  Notions of guardedness serve to delineate the admissibility of cycles, e.g.
in recursion, corecursion, iteration, or tracing. We introduce an abstract
notion of guardedness structure on a symmetric monoidal category, along with a
corresponding notion of guarded traces, which are defined only if the cycles
they induce are guarded. We relate structural guardedness, determined by
propagating guardedness along the operations of the category, to geometric
guardedness phrased in terms of a diagrammatic language. In our setup, the
Cartesian case (recursion) and the co-Cartesian case (iteration) become
completely dual, and we show that in these cases, guarded tracedness is
equivalent to presence of a guarded Conway operator, in analogy to an
observation on total traces by Hasegawa and Hyland. Moreover, we relate guarded
traces to unguarded categorical uniform fixpoint operators in the style of
Simpson and Plotkin. Finally, we show that partial traces based on
Hilbert-Schmidt operators in the category of Hilbert spaces are an instance of
guarded traces.
