Idea

A steady function is one equipped with a certain sort of “witness of constancy”. However, in higher category theory and homotopy theory, it is debatable whether or not this witness really exhibits “constancy”, hence the use of a different word. (The word “steady” was suggested by Andrej Bauer.)

Definition

In homotopy type theory, a function $f:A\to B$ is steady if we have a term of type

$\prod_{(x,y:A)} (f x = f y).$

By regarding homotopy type theory as the internal logic of an (∞,1)-topos, we obtain a definition that makes sense in any higher category with binary products: a morphism $f:A\to B$ is steady if the two composites $A\times A \rightrightarrows A \xrightarrow{f} B$ are equivalent.

Relationship to constancy

If $f$ is constant in the sense that it factors through the terminal object (i.e. we have $f = \lambda x. b$ for some $b:B$), then $f$ is obviously steady. The converse holds if we know that the domain $A$ is inhabited, for if $a_0:A$, then $f a = f a_0$ for all $a:A$. However, the identity function of the empty type is steady, yet not equal to $\lambda x.b$ for any $b:\emptyset$ (since no such $b$ exists).

More generally, if $f$ factors through the propositional truncation ${\|A\|}$, then it is steady, since any two elements of ${\|A\|}$ are equal (i.e. it is an h-proposition). In fact, this is true if $f$ factors through any h-proposition (in which case it in fact also factors through ${\|A\|}$, by the universal property of the latter).

The converse to this last implication does hold for some specific $f:A\to B$, such as:

• If $B$ is an h-set. For then $f$ factors through the 0-truncation ${\|A\|_0}$, and the set-coequalizer of the two projections ${\|A\|_0} \times {\|A\|_0} \to {\|A\|_0}$ is the propositional truncation.

• If $A$ has split support. For then we have a composite ${\|A\|} \to A \xrightarrow{f} B$, whose restriction to $A$ is equal to $f$ by steadiness.

• If $A=P+Q$, with $P$ and $Q$ h-propositions. For then ${\|A\|}$ is the join $P*Q$ of $P$ and $Q$, i.e. the pushout of the two projections $P \leftarrow P\times Q \to Q$. The universal property of this pushout says exactly that any steady map $P+Q\to B$ factors through $P*Q = \Vert P+Q\Vert$.

• If $A=B$ (see below).

However, it can fail in general, even when $A$ is merely inhabited (i.e. ${\|A\|}= 1$). For instance, let $A=P+Q+R$ for h-propositions $P$, $Q$, and $R$, and let $B$ be the triple pushout of $P$, $Q$, and $R$ under $P\times Q$, $P\times R$, and $Q\times R$. Then there is a steady map $f:A\to B$, but there exist models in which ${\|A\|} = 1$ but $B$ has no global element]. The most straightforward such model is [[presheaves? on the poset of proper subsets of $\{a,b,c\}$, with $P=\{a,b\}$, $Q=\{b,c\}$, and $R=\{a,c\}$. In this model, we have $B(S) = 1$ for all nonempty proper subsets $S$, while $B(\emptyset) = S^1$, and $B$ has no global sections.

See this discussion.

In general, being steady may be regarded as an “incoherent approximation” to constancy in the sense of factoring through an h-proposition. Indeed for a set $A$, its propositional truncation is the set-coequalizer of $A\times A\rightrightarrows A$. However, in general such a construction requires the realization of a whole simplicial diagram (the simplicial kernel of the map $A\to 1$).

While an arbitrary steady function is not very coherent, a steady endofunction $f:A\to A$ has some extra degree of “coherence”, as witnessed by the following results of (KECA).

Lemma

If $f:A\to A$ is steady, then the type $Fix(f) \coloneqq \sum_{x:A} (f x = x)$ is an h-proposition, and equivalent to ${\|A\|}$.

Proof

Suppose $H: \prod_{(x,y:A)} (f x = f y)$, and let $(a,p),(b,q):Fix(f)$; we want to show $(a,p)=(b,q)$. Let $r:a=b$ be the concatenated path

$a \xrightarrow{p^{-1}} f a \xrightarrow{H_{a,a}^{-1}} f a \xrightarrow{H_{a,b}} f b \xrightarrow{q} b.$

It will suffice to show that $p \bullet r = ap_f(r) \bullet q$, where $ap_f$ denotes the action of $f$ on paths. However, the dependent action of $H$ on paths implies that $H_{x,y} \bullet ap_f(s) = H_{x,y'}$ for any $x:A$ and any $s:y=y'$, and in particular $ap_f(r) = H_{a,a}^{-1} \bullet H_{a,b}$. From this $p \bullet r = ap_f(r) \bullet q$ is immediate. Thus, $Fix(f)$ is an h-proposition.

Now we have a map $g:A\to Fix(f)$ defined by $g(x) \coloneqq (f x, H_{f x,x})$, so by the universal property of ${\|A\|}$, we have ${\|A\|} \to Fix(f)$. On the other hand, we have the first projection $pr_1:Fix(f) \to A$, and hence $Fix(f) \to {\|A\|}$. Thus, these two h-propositions are equal.

Theorem

A type $A$ has split support if and only if it admits a steady endomap $f:A\to A$.

Proof

Given ${\|A\|} \to A$, the composite $A\to {\|A\|} \to \A$ is steady. Conversely, if $f$ is steady, $Fix(f) = {\|A\|}$ by the lemma, so $pr_1:Fix(f) \to A$ splits the support of $A$.

Note that $pr_1 \circ g = f$, so that if we start with a steady endomap of $A$, deduce a splitting of the support of $A$, and then reconstruct a steady endomap, we obtain the same map. However, the proof of steadiness is generally different from the one we began with, so this “logical equivalence” is not an equivalence of types.

Corollary

A type $A$ is an h-set if and only if every identity type $x=_A y$ admits a steady endomap.

Proof

We know that $A$ is an h-set just when all $x=_A y$ have split support.

References

Last revised on October 12, 2013 at 20:54:29. See the history of this page for a list of all contributions to it.