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We consider three monads on$\\mathsf{Top}$, the category of topological spaces, which formalize topological aspects of probability and possibility in categorical terms. The first one is the Hoare hyperspace monad H , which assigns to every space its space of closed subsets equipped with the lower Vietoris topology. The second one is the monad V of continuous valuations, also known as the extended probabilistic powerdomain. We construct both monads in a unified way in terms of double dualization. This reveals a close analogy between them and allows us to prove that the operation of taking the support of a continuous valuation is a morphism of monads$V \\to H$. In particular, this implies that every H -algebra (topological complete semilattice) is also a V -algebra. We show that V can be restricted to a submonad of$\\tau$-smooth probability measures on$\\mathsf{Top}$. By composing these morphisms of monads, we obtain that taking the supports of$\\tau$-smooth probability measures is also a morphism of monads.

Let A be a 2-category with suitable opcomma objects and pushouts. We give a direct proof that, provided that the codensity monad of a morphism p exists and is preserved by a suitable morphism, the factorization given by the lax descent object of the two-dimensional cokernel diagram of p is up to isomorphism the same as the semantic factorization of p, either one existing if the other does. The result can be seen as a counterpart account to the celebrated Bénabou–Roubaud theorem. This leads in particular to a monadicity theorem, since it characterizes monadicity via descent. It should be noted that all the conditions on the codensity monad of p trivially hold whenever p has a left adjoint and, hence, in this case, we find monadicity to be a two-dimensional exact condition on p, namely, to be an effective faithful morphism of the 2-category A .

The theory of presentations of enriched monads was developed by Kelly, Power, and Lack, following classic work of Lawvere, and has been generalized to apply to subcategories of arities in recent work of Bourke–Garner and the authors. We argue that, while theoretically elegant and structurally fundamental, such presentations of enriched monads can be inconvenient to construct directly in practice, as they do not directly match the definitional procedures used in constructing many categories of enriched algebraic structures via operations and equations. Retaining the above approach to presentations as a key technical underpinning, we establish a flexible formalism for directly describing enriched algebraic structure borne by an object of a V -category C in terms of parametrized J - ary operations and diagrammatic equations for a suitable subcategory of arities J ↪ C . On this basis we introduce the notions of diagrammatic J - presentation and J - ary variety , and we show that the category of J -ary varieties is dually equivalent to the category of J -ary V -monads. We establish several examples of diagrammatic J -presentations and J -ary varieties relevant in both mathematics and theoretical computer science, and we define the sum and tensor product of diagrammatic J -presentations. We show that both J - relative monads and J - pretheories give rise to diagrammatic J -presentations that directly describe their algebras. Using diagrammatic J -presentations as a method of proof, we generalize the pretheories-monads adjunction of Bourke and Garner beyond the locally presentable setting. Lastly, we generalize Birkhoff’s Galois connection between classes of algebras and sets of equations to the above setting.

50 I. 50 II. 52 III. 53 IV. 66 V. 71 VI. 72 VII. 74 75 References 75 SUMMARY: Cryptospores, recovered from Ordovician through Devonian rocks, differ from trilete spores in possessing distinctive configurations (i.e. hilate monads, dyads, and permanent tetrads). Their affinities are contentious, but knowledge of their relationships is essential to understanding the nature of the earliest land flora. This review brings together evidence about the source plants, mostly obtained from spores extracted from minute, fragmented, yet exceptionally anatomically preserved fossils. We coin the term ‘cryptophytes’ for plants that produced the cryptospores and show them to have been simple terrestrial organisms of short stature (i.e. millimetres high). Two lineages are currently recognized. Partitatheca shows a combination of characters (e.g. spo‐rophyte bifurcation, stomata, and dyads) unknown in plants today. Lenticulatheca encompasses discoidal sporangia containing monads formed from dyads with ultrastructure closer to that of higher plants, as exemplified by Cooksonia. Other emerging groupings are less well characterized, and their precise affinities to living clades remain unclear. Some may be stem group embryophytes or tracheophytes. Others are more closely related to the bryophytes, but they are not bryophytes as defined by extant representatives. Cryptophytes encompass a pool of diversity from which modern bryophytes and vascular plants emerged, but were competitively replaced by early tracheophytes. Sporogenesis always produced either dyads or tetrads, indicating strict genetic control. The long‐held consensus that tetrads were the archetypal condition in land plants is challenged.

