Quasi-categoryIn mathematics, more specifically , a quasi-category (also called quasicategory, weak Kan complex, inner Kan complex, infinity category, ∞-category, Boardman complex, quategory) is a generalization of the notion of a . The study of such generalizations is known as . Quasi-categories were introduced by . André Joyal has much advanced the study of quasi-categories showing that most of the usual basic and some of the advanced notions and theorems have their analogues for quasi-categories.
Higher category theoryIn mathematics, higher category theory is the part of at a higher order, which means that some equalities are replaced by explicit arrows in order to be able to explicitly study the structure behind those equalities. Higher category theory is often applied in algebraic topology (especially in homotopy theory), where one studies algebraic invariants of spaces, such as their fundamental . An ordinary has and morphisms, which are called 1-morphisms in the context of higher category theory.
Simplicial setIn mathematics, a simplicial set is an object composed of simplices in a specific way. Simplicial sets are higher-dimensional generalizations of directed graphs, partially ordered sets and . Formally, a simplicial set may be defined as a contravariant functor from the to the . Simplicial sets were introduced in 1950 by Samuel Eilenberg and Joseph A. Zilber. Every simplicial set gives rise to a "nice" topological space, known as its geometric realization.
Derived categoryIn mathematics, the derived category D(A) of an A is a construction of homological algebra introduced to refine and in a certain sense to simplify the theory of derived functors defined on A. The construction proceeds on the basis that the of D(A) should be chain complexes in A, with two such chain complexes considered isomorphic when there is a chain map that induces an isomorphism on the level of homology of the chain complexes. Derived functors can then be defined for chain complexes, refining the concept of hypercohomology.
CatThe cat (Felis catus) is a domestic species of small carnivorous mammal. It is the only domesticated species in the family Felidae and is commonly referred to as the domestic cat or house cat to distinguish it from the wild members of the family. Cats are commonly kept as house pets but can also be farm cats or feral cats; the feral cat ranges freely and avoids human contact. Domestic cats are valued by humans for companionship and their ability to kill vermin. About 60 cat breeds are recognized by various cat registries.
MapA map is a symbolic depiction emphasizing relationships between elements of some space, such as objects, regions, or themes. Many maps are static, fixed to paper or some other durable medium, while others are dynamic or interactive. Although most commonly used to depict geography, maps may represent any space, real or fictional, without regard to context or scale, such as in brain mapping, DNA mapping, or computer network topology mapping.
Feral catA feral cat or a stray cat is an unowned domestic cat (Felis catus) that lives outdoors and avoids human contact: it does not allow itself to be handled or touched, and usually remains hidden from humans. Feral cats may breed over dozens of generations and become an aggressive local apex predator in urban, savannah and bushland environments. Some feral cats may become more comfortable with people who regularly feed them, but even with long-term attempts at socialization, they usually remain aloof and are most active after dusk.
Black catA black cat is a domestic cat with black fur that may be a mixed or specific breed, or a common domestic cat of no particular breed. The Cat Fanciers' Association (CFA) recognizes 22 cat breeds that can come with solid black coats. The Bombay breed is exclusively black. All-black fur pigmentation is slightly more prevalent in male cats than female cats. Most black cats have golden irises due to their high melanin pigment content. In popular myths, witches are believed to be associated with black cats.
Persian catThe Persian cat, also known as the Persian longhair, is a long-haired breed of cat characterized by a round face and short muzzle. The first documented ancestors of Persian cats might have been imported into Italy from Khorasan as early as around 1620, however this has not been proven. Instead there is stronger evidence for a longhaired cat breed being exported from Afghanistan and Iran from the 19th century onwards. Widely recognized by cat fancy since the late 19th century, Persian cats were first adopted by the British, and later by American breeders after World War II.
Map layoutMap layout, also called map composition or (cartographic) page layout, is the part of cartographic design that involves assembling various map elements on a page. This may include the map image itself, along with titles, legends, scale indicators, inset maps, and other elements. It follows principles similar to page layout in graphic design, such as balance, gestalt, and visual hierarchy. The term map composition is also used for the assembling of features and symbols within the map image itself, which can cause some confusion; these two processes share a few common design principles but are distinct procedures in practice.
