|
Isofibration
In mathematics, especially in homotopy theory, a left fibration of simplicial sets is a map that has the right lifting property with respect to the horn inclusions \Lambda^n_i \subset \Delta^n, 0 \le i < n. A right fibration is defined similarly with the condition . A is one with the right lifting property with respect to every horn inclusion; hence, a Kan fibration is exactly a map that is both a left and right fibration. Examples A right fibration is a cartesian fibration such that each fiber is a . In particular, a[...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Homotopy Theory
In mathematics, homotopy theory is a systematic study of situations in which Map (mathematics), maps can come with homotopy, homotopies between them. It originated as a topic in algebraic topology, but nowadays is learned as an independent discipline. Applications to other fields of mathematics Besides algebraic topology, the theory has also been used in other areas of mathematics such as: * Algebraic geometry (e.g., A1 homotopy theory, A1 homotopy theory) * Category theory (specifically the study of higher category theory, higher categories) Concepts Spaces and maps In homotopy theory and algebraic topology, the word "space" denotes a topological space. In order to avoid Pathological (mathematics), pathologies, one rarely works with arbitrary spaces; instead, one requires spaces to meet extra constraints, such as being Category of compactly generated weak Hausdorff spaces, compactly generated weak Hausdorff or a CW complex. In the same vein as above, a "Map (mathematics), ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cofibration
In mathematics, in particular homotopy theory, a continuous mapping between topological spaces :i: A \to X, is a ''cofibration'' if it has the homotopy extension property with respect to all topological spaces S. That is, i is a cofibration if for each topological space S, and for any continuous maps f, f': A\to S and g:X\to S with g\circ i=f, for any homotopy h : A\times I\to S from f to f', there is a continuous map g':X \to S and a homotopy h': X\times I \to S from g to g' such that h'(i(a),t)=h(a,t) for all a\in A and t\in I. (Here, I denotes the unit interval ,1/math>.) This definition is formally dual to that of a fibration, which is required to satisfy the homotopy lifting property with respect to all spaces; this is one instance of the broader Eckmann–Hilton duality in topology. Cofibrations are a fundamental concept of homotopy theory. Quillen has proposed the notion of model category as a formal framework for doing homotopy theory in more general categories; a m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Princeton University Press
Princeton University Press is an independent publisher with close connections to Princeton University. Its mission is to disseminate scholarship within academia and society at large. The press was founded by Whitney Darrow, with the financial support of Charles Scribner, as a printing press to serve the Princeton community in 1905. Its distinctive building was constructed in 1911 on William Street in Princeton. Its first book was a new 1912 edition of John Witherspoon's ''Lectures on Moral Philosophy.'' History Princeton University Press was founded in 1905 by a recent Princeton graduate, Whitney Darrow, with financial support from another Princetonian, Charles Scribner II. Darrow and Scribner purchased the equipment and assumed the operations of two already existing local publishers, that of the ''Princeton Alumni Weekly'' and the Princeton Press. The new press printed both local newspapers, university documents, '' The Daily Princetonian'', and later added book publishing ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Small Object Argument
In mathematics, especially in category theory, Quillen’s small object argument, when applicable, constructs a factorization of a morphism in a functorial way. In practice, it can be used to show some class of morphisms constitutes a weak factorization system in the theory of model categories. The argument was introduced by Quillen to construct a model structure on the category of (reasonable) topological spaces. The original argument was later refined by Garner. Statement Let C be a category that has all small colimits. We say an object x in it is compact with respect to an ordinal \omega if \operatorname(x, -) commutes with an \omega-filterted colimit. In practice, we fix \omega and simply say an object is compact if it is so with respect to that fixed \omega. If F is a class of morphismms, we write l(F) for the class of morphisms that satisfy the left lifting property with respect to F. Similarly, we write r(F) for the right lifting property. Then Example: presheaf Her ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Universal Cartesian Fibration
