I wanted to mention that at one point Zalamea basically says that Peirce is doing category theory, or category theory is Peircean. — fishfry
I don't really see that myself as category theory seeks a closed structure preserving relation whereas semiosis is open ended both in being grounded in spontaneity and hierarchically elaborative. The spirit seems quite different as even though Peirce appears to be proposing rigid categories (and indeed goes overboard in turning his trichotomy into a hierarchy of 66 classes of sign), essentially the whole structure is quite fluid and approximate - more always a process than a structure as such.
So category theory seeks an analytic foundations whereas semiosis seeks a synthetic one. One is about the tight circle of a conservation principle where you can move about among different versions of the same thing without information being lost (the essential structure always preserved), while the other is an open story about how information actually gets created ... from "nothing".
They may still relate. But probably as Peirce telling the developmental tale of how any exact structure can come to be, and then category theory as a tale of that developed general structure.
So perhaps a connection. But coming at it from quite different metaphysical directions. So foundationally different as projects.
I have to say that I have a somewhat negative view of category theory because it seems to add so little to the practice of science. In particular, two rather brilliant people - Robert Rosen in mathematical biology and John Baez in mathematical physics - have tried to apply it in earnest to real world modelling (life itself, and particle physics). Yet the results feel stilted. Nothing very fruitful was achieved.
By contrast, semiosis just slots straight into the natural sciences. It makes instant sense.
Category theory is dyadic and associative - which is not wrong but, to me, the flattened mechanical view of reality. It is structure frozen out of the developmental processes from which - in nature - it must instead emerge as a limit.
Then semiosis is the three dimensional and dynamical view of reality - organic in that it captures the further axis which speaks to a fundamental instability of nature, and hence the need for emergent development of regulating structure.
The switch from a presumption of foundational stability to foundational instability is something I want to emphasise. That is the Heraclitean shift in thought. Regularity has to emerge to stabilise things. And yet regularity still needs vague or unstable foundations. The world can't be actually frozen in time.
And this connects back to models of the continuum. The mathematician wants to have a number line that can be cut - and the cut is stable. The number line is a passive entity that simply accepts any mark we try to make. It is a-causal - in exactly the same mechanical fashion that Newton imagined the atomism of masses free to do their causal thing within the passive backdrop of an a-causal void.
But Peirceanism would say the opposite. The number line - like the quantum vacuum - is alive with a zero point energy. It sizzles and crackles with possibility. On the finest scale, it becomes impossible to work with due to its fundamental instability.
And regular maths seems to understand that unconsciously. That is why it approaches the number line with a system of constraints. As Zalamea describes, the strategy to approach the reals is via the imposition of a succession of distinctions - the operations of difference, proportion and then finally (in some last gasp desperation) the waving hand of future convergence.
So maths tames the number line by a series of constraining steps. It minimises its indeterminism or dynamism, and looks up feeling relieved. Its world is now safe to get on with arithmetic.
But the Peircean revolution is about seeing this for what it really is. Maths just wants to shrink instability out of sight. Peirce says no. Let's turn our metaphysics around so that it becomes an account of this whole thing - the instability that is fundamental and the semiotic machinery that arises to tame it. Maths itself needs to be understood as a semiotic exercise.
So that would be where semiotics stands in regards to category theory. It is the bigger view that explains why mathematicians might strive to extract some rigid final frozen closed sense of essential mathematical structure from the wildly tossing seas of pure and unbounded possibility.
I would note the interesting contrast with fundamental physics where the crisis is instead quantum instability. In seeking a solid atomistic foundation, at a certain ultimate Planck scale, suddenly everything went as pear-shaped as could be imagined. Reality became just fundamentally weird and impossible.
But that is too much hyperventilation in the other direction. Just looking around we can see the fact that existence itself is thoroughly tamed quantum indeterminacy. The Universe as it is (especially now that it is so close to its heat death) is classical to a very high degree. So all that quantum weirdness is in fact pretty much completely collapsed in practice. Instability has been constrained by its own emergent classical limit (its own sum over possibility).
So where maths is too cosy in believing in its classicality, physics is too hung up on its discovery of basic instability. Both have gone overboard in complementary directions.
Semiosis is then the metaphysics that stands in the middle and can relate the determinate to the indeterminate in logical fashion. Especally as pansemiosis - the nascent field of dissipative structure theory - it is the quantum interpretation that finally makes sense.
Hot damn! ;)