symmetric monoidal (∞,1)-category of spectra
An initial algebra for an endofunctor $F$ on a category $C$ is an initial object in the category of algebras of $F$. These play a role in particular as the categorical semantics for inductive types.
The concept of an algebra of an endofunctor is arguably somewhat odd, a more natural concept being that of an algebra over a monad. However, the former can often be reduced to the latter.
If $\mathcal{C}$ is a complete category, then the category of algebras of an endofunctor $F : \mathcal{C} \to \mathcal{C}$ is equivalent to the category of algebras over a monad of the free monad on $F$, if the latter exists.
The proof is fairly straightforward, see for instance (Maciej) or at free monad.
The existence of free monads, on the other hand, can be a tricky question. One general technique is the transfinite construction of free algebras.
If $F$ has an initial algebra $\alpha: F(X) \to X$, then $X$ is isomorphic to $F(X)$ via $\alpha$.
In this sense, $X$ is a fixed point of $F$. Being initial, $X$ is the smallest fixed point of $F$ in that there is a map from $X$ to any other fixed point (indeed, any other algebra), and this map is an injection if $C$ is Set.
The dual concept is terminal coalgebra, which is the largest fixed point of $F$.
Given an initial algebra structure $\alpha: F(X) \to X$, define an algebra structure on $F(X)$ to be $F(\alpha): F(F(X)) \to F(X)$. By initiality, there exists an $F$-algebra map $i: X \to F(X)$, so that
commutes. Now it is trivial, in fact tautological that $\alpha$ is itself an $F$-algebra map $F(X) \to X$. Thus $\alpha \circ i = 1_X$, since both sides of the equation are $F$-algebra maps $X \to X$ and $X$ is initial. As a result, $F(\alpha) \circ F(i) = 1_{F(X)}$, so that $i \circ \alpha = 1_{F(X)}$ according to the commutative square. Hence $\alpha$ is an isomorphism, with inverse $i$.
In many cases, initial algebras can be constructed in recursive fashion, using the following special case of a theorem due to Adámek.
Let $C$ be a category with an initial object $0$ and transfinite composition of lengh $\omega$, hence colimits of sequences $\omega \to C$ (where $\omega$ is the first infinite ordinal), and suppose $F: C \to C$ preserves colimits of $\omega$-chains. Then the colimit $\gamma$ of the chain
carries a structure of initial $F$-algebra.
The $F$-algebra structure $F\gamma \to \gamma$ is inverse to the canonical map $\gamma \to F\gamma$ out of the colimit (which is invertible by the hypothesis on $F$). The proof of initiality may be extracted by dualizing the corresponding proof given at terminal coalgebra.
This approach can be generalized to the transfinite construction of free algebras.
Initial algebras of endofunctors are the categorical semantics of extensional inductive types. The generalization to weak initial algebras? captures the notion in intensional type theory and homotopy type theory.
The archetypical example of an initial algebra is the set of natural numbers.
Let $\mathcal{T}$ be topos and let $F : \mathcal{T} \to \mathcal{T}$ the functor given by
(hence the functor underlying the “maybe monad”). Then an initial algebra over $F$ is precisely a natural number object $\mathbb{N}$ in $\mathcal{T}$.
By definition, an $F$-algebra is an object $X$ equipped with a morphism
hence equivalently with a point $0 : * \to X$ and an endomorphism $s : X \to X$. This being inital means that for $(0_Y, s_Y) : * \coprod Y \to Y$ any other morphism, there is a unique morphism $f : X \to Y$ such that the diagram
commutes. This is the very definition of natural number object $X = \mathbb{N}$.
Theorem 2 applies in particular to any functor $F: Set \to Set$ which is a colimit of finitely representable functors $hom(n, -): X \mapsto X^n$, as in the following examples.
Let $A$ be a set, and let $F: Set \to Set$ be the functor $F(X) = 1 + A \times X$. Then the initial $F$-algebra is $A^*$, the free monoid on $A$. The $F$-algebra structure is
where $e: 1 \to A^*$ is the identity and $m|: A \times A^* \to A^*$ is the restriction of the monoid multiplication along the evident inclusion $i \times 1: A \times A^* \to A^* \times A^*$.
This “fixed point” of $F$ can be thought of as the result of the (slightly nonsensical) calculation
which can be made rigorous by interpreting the initial equality as defining the solution $X$ by recursion, and applying the theorem above.
Let $F: Set \to Set$ be the functor $F(X) = 1 + X^2$. Then the initial $F$-algebra is the set $T$ of isomorphism classes of finite (planar, rooted) binary trees. The $F$-algebra structure is
where $r: 1 \to T$ names the tree consisting of just a root vertex, and $j: T^2 \to T$ creates a tree $t \vee t'$ from two trees $t$, $t'$ by joining their roots to a new root, so that the root of $t$ becomes the left child and the root of $t'$ the right child of the new root.
The recursive equation
would seem to imply that the structure $T$ behaves something like a structural “sixth root of unity”, and indeed the structural isomorphism $T \cong F(T)$ allows one to exhibit an isomorphism
constructively, as famously explored in the paper by Andreas Blass, Seven Trees in One.
Let $F: Set \to Set$ be the functor $F(X) = X^*$ (the free monoid from an earlier example). Then the initial $F$-algebra is the set of isomorphism classes of finite planar rooted trees (not necessarily binary as in the previous example).
Let $C$ be the coslice category $\mathbb{Z} \downarrow Ab$, and let $F: C \to C$ be the functor which pushes out along the multiplication-by-$p$ map $p \cdot -: \mathbb{Z} \to \mathbb{Z}$. Then the initial $F$-algebra is the Pruefer group $\mathbb{Z}[p^{-1}]/\mathbb{Z}$. See the discussion at the n-Category Café, starting here.
Let $Ban$ be the category of complex Banach spaces and maps of norm bounded above by $1$, and let $F: \mathbb{C} \downarrow Ban \to \mathbb{C} \downarrow Ban$ be the squaring functor $X \mapsto X \times X$. Then the initial $F$-algebra is $L^1([0, 1])$ (with respect to the usual Lebesgue measure). This result is due to Tom Leinster; see this MathOverflow discussion.
inductive type, initial algebra of an endofunctor
higher inductive type, initial algebra of a presentable ∞-monad
A textbook account of the basic theory is in chapter 10 of
A brief review of some basics with an eye towards inductive types is in section 2 of
The relation to free monads is discussed in