∞-Categories for the Working Mathematician
نویسندگان
چکیده
homotopy theory C.1. Lifting properties, weak factorization systems, and Leibniz closure C.1.1. Lemma. Any class of maps characterized by a right lifting property is closed under composition, product, pullback, retract, and limits of towers; see Lemma C.1.1. Proof. For now see [17, 11.1.4] and dualize. On account of the dual of Lemma C.1.1, any set of maps in a cocomplete category “cellularly generates” a larger class of maps with the same left lifting property. C.1.2. Definition. The class of maps cellularly generated by a set of maps is comprised of those maps obtained as sequential composites of pushouts of coproducts of those maps. C.2. Simplicial sets and markings C.2.1. Definition. Write Δ≤n ⊂ Δ for the full subcategory of the simplex category of 1.1.1 spanned by the ordinals [0], ... , [n]. Restriction and left and right Kan extension define adjunctions Set op Set op ≤n resn rann ⊥ lann ⊥ inducing an idempotent comonad skn ≔ lann ∘ resn and an idempotent monad coskn ≔ rann ∘ resn on SSet that are adjoint skn ⊣ coskn. The counit and unit of this comonad and monad define canonical maps sknX X cosknX ε η relating a simplicial setX with its n-skeleton and n-coskeleton. We sayX is n-skeletal or n-coskeletal if the former or latter of these maps, respectively, is an isomorphism. The next lemma proves that the monomorphisms are cellularly generated by the simplex boundary inclusions ∂Δ[n] ↪ Δ[n] for n ≥ 0. C.2.2. Lemma. Any monomorphism of simplicial sets decomposes canonically as a sequential composite of pushouts of coproducts of the maps ∂Δ[n] ↪ Δ[n] for n ≥ 0. Proof. An exercise, for now. C.2.3. Definition ((left-/right-/inner-)anodyne extensions). • The set of horn inclusions Λk[n] ↪ Δ[n] for n ≥ 1 and 0 ≤ k ≤ n cellularly generates the anodyne extensions.
منابع مشابه
. A G ] 1 7 O ct 2 00 1 Derived categories for the working mathematician
It is becoming increasingly difficult for geometers and even physicists to avoid papers containing phrases like “triangulated category”, not to mention derived functors. I will give some motivation for such things from algebraic geometry, and show how the concepts are already familiar from topology. This gives a natural and simple way to look at cohomology and other scary concepts in homologica...
متن کامل0 Derived categories for the working mathematician
It is becoming increasingly difficult for geometers and even physicists to avoid papers containing phrases like “triangulated category”, not to mention derived functors. I will give some motivation for such things from algebraic geometry, and show how the concepts are already familiar from topology. This gives a natural and simple way to look at cohomology and other scary concepts in homologica...
متن کاملThe Joy of String Diagrams
In the past recent years, I have been using string diagrams to teach basic category theory (adjunctions, Kan extensions, but also limits and Yoneda embedding). Using graphical notations is undoubtedly joyful, and brings us close to other graphical syntaxes of circuits, interaction nets, etc... It saves us from laborious verifications of naturality, which is built-in in string diagrams. On the o...
متن کاملExploring the lived experiences of nurses working in the COVID-19 ward of Shahid Sadoughi Hospital, Yazd, Iran: A qualitative study
Introduction: The outbreak of COVID-19 has created a global health emergency worldwide due to its rapid transmission, which is a notable feature of the virus. This contagious disease leads to physical health problems and several psychological disorders. Nurses are at the forefront of fighting against this disease, so this study was attempted to discover the lived experiences of nurses working i...
متن کاملAlcor: A user interface for Mizar
The Alcor user interface to the Mizar library is intended to provide a test bed for exploring how a mathematician might interact with mathematical knowledge management tools. Specifically, how can a mathematician whilst working on or writing up mathematics look up relevant mathematical knowledge without interrupting their workflow? We describe how a specific interaction style has been used to i...
متن کامل