Universal is the adjective for universe. Universal may also refer to: Companies * NBCUniversal, a media and entertainment company that is a subsidiary of Comcast ** Universal Animation Studios, an American Animation studio, and a subsidiary of NBCUniversal ** Universal TV, a television channel owned by NBCUniversal ** Universal Kids, an American current television channel, formerly known as Sprout, owned by NBCUniversal ** Universal Pictures, an American film studio, and a subsidiary of NBCUniversal ** Universal Television, a television division owned by NBCUniversal Content Studios ** Universal Destinations & Experiences, the theme park unit of NBCUniversal * Universal Airlines (other) * Universal Avionics, a manufacturer of flight control components * Universal Corporation, an American tobacco company * Universal Display Corporation, a manufacturer of displays * Universal Edition, a classical music publishing firm, founded in Vienna in 1901 * Universal Entertainme ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Higher Topos Theory
''Higher Topos Theory'' is a treatise on the theory of ∞-categories written by American mathematician Jacob Lurie. In addition to introducing Lurie's new theory of ∞-topoi, the book is widely considered foundational to higher category theory. Since 2018, Lurie has been transferring the contents of ''Higher Topos Theory'' (along with new material) to Kerodon, an "online resource for homotopy-coherent mathematics" inspired by the Stacks Project. Topics ''Higher Topos Theory'' covers two related topics: ∞-categories and ∞-topoi (which are a special case of the former). The first five of the book's seven chapters comprise a rigorous development of general ∞-category theory in the language of quasicategories, a special class of simplicial set which acts as a model for ∞-categories. The path of this development largely parallels classical category theory, with the notable exception of the ∞-categorical Grothendieck construction; this correspondence, which Lurie refers to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Twisted Diagonal (simplicial Sets)
In higher category theory in mathematics, the twisted diagonal of a simplicial set (for ∞-categories also called the twisted arrow ∞-category) is a construction, which generalizes the twisted diagonal of a category to which it corresponds under the nerve construction. Since the twisted diagonal of a category is the category of elements of the Hom functor, the twisted diagonal of an ∞-category can be used to define the Hom functor of an ∞-category. Twisted diagonal with the join operation For a simplicial set A define a bisimplicial set and a simplicial set with the opposite simplicial set and the join of simplicial sets by:Cisinski 2019, 5.6.1. : \mathbf(A)_ =\operatorname((\Delta^m)^\mathrm*\Delta^n,A), : \operatorname(A) =\delta^*(\mathbf(A)). The canonical morphisms (\Delta^m)^\mathrm\rightarrow(\Delta^m)^\mathrm*\Delta^n\leftarrow\Delta^n induce canonical morphisms \mathbf(A)\rightarrow A^\mathrm\boxtimes A and \operatorname(A)\rightarrow A^\mathrm\times A. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Simplicial Homotopy Class
In algebraic topology, a simplicial homotopy is an analog of a homotopy between topological spaces for simplicial sets. Precisely,pg 23 if :f, g: X \to Y are maps between simplicial sets, a simplicial homotopy from ''f'' to ''g'' is a map :h: X \times \Delta^ \to Y such that the restriction of h along X \simeq X \times \Delta^ \overset\hookrightarrow X \times \Delta^ is f and the restriction along 1 is g; se In particular, f(x) = h(x, 0) and g(x) = h(x, 1) for all ''x'' in ''X''. Using the adjunction :\operatorname(X \times \Delta^1, Y) = \operatorname(\Delta^1 \times X, Y) = \operatorname(\Delta^1, \underline(X, Y)), the simplicial homotopy h can also be thought of as a path in the simplicial set \underline(X, Y). A simplicial homotopy is in general not an equivalence relation. However, if \underline(X, Y) is a Kan complex (e.g., if Y is a Kan complex), then a homotopy from f : X \to Y to g : X \to Y is an equivalence relation. Indeed, a Kan complex is an ∞-groupoid; i.e., every ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
