---
abstract: 'This paper provides geometrical descriptions of the Frobenius monad freely generated by a single object. These descriptions are related to results connecting Frobenius algebras and topological quantum field theories. In these descriptions, which are based on coherence results for self-adjunctions (adjunctions where an endofunctor is adjoint to itself), ordinals in $\varepsilon_0$ play a prominent role. The paper ends by considering how the notion of Frobenius algebra induces the collapse of the hierarchy of ordinals in $\varepsilon_0$, and by raising the question of the exact categorial abstraction of the notion of Frobenius algebra.'
author:
- |
    and [Zoran Petri' c]{}\
    [Mathematical Institute, SANU]{}\
    [Knez Mihailova 36, p.f. 367, 11001 Belgrade, Serbia]{}\
    [email: {kosta, zpetric}@mi.sanu.ac.yu]{}
title: Ordinals in Frobenius Monads
---

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[*Mathematics Subject Classification* ([*2000*]{}): $\;$03G30, 03E10, 16H05, 16W30, 18C15, 18D10, 55N22]{}

[: Frobenius monad, Frobenius algebra, self-adjunction, bijunction, coherence, split equivalence, transfinite ordinal, separable algebra, cobordism, topological quantum field theory]{}

Introduction
============

The purpose of this paper is to connect two seemingly distant and unrelated topics: Frobenius algebras and ordinals contained in the infinite denumerable ordinal $\varepsilon_0$ (namely, the least ordinal $\xi$ such that ${\omega^\xi=\xi}$). Frobenius algebras play an important role in topology, mathematical physics and 