed it in so many words before Urs brought it up last May.

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?

Well, actually any of (2) through (5) are usable here, since they all produce equivalent types. It’s true that (2) appears e.g. in Rezk’s original definition of completeness for Segal spaces, but I think (3) would do just as well there. (4) and (5) don’t make as much sense in classical homotopy theory, but are fine in type theory.

Voevodsky’s original definition of the univalence axiom was stated in terms of (4) (or equivalently (5)), which he called “weak equivalence”. I’ve been very much against calling them “weak”, however, because in type theory the definition is equivalent to, say, (2) and hence there is not really anything weak about it. In particular, they always have homotopy inverses.

That said, the new version of the HoTT Coq library is moving away from (4) towards using (2) as the basic structure, mainly for technical reasons: one of the things you want to do most with an equivalence is to extract its inverse, and (2) includes that inverse as an explicit datum whereas with (4) there is more unpacking to be done. I also agree that the notion which we use most day-to-day should be the one dignified with the short term “equivalence”, so it seems that you’ll probably get your wish. If necessary to distinguish (2) from the other definitions, I think it’s probably most informative to call (2) an “adjoint equivalence”, as we have in the past.

I would still like to have good names for the other notions, however. I’ll wait for a few more responses before I chime in with my thoughts on those.

Posted by: Mike Shulman on December 21, 2012 12:31 AM | Permalink | Reply to this

### Re: Names For Equivalences

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)”

Posted by: M. on December 21, 2012 11:37 AM | Permalink | Reply to this

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