The stochastic interpretation of Parikh＇s game logic should not follow the usual pattern of Kripke models, which in turn are based on the Kleisli morphisms for the Giry monad, rather, a specific and more general approach to proba- bilistic nondeterminism is required. We outline this approach together with its probabilistic and measure theoretic basis, in- troducing in a leisurely pace the Giry monad and its Kleisli morphisms together with important techniques for manipu- lating them. Proof establishing specific techniques are given, and pointers to the extant literature are provided. After working through this tutorial, the reader should find it easier to follow the original literature in this and related areas, and it should be possible for her or him to appreciate measure theoretic arguments for original work in the areas of Markov transition systems, and stochastic effectivity func- tions.

It is well known that classical varieties of $\\Sigma$ -algebras correspond bijectively to finitary monads on $\\mathsf{Set}$ . We present an analogous result for varieties of ordered $\\Sigma$ -algebras, that is, categories of algebras presented by inequations between $\\Sigma$ -terms. We prove that they correspond bijectively to strongly finitary monads on $\\mathsf{Pos}$ . That is, those finitary monads which preserve reflexive coinserters. We deduce that strongly finitary monads have a coinserter presentation, analogous to the coequalizer presentation of finitary monads due to Kelly and Power. We also show that these monads are liftings of finitary monads on $\\mathsf{Set}$ . Finally, varieties presented by equations are proved to correspond to extensions of finitary monads on $\\mathsf{Set}$ to strongly finitary monads on $\\mathsf{Pos}$ .

For a quantale I , the unit interval endowed with a continuous triangular norm, we introduce the canonical, op-canonical and Kleisli extensions of the conical I -semifilter monad to I - Rel . It is proved that the op-canonical extension coincides with the Kleisli extension.

The advent of embryophytes (land plants) is among the most important evolutionary breakthroughs in Earth history. It irreversibly changed climates and biogeochemical processes on a global scale; it allowed all eukaryotic terrestrial life to evolve and to invade nearly all continental environments. Before this work, the earliest unequivocal embryophyte traces were late Darriwilian (late Middle Ordovician; c. 463-461 million yr ago (Ma)) cryptospores from Saudi Arabia and from the Czech Republic (western Gondwana). Here, we processed Dapingian (early Middle Ordovician, c. 473-471 Ma) palynological samples from Argentina (eastern Gondwana). We discovered a diverse cryptospore assemblage, including naked and envelope-enclosed monads and tetrads, representing five genera. Our discovery reinforces the earlier suggestion that embryophytes first evolved in Gondwana. It indicates that the terrestrialization of plants might have begun in the eastern part of Gondwana. The diversity of the Dapingian assemblage implies an earlier, Early Ordovician or even Cambrian, origin of embryophytes. Dapingian to Aeronian (Early Silurian) cryptospore assemblages are similar, suggesting that the rate of embryophyte evolution was extremely slow during the first c. 35-45 million yr of their diversification. The Argentinean cryptospores predate other cryptospore occurrences by c. 8-12 million yr, and are currently the earliest evidence of plants on land.

We propose that entropy is a universal co-homological class in a theory associated to a family of observable quantities and a family of probability distributions. Three cases are presented: (1) classical probabilities and random variables; (2) quantum probabilities and observable operators; (3) dynamic probabilities and observation trees. This gives rise to a new kind of topology for information processes, that accounts for the main information functions: entropy, mutual-informations at all orders, and Kullback–Leibler divergence and generalizes them in several ways. The article is divided into two parts, that can be read independently. In the first part, the introduction, we provide an overview of the results, some open questions, future results and lines of research, and discuss briefly the application to complex data. In the second part we give the complete definitions and proofs of the theorems A, C and E in the introduction, which show why entropy is the first homological invariant of a structure of information in four contexts: static classical or quantum probability, dynamics of classical or quantum strategies of observation of a finite system.

Given a sheaf on a projective space Pn{\\mathbf P}^n, we define a sequence of canonical and effectively computable Chow complexes on the Grassmannians of planes in Pn{\\mathbf P}^n, generalizing the well-known Beilinson monad on Pn{\\mathbf P}^n. If the sheaf has dimension kk, then the Chow form of the associated kk-cycle is the determinant of the Chow complex on the Grassmannian of planes of codimension k+1k+1. Using the theory of vector bundles and the canonical nature of the complexes, we are able to give explicit determinantal and Pfaffian formulas for resultants in some cases where no polynomial formulas were known. For example, the Horrocks–Mumford bundle gives rise to a polynomial formula for the resultant of five homogeneous forms of degree eight in five variables.