Problem 1.(The Mapping Cone is a Homotopy Limit and a Homotopy Colimit) Let A be an Abelian category. Let A● and B● be objects in Ch●(A). Let f ● ∶ A● → B● be a morphism in Ch●(A). For every object T ● of Ch●(A), a left homotopy annihilator to T ● is a pair (Q●, σ●) of a morphism Q● ∶ B● → T ● in Ch●(A) and a nullhomotopy (σn ∶ An → T n−1)n∈Z of Q● ○ f ●, i.e., for every n ∈ Z, Q ○ f = dn−1 T ○...

We show that although the algebraic semantics of place/transition Petri nets under the collective token philosophy can be fully explained in terms of strictly symmetric monoidal categories, the analogous construction under the individual token philosophy is not completely satisfactory, because it lacks universality and also functoriality. We introduce the notion of pre-net to recover these aspe...

In this paper we study quotients of posets by group actions. In order to define the quotient correctly we enlarge the considered class of categories from posets to loopfree categories: categories without nontrivial automorphisms and inverses. We view group actions as certain functors and define the quotients as colimits of these functors. The advantage of this definition over studying the quoti...

We identify the topological Hochschild homology (THH) of the Thom spectrum associated to an E∞ classifying map X → BG, for G an appropriate group or monoid (e.g. U , O, and F ). We deduce the comparison from the observation of McClure, Schwanzl, and Vogt that THH of a cofibrant commutative S-algebra (E∞ ring spectrum) R can be described as an indexed colimit together with a verification that th...

Two notions are becoming increasingly important in ontological engineering: modularity and heterogeneity. Whilst heterogeneity allows ontologies to be specified in different logical formalisms and thus calls for sophisticated integration and combination techniques, various notions of modularity have been envisaged in the literature in order to support maintenance and reuse of (parts of) ontolog...

It is a known fact that the subobjects of an object in an adhesive category form a distributive lattice. Building on this observation, in the paper we show how the representation theorem for finite distributive lattices applies to subobject lattices. In particular, we introduce a notion of irreducible object in an adhesive category, and we prove that any finite object of an adhesive category ca...

This work is a first step toward the reconciliation of the two main approaches to composition in system modeling, namely the categorical one and the algebraic one. In particular, we present a mapping from CommUnity, which uses the categorical approach based on colimits, into the Tile Model, which uses algebraic operators for composition. Our results include a standard decomposition for CommUnit...

Abstract Simplicial presheaves on cartesian spaces provide a general notion of smooth spaces. There is corresponding version the singular complex functor, which maps to simplicial sets. We consider localisation (projective or injective) model category at morphisms become weak equivalences under functor. prove that this agrees with motivic-style $$\mathbb {R}$$ <mml:math xmlns:mml="http://www.w3...

In large software development projects, there is always a need for refactoring and optimization of the design. Usually designs are represented using UML diagrams (e.g., class diagrams). A engineering team may create multiple versions satisfying some external constraints. cases, subdiagrams developed can be selected combined into one diagram. It difficult to perform this task manually since manu...

Mazzola et al. propose a metaphor to describe the process by means of which an open question is solved in a creative way, likening it to the manner by which, due to the Yoneda Lemma of category theory, the internal structure of any object is completely characterised by a diagram of certain selected objects and morphisms. Building upon this metaphor, we explore how the creativity underlying conc...