A fixed point (or fixpoint) of an endofunction $f \colon X \to X$ is an element $x \in X$ such that $f(x) = x$. Similarly, if $C$ is a category with a terminal object $1$ and an endomorphism $f \colon X \to X$, then a fixed point of $f$ is an element $x \colon 1 \to X$ such that $f \circ x = x$.
More generally still, we can speak of the same notion but replacing global elements $x \colon 1 \to X$ by generalized elements $x \colon U \to X$, where again $x$ is a fixed point of $f \colon X \to X$ if $f \circ x = x$.
The same notion undergoes further generalization by categorification. For example, if $F \colon C \to C$ is an endofunctor, then an object $c$ of $C$ is called a “fixed point” of $F$ is there is an isomorphism $F(c) \cong c$ (although usually, a fixed point of a functor is an object together with a specified such isomorphism).
The importance of fixed points all throughout mathematics is difficult to overstate. They are particularly important in analysis, topology, lattice theory, set theory, and category theory. Fixed points of endofunctors frequently arise as solutions to “recursive equations”, especially in the form of initial algebras and terminal coalgebras.
A typical application of fixed points in analysis is through the following general theorem:
Suppose $X$ is an (inhabited) complete metric space with distance function $d$ and $f \colon X \to X$ is a function with Lipschitz constant $r$ less than $1$, i.e., $d(f(x), f(y)) \leq r \cdot d(x, y)$ for all $x, y \in X$. Then $f$ has a unique fixed point.
The proof is very easy: starting with an initial point $x_0$, the sequence defined by $x_{n+1} = f(x_n)$ is readily seen to be a Cauchy sequence which converges to some point $x$. The sequence $x_{n+1} = f(x_n)$ converges to both $x$ and $f(x)$, so $x = f(x)$. If $x$ and $y$ are fixed points, then $0 \leq d(x, y) = d(f(x), f(y)) \leq r \cdot d(x, y)$, so
whence $0 \leq d(x, y) \leq 0$ and therefore $x = y$.
By a suitable choice of space $X$ and endomorphism $f$, this theorem can be used to establish existence and uniqueness of solutions of differential equations (this should be expanded upon).
Brouwer’s theorem
Schauder
Kakutani
Tychonoff
Lefschetz
Atiyah-Singer; Wood’s Hole
If $L$ is a complete lattice, then every monotone (i.e., order-preserving) map $f \colon L \to L$ has a fixed point (in fact, the fixed points of $f$ form a complete lattice).
The algebras for the functor $f \colon L \to L$ are elements $x$ such that $f(x) \leq x$. Limits in $Alg(L)$ are preserved an reflected by the forgetful functor or inclusion $Alg(L) \to L$, hence $Alg(L)$ is complete if $L$ is. In particular, $Alg(L)$ has an initial object, and initial algebras are fixed points.
(In more down-to-earth terms, let $t$ be the infimum of $A = \{x: f(x) \leq x\}$. Then for every $x \in A$, we have $f(t) \leq f(x) \leq x$, hence $f(t) \leq t$. We therefore also have $f(f(t)) \leq f(t)$, so that $f(t) \in A$. But then it follows that $t \leq f(t)$, whence $t = f(t)$. Clearly this $t$ is a least fixed point of $f$.)
To show completeness of $fix(f)$, suppose given $S \subseteq fix(f)$, and let $s$ be the supremum of $S$ in $L$. Then the principal upward-closed set $s \uparrow$ generated by $s$ is a complete lattice. Also, $f$ restricts to a map $s \uparrow \to s \uparrow$ (for if $s \leq x$, then $p \leq x$ for all $p \in S$, whence $p = f(p) \leq f(x)$ for all $p \in S$, so that $s \leq f(x)$). The least fixed point of $f: s \uparrow \to s \uparrow$ is the supremum of $S$ in $fix(f)$.
A virtual corollary of this theorem is the Cantor-Schroeder-Bernstein theorem.
The following significantly strengthens the Knaster-Tarski theorem, and is based on the notion of an ipo (inductive partial order; see Paul Taylor’s book), i.e., a poset with a bottom element and admitting joins of directed subsets.
If $L$ is an ipo, then every monotone map $f \colon L \to L$ has a (least) fixed point.
Consider the smallest $S$ among subsets of $L$ which contain $\bot$, are closed under $f$, and closed under directed joins in $L$. Then the restriction $f \colon S \to S$ satisfies $1_S \leq f$, i.e., $f$ is inflationary. (For the set of all elements $x \in L$ such that $x \leq f(x)$ is another such subset, so $S$ is contained in it.) Thus it suffices to show that every inflationary map on a poset $S$ with a bottom element and directed joins has a fixed point.
The collection $I$ of inflationary monotone maps on $S$ is an ipo: its bottom element is $1_S$, and directed joins are computed pointwise. Moreover, $I$ is itself directed: if $f, g \in I$, then $f \circ g \in I$ dominates them both. Thus the directed join $t$ of the collection exists; it is the maximal element $t$ of $I$. It follows that that $f \circ t \leq t$, but also $t \leq f \circ t$ since $f$ is inflationary. So $f \circ t = t$, and in particular $t(\bot) \in S$ is a fixed point of $f$.
This $t(\bot)$ is a least fixed point of $f: L \to L$. For if $x$ is any fixed point, the downward-closed set $L \downarrow x$ contains $\bot$, is closed under $f$, and is closed under directed unions, and therefore $S \subseteq L \downarrow x$. Hence $t(\bot) \in S$ satisfies $t(\bot) \leq x$.
