Eilenberg-Kelly-MacLane graphs [EK66, KM71] elegantly describe certain morphisms of closed categories. This paper shows that little more is needed to present the free star-autonomous category [Bar79] generated by a category, for a full coherence theorem: the commutativity of diagrams of canonical maps is decidable. Given a set A = {a, b, . . .} of generators, we define the category of A-linking...

This is a survey paper on the implication of Yoneda lemma, named after Japanese mathematician Nobuo Yoneda, to category theory. We prove Yoneda lemma. We use Yoneda lemma to prove that each of the notions universal morphism, universal element, and representable functor subsumes the other two. We prove that a category is anti-equivalent to the category of its representable functors as a corollar...

0.1. The kinds of homotopy theories under consideration in this paper are Waldhausen ∞-categories [2, Df. 2.7]. (We employ the quasicategory model of∞-categories for technical convenience.) These are ∞-categories with a zero object and a distinguished class of morphisms (called cofibrations or ingressive morphisms) that satisfies the following conditions. (0.1.1) Any equivalence is ingressive. ...

We exhibit an algorithm to decide if the fixed-points of a morphism avoid (long) abelian repetitions and we use it to show that long abelian squares are avoidable over the ternary alphabet. This gives a partial answer to one of Mäkelä's questions. Our algorithm can also decide if a morphism avoids additive repetitions or k-abelian repetitions and we use it to show that long 2-abelian square are...

The behaviour and interaction of finite limits (products, pullbacks and equalisers) and colimits (coproducts and coequalisers) in the category of sets is illustrated in a “hands on” way by giving the interpretation of a simple imperative language in terms of partial functions between sets of states. We show that the interpretation is a least fixed point and satisfies the usual proof rule for lo...

It was the idea of Calvin Elgot [4] to use Lawvere theories for the study of the semantics of recursion. He introduced iterative theories as those Lawvere theories in which every ideal morphism e : n → n+p (representing a system of recursive equations in n variables and p parameters) has a unique solution, i. e., a unique morphism e† such that the equation e† = [e†, idp] ·e holds. Elgot proved ...

A braided monoidal category may be considered a $3$-category with one object and $1$-morphism. In this paper, we show that, more generally, $3$-categories $1$-morphisms given by elements of group $G$ correspond to $G$-crossed categories, certain mathematical structures which have emerged as important invariants low-dimensional quantum field theories. More precisely, that the 4-category $\mathca...

1. Preliminaries. A functor from a category of combinatorial geometries, or equivalently a category of geometric lattices, to a category of commutative algebras will be described, and some properties of this functor will be investigated. In particular, a cohomology will be associated to each point of a geometry and will be derived from the associated algebra. If (G, S) is a geometry on a set S ...

A function complex is introduced on the category of flows so that the model category of flows becomes a simplicial model category. This allows us to show that the homotopy branching space of the cone of a morphism of flows is homotopy equivalent to the cone of its image by the homotopy branching space functor. The crux of the proof is that the homotopy branching space of the terminal flow is co...

In this paper we study two types of descent in the category Berkovich analytic spaces: flat and with respect to an extension ground field. Quite surprisingly, deepest results direction seem be second type, including properties being a good space morphism without boundary.