4,171 research outputs found
Categorical Pullbacks
The main purpose of this article is to introduce the categorical concept of pullback in Mizar. In the first part of this article we redefine homsets, monomorphisms, epimorpshisms and isomorphisms [7] within a free-object category [1] and it is shown there that ordinal numbers can be considered as categories. Then the pullback is introduced in terms of its universal property and the Pullback Lemma is formalized [15]. In the last part of the article we formalize the pullback of functors [14] and it is also shown that it is not possible to write an equivalent definition in the context of the previous Mizar formalization of category theory [8].Via del Pero 102, 54038 Montignoso, ItalyJiri Adamek, Horst Herrlich, and George E. Strecker. Abstract and Concrete Categories: The Joy of Cats. Dover Publication, New York, 2009.Grzegorz Bancerek. Cardinal numbers. Formalized Mathematics, 1(2):377–382, 1990.Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91–96, 1990.Grzegorz Bancerek. The well ordering relations. Formalized Mathematics, 1(1):123–129, 1990.Grzegorz Bancerek. Zermelo theorem and axiom of choice. Formalized Mathematics, 1 (2):265–267, 1990.Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107–114, 1990.Francis Borceaux. Handbook of Categorical Algebra I. Basic Category Theory, volume 50 of Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge, 1994.Czesław Byliński. Introduction to categories and functors. Formalized Mathematics, 1 (2):409–420, 1990.Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1): 55–65, 1990.Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153–164, 1990.Czesław Byliński. Partial functions. Formalized Mathematics, 1(2):357–367, 1990.Czesław Byliński. Some basic properties of sets. Formalized Mathematics, 1(1):47–53, 1990.Agata Darmochwał. Finite sets. Formalized Mathematics, 1(1):165–167, 1990.F. William Lawvere. Functorial semantics of algebraic theories and some algebraic problems in the context of functorial semantics of algebraic theories. Reprints in Theory and Applications of Categories, 5:1–121, 2004.Saunders Mac Lane. Categories for the Working Mathematician, volume 5 of Graduate Texts in Mathematics. Springer Verlag, New York, Heidelberg, Berlin, 1971.Beata Padlewska. Families of sets. Formalized Mathematics, 1(1):147–152, 1990.Marco Riccardi. Object-free definition of categories. Formalized Mathematics, 21(3): 193–205, 2013. doi:10.2478/forma-2013-0021.Andrzej Trybulec. Enumerated sets. Formalized Mathematics, 1(1):25–34, 1990.Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67–71, 1990.Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1 (1):73–83, 1990.Edmund Woronowicz. Relations defined on sets. Formalized Mathematics, 1(1):181–186, 1990
t-structures are normal torsion theories
We characterize -structures in stable -categories as suitable
quasicategorical factorization systems. More precisely we show that a
-structure on a stable -category is
equivalent to a normal torsion theory on , i.e. to a
factorization system where both classes
satisfy the 3-for-2 cancellation property, and a certain compatibility with
pullbacks/pushouts.Comment: Minor typographical corrections from v1; 25 pages; to appear in
"Applied Categorical Structures
Some remarks on pullbacks in Gumm categories
We extend some properties of pullbacks which are known to hold in a Mal'tsev
context to the more general context of Gumm categories. The varieties of
universal algebras which are Gumm categories are precisely the congruence
modular ones. These properties lead to a simple alternative proof of the known
property that central extensions and normal extensions coincide for any Galois
structure associated with a Birkhoff subcategory of an exact Goursat category.Comment: 12 page
Universal Constructions for (Co)Relations: categories, monoidal categories, and props
Calculi of string diagrams are increasingly used to present the syntax and
algebraic structure of various families of circuits, including signal flow
graphs, electrical circuits and quantum processes. In many such approaches, the
semantic interpretation for diagrams is given in terms of relations or
corelations (generalised equivalence relations) of some kind. In this paper we
show how semantic categories of both relations and corelations can be
characterised as colimits of simpler categories. This modular perspective is
important as it simplifies the task of giving a complete axiomatisation for
semantic equivalence of string diagrams. Moreover, our general result unifies
various theorems that are independently found in literature and are relevant
for program semantics, quantum computation and control theory.Comment: 22 pages + 3 page appendix, extended version of arXiv:1703.0824
An embedding theorem for adhesive categories
Adhesive categories are categories which have pushouts with one leg a
monomorphism, all pullbacks, and certain exactness conditions relating these
pushouts and pullbacks. We give a new proof of the fact that every topos is
adhesive. We also prove a converse: every small adhesive category has a fully
faithful functor in a topos, with the functor preserving the all the structure.
Combining these two results, we see that the exactness conditions in the
definition of adhesive category are exactly the relationship between pushouts
along monomorphisms and pullbacks which hold in any topos.Comment: 8 page
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