One may mimic the last part of the proof of the Knaster-Tarski theorem to show that if $L$ is an ipo and $f$ is monotone, then $fix(f)$ (with the order inherited from $L$) is also an ipo.
In at least some of the “lower” levels of the hierarchy of countable ordinals, leading up to the Feferman-Schütte ordinal, fixed points of continuous operators on the first uncountable ordinal play a central role. More information may be found at countable ordinal.
Critical points of elementary embeddings (non-fixed points)
Various classical fixed-point theorems for monotone functions on posets can be “categorified” to give appropriate fixed-point theorems for endofunctors on categories. An example is that Kleene’s fixed-point theorem generalizes to Adamek’s fixed-point theorem:
Let $C$ be a category with an initial object $0$ and colimits of $\kappa$-directed diagrams for some regular cardinal $\kapa$, and suppose $F \colon C \to C$ preserves $\kappa$-directed colimits. Then $F$ has an initial algebra (which by Lambek’s theorem is a fixed point of $F$).
Regarding $\kappa$ as an ordinal $\{\alpha \lt \kappa\}$ (hence a poset, hence a category), define a functor $G \colon \kappa \to C$ recursively: on objects, put $G(\emptyset) = 0$, $G(\alpha + 1) = F(G(\alpha))$, and $G(\beta) = colim_{\alpha \lt \beta} G(\alpha)$ for $\beta$ a limit ordinal. On morphisms $\alpha \lt \beta$,
$G(\emptyset \lt \beta)$ is the unique map $0 \to G(\beta)$;
For $\beta$ a limit ordinal, $G(\alpha \lt \beta)$ is a component of the cocone diagram that defines $G(\beta)$ as a colimit;
G(\alpha + 1 \lt \beta + 1) = F(G(\alpha \lt \beta))$;
For $\alpha$ a limit ordinal, $G(\alpha \lt \beta + 1)$ is the unique map $colim_{\gamma \lt \alpha} G(\gamma) \to G(\beta +1$ corresponding to the cocone from the diagram $G| \colon \alpha \to C$ to $G(\beta+1)$.
By assumption, $colim G$ exists in $C$, and this colimit is preserved by $F$. (To be continued.)
A typical application is where $C$ is a $\kappa$-accessible category and $F \colon C \to C$ is a $\kappa$-accessible functor. A concise statement is that accessible endofunctors on accessible categories with an initial object possess categorical fixed points (in fact, the fixed points form another accessible category with an initial object).
An arguably more elegant viewpoint on this is given in the following theorem and proof.
Suppose $C$ is locally presentable, i.e., accessible and complete/cocomplete, and suppose $F \colon C \to C$ is an accessible functor. Then the category of $F$-algebras is also locally presentable. In particular, there exists an initial $F$-algebra. Moreover, the category of fixed points, i.e., the category of $F$-algebras $X$ for which the structure $F X \to X$ is an isomorphism is also locally presentable (in particular, cocomplete and complete).
$Alg_F$ is an inserter construction, i.e., the data consisting of the evident 1-cell $U \colon Alg_F \to C$ and 2-cell $F \circ U \to U$ is universal among all such data in the 2-category $Cat$. Since $Acc$ is closed under 2-limits in $Cat$ such as inserters (see Makkai-Paré, theorem 5.1.6), the category $Alg_F$ is accessible. It is also complete since $C$ is complete (being locally presentable) and the underlying functor $U \colon Alg_F \to F$ reflects limits. Since $Alg_F$ is accessible and complete, it is also cocomplete, hence contains an initial object.
Similarly, the category of fixed points $i \colon Fix(F) \to C$ is formed as an iso-inserter construction and is therefore accessible. If $D \colon J \to Fix(F)$ is a diagram of fixed points and $c \in Ob(C)$ is the colimit of $i \circ D$, then $F$ induces a functor (which we give the same name) $F\colon c \downarrow C \to c \downarrow C$, because for any object $c \to d$ in $c \downarrow C$, we have a cocone $i \circ D \cong F \circ i \circ D \to F(d)$, factoring uniquely through an arrow $c \to F(d)$. Since $c \downarrow C$ is a locally presentable category, the accessible functor $F$ acting on it has an initial algebra $(c \to c', (c \to F(c')) \to (c \to c'))$, as we argued before. This is the colimit of $D$ in $Fix(F)$, as is easily checked. Therefore $Fix(F)$ is cocomplete and accessible.
Boundedness conditions, such as those given as hypotheses in the previous two theorems, are generally required to establish existence of categorical fixed points. The example of the covariant power-set functor on $Set$ shows that a naive generalization of the Knaster-Tarski theorem from complete lattices to complete/cocomplete categories is hopelessly false.
Paul Taylor has built an elegant theory of locating certain initial algebras inside final coalgebras, via his notion of well-founded coalgebras. This too can be “categorified” (to be continued).
Let $E$ be a topos, and let $F \colon E \to E$ be a left-exact idempotent functor. Then the category of $F$-coalgebras $X$ whose structure maps $\theta \colon X \to F(X)$ are isomorphisms is a topos.
Todd: Here “idempotent” involves a coassociativity condition. To be related to structures over a modal operator, as at my web, or to diads a la Toby Kenney.
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Claudio Hermida, Bart Jacobs, Structural induction and coinduction in a fibrational setting, Information and Computation 145 (1997), 107-152. (citeseer)
Michael Makkai and Robert Paré, Accessible Categories: The Foundations of Categorical Model Theory, Contemporary Mathematics 104, AMS (1989).