RESEARCH STATEMENT My current research interests lie in ...

2 STEFAN MILIUS2. Recent Research2.1. Completely Iterative Algebras. My paper [34] concerns a generalization and extension of the classicalwork of Elgot and Nelson using ideas from the theory of coalgebras, see e. g. [2, 24, 27, 42] for introductorytexts. The basic observation is that infinite Σ-trees form the terminal coalgebra for the polynomial endofunctorassociated to the signature Σ. Then one can prove the classical results using category-theoreticmethods, i. e., the universal property of a terminal coalgebra is sufficient to obtain those result.More precisely, in lieu of a signature one works with an endofunctor H of Set (or a more general categorysatisfying some rather mild side conditions). Then a (co-)algebra is an underlying set A together with a mapfrom HA to A (or from A to HA, respectively). I introduce the concept of completely iterative algebras(cia). For a signature Σ the algebra T Σ X of all Σ-trees over the set X of generators is completely iterative.Other important examples stem from the realm of algebras on complete metric spaces. We already saw thegolden ratio and the Cantor space as unique solutions of recursive equations. The respective cias are carriedby the interval [1, 2] and by the set of non-empty closed subsets of [0, 1] with the Hausdorff metric. A basicopening result of the theory of cias is theTheorem 2.1. [34] A free cia on a set X is precisely a terminal coalgebra for the functor H( ) + X.The importance of this result, bringing algebraic and coalgebraic methods under one roof, lies in applications:terminal coalgebras often serve as domains for semantics of recursive definition; infinite trees, streams,formal languages accepted by sequential automata form terminal coalgebras. The characterization of terminalcoalgebras as free algebras of some sort opens the door for coalgebraic tools in algebraic semantics.And so we work with endofunctors that have enough terminal coalgebras, i. e., for every set X there exists aterminal coalgebra T X for H( ) + X. This is a mild assumption. For example, every accessible endofunctorof Makkai and Paré [30] has enough terminal coalgebras.Theorem 2.2. [1, 34] Let H have enough terminal coalgebras T X. Then T is the object assignment of amonad, and this monad is the free completely iterative monad on H.This result is an analogue of Nelson’s result on iterative algebras; the above theorem states that free ciasyield a free completely iterative theory. At the same time our result is more general and we feel that ourproof is clearer using category-theoretic language and tools. For the category-theoretic semantics of recursivefunction definitions Theorem 2.2 is an important prerequisite. In the special case of a polynomial endofunctorarising from a signature, the universal property of the monad T captures second-order substitution, i. e.,substitution of infinite trees for operation symbols. This notion is necessary to formulate the notion of asolution of a recursive function definition in the appropriate generality. We believe that this is the first timesecond-order substitution is understood by a categorical universal property.2.2. Iterative Algebras. In joint work with Adámek and Velebil [4, 5, 6] we deal with iterative algebrasand iterative theories. Recall that here we are concerned with unique solutions of finite systems of recursiveequations. Hence, the suitable category-theoretic environment for this study are locally finitely presentablecategories of Gabriel and Ulmer [21], e. g., sets, posets, graphs, finitary varieties of algebras, etc. In this settingwe introduce iterative algebras for an endofunctor. All completely iterative algebras for an endofunctorare, of course, iterative. Rational trees for a signature Σ form an iterative algebra which is not completelyiterative (for the polynomial endofunctor arising from Σ).Theorem 2.3. [6] For a finitary endofunctor H a free iterative algebra RX on an object X is obtained asthe colimit of all coalgebras for H( ) + X with a finitely presentable carrier.The existence of free iterative algebras is easy to prove from general category-theoretic results. However,the coalgebraic construction is a non-trivial and crucial result. All our subsequent results on iterativity relyon it. Furthermore, it yields explicit descriptions of free iterative algebras. For example, the constructionexplains that for a signature Σ free iterative algebras are rational Σ-trees as expected. The main resultof [4, 6] isTheorem 2.4. For every finitary endofunctor of a locally finitely presentable category the monad of freeiterative algebras is a free iterative monad.