Sphynx catThe Sphynx cat (pronounced , ˈsfɪŋks) also known as the Canadian Sphynx, is a breed of cat known for its lack of fur. Hairlessness in cats is a naturally occurring genetic mutation, and the Sphynx was developed through selective breeding of these animals, starting in the 1960s. According to breed standards, the skin should have the texture of chamois leather, as it has fine hairs, or the cat may be completely hairless. Whiskers may be present, either whole or broken, or may be totally absent.
World mapA world map is a map of most or all of the surface of Earth. World maps, because of their scale, must deal with the problem of projection. Maps rendered in two dimensions by necessity distort the display of the three-dimensional surface of the earth. While this is true of any map, these distortions reach extremes in a world map. Many techniques have been developed to present world maps that address diverse technical and aesthetic goals. Charting a world map requires global knowledge of the earth, its oceans, and its continents.
Homotopy categoryIn mathematics, the homotopy category is a built from the category of topological spaces which in a sense identifies two spaces that have the same shape. The phrase is in fact used for two different (but related) categories, as discussed below. More generally, instead of starting with the category of topological spaces, one may start with any and define its associated homotopy category, with a construction introduced by Quillen in 1967. In this way, homotopy theory can be applied to many other categories in geometry and algebra.
Road mapA road map, route map, or street map is a map that primarily displays roads and transport links rather than natural geographical information. It is a type of navigational map that commonly includes political boundaries and labels, making it also a type of political map. In addition to roads and boundaries, road maps often include points of interest, such as prominent businesses or buildings, tourism sites, parks and recreational facilities, hotels and restaurants, as well as airports and train stations.
Set theorySet theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly concerned with those that are relevant to mathematics as a whole. The modern study of set theory was initiated by the German mathematicians Richard Dedekind and Georg Cantor in the 1870s. In particular, Georg Cantor is commonly considered the founder of set theory.
Topographic mapIn modern mapping, a topographic map or topographic sheet is a type of map characterized by large-scale detail and quantitative representation of relief features, usually using contour lines (connecting points of equal elevation), but historically using a variety of methods. Traditional definitions require a topographic map to show both natural and artificial features. A topographic survey is typically based upon a systematic observation and published as a map series, made up of two or more map sheets that combine to form the whole map.
Model categoryIn mathematics, particularly in homotopy theory, a model category is a with distinguished classes of morphisms ('arrows') called 'weak equivalences', 'fibrations' and 'cofibrations' satisfying certain axioms relating them. These abstract from the category of topological spaces or of chain complexes ( theory). The concept was introduced by . In recent decades, the language of model categories has been used in some parts of algebraic K-theory and algebraic geometry, where homotopy-theoretic approaches led to deep results.
Triangulated categoryIn mathematics, a triangulated category is a with the additional structure of a "translation functor" and a class of "exact triangles". Prominent examples are the of an , as well as the . The exact triangles generalize the short exact sequences in an abelian category, as well as fiber sequences and cofiber sequences in topology. Much of homological algebra is clarified and extended by the language of triangulated categories, an important example being the theory of sheaf cohomology.
Loop spaceIn topology, a branch of mathematics, the loop space ΩX of a pointed topological space X is the space of (based) loops in X, i.e. continuous pointed maps from the pointed circle S1 to X, equipped with the compact-open topology. Two loops can be multiplied by concatenation. With this operation, the loop space is an A∞-space. That is, the multiplication is homotopy-coherently associative. The set of path components of ΩX, i.e. the set of based-homotopy equivalence classes of based loops in X, is a group, the fundamental group π1(X).
QuestionA question is an utterance which serves as a request for information. Questions are sometimes distinguished from interrogatives, which are the grammatical forms typically used to express them. Rhetorical questions, for instance, are interrogative in form but may not be considered bona fide questions, as they are not expected to be answered. Questions come in a number of varieties. Polar questions are those such as the English example "Is this a polar question?", which can be answered with "yes" or "no".