∞-category Of Kan Complexes
In mathematics, Kan complexes and Kan fibrations are part of the theory of simplicial sets. Kan fibrations are the fibrations of the standard model category structure on simplicial sets and are therefore of fundamental importance. Kan complexes are the fibrant objects in this model category. The name is in honor of Daniel Kan. For various kinds of fibrations for simplicial sets, see Fibration of simplicial sets. Definitions Definition of the standard n-simplex For each ''n'' ≥ 0, recall that the standard n-simplex, \Delta^n, is the representable simplicial set :\Delta^n(i) = \mathrm_ ( Applying the geometric realization functor to this simplicial set gives a space homeomorphic to the topological standard n-simplex: the convex subspace of \mathbb^ consisting of all points (t_0,\dots,t_n) such that the coordinates are non-negative and sum to 1. Definition of a horn For each ''k'' ≤ ''n'', this has a subcomplex \Lambda^n_k, the ''k''-th horn ins ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Strong Deformation Retract
In topology, a retraction is a continuous mapping from a topological space into a subspace that preserves the position of all points in that subspace. The subspace is then called a retract of the original space. A deformation retraction is a mapping that captures the idea of ''continuously shrinking'' a space into a subspace. An absolute neighborhood retract (ANR) is a particularly well-behaved type of topological space. For example, every topological manifold is an ANR. Every ANR has the homotopy type of a very simple topological space, a CW complex. Definitions Retract Let ''X'' be a topological space and ''A'' a subspace of ''X''. Then a continuous map :r\colon X \to A is a retraction if the restriction of ''r'' to ''A'' is the identity map on ''A''; that is, r(a) = a for all ''a'' in ''A''. Equivalently, denoting by :\iota\colon A \hookrightarrow X the inclusion, a retraction is a continuous map ''r'' such that :r \circ \iota = \operatorname_A, that is, the composition ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Geometric Realization Of A Simplicial Set
In mathematics, a simplicial set is a sequence of sets with internal order structure ( abstract simplices) and maps between them. Simplicial sets are higher-dimensional generalizations of directed graphs. Every simplicial set gives rise to a "nice" topological space, known as its geometric realization. This realization consists of geometric simplices, glued together according to the rules of the simplicial set. Indeed, one may view a simplicial set as a purely combinatorial construction designed to capture the essence of a topological space for the purposes of homotopy theory. Specifically, the category of simplicial sets carries a natural model structure, and the corresponding homotopy category is equivalent to the familiar homotopy category of topological spaces. Formally, a simplicial set may be defined as a contravariant functor from the simplex category to the category of sets. Simplicial sets were introduced in 1950 by Samuel Eilenberg and Joseph A. Zilber. Simplicial s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Simplicial Homotopy Equivalence
In algebraic topology, a simplicial homotopy is an analog of a homotopy between topological spaces for simplicial sets. Precisely,pg 23 if :f, g: X \to Y are maps between simplicial sets, a simplicial homotopy from ''f'' to ''g'' is a map :h: X \times \Delta^ \to Y such that the restriction of h along X \simeq X \times \Delta^ \overset\hookrightarrow X \times \Delta^ is f and the restriction along 1 is g; se In particular, f(x) = h(x, 0) and g(x) = h(x, 1) for all ''x'' in ''X''. Using the adjunction :\operatorname(X \times \Delta^1, Y) = \operatorname(\Delta^1 \times X, Y) = \operatorname(\Delta^1, \underline(X, Y)), the simplicial homotopy h can also be thought of as a path in the simplicial set \underline(X, Y). A simplicial homotopy is in general not an equivalence relation. However, if \underline(X, Y) is a Kan complex (e.g., if Y is a Kan complex), then a homotopy from f : X \to Y to g : X \to Y is an equivalence relation. Indeed, a Kan complex is an ∞-groupoid; i.e., every ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
