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Let $M_{\bullet}$ be a simplicial manifold. There are two ways to computing its cohomology. Consider the cosimplicial module $A_{DR}(M)$. It defines a functor $A_{DR}(M_{\bullet})\: : \: \boldsymbol{\ …

Do you know some reference about the notion of parallel transport for simplicial manifolds?

Let $X$ be a topological space and let $PX$ be its space of paths. Let $I=[0,1]$ with coordinate $s$. There is an homotopy $$ F\: : \: I\times PM\to PM $$ Defined by $F(s,y)(t):=y(st)$. This map is an …

Let $K$ be a simplicial set and let $\Delta K$ be the category of simplices, i.e the category where the objects are simplicial maps $$ \Delta[n]\to K $$ and the maps $\phi\: : \: (\Delta[n]\to K)\to …

Let $\mathcal{C}$, $\mathcal{D}$ be two small categories. Let $f\: : \: \mathcal{C}\to \mathcal{D}$ be a functor. Then it induces a functor $$ f^{*}\: : \: sPsh(\mathcal{D})\to sPsh(\mathcal{C}) $$ v …

I'm reading the Hirschhorn's book model categories and their localization and I have a question about frames and resolutions. Following the book (definition 16.6.1) a cosimplicial frame on an object …

Consider the Hovey's definition of cosimplicial frame. A cosimplicial frame $\tilde{X}^{*}$ on an object $X\in \mathcal{M}$ is a factorization $p^{*}(X)\to \tilde{X}^{*}\to ccX$ of the fold map $p^{*} …

Let $X,Y$ be two simplicial presheaves on a small category $\mathcal{C}$, let $*$ be the final simplicial presheaf. Consider the category of simplicial presheaves equipped with its projective model st …

Let $G$ be a finitely generated group, then its action groupoid $BG$ is a simplicial set. In fact $BG$ is the nerve of a groupoid where the set of objects is given by a point $*$ and the set of maps i …

Let $Psh(\mathcal{C})$ be the category of simplicial presheaves equipped with the projective model structure. The cartesian product between two representables presheaves is clearly again representable …

Let $\Omega^{*}_{\text{poly}}\: : \: sSet\to dg_{\geq 0}Comm_{+}$ be the polynomial De Rahm functor on simplicial sets, where the codomain is the category of commutative differential graded algebras o …

Given a connected topological space $X$, its space of path $PX$ is again a topological space. On the other hand, for a simplicial set $K_{\bullet}$, its path space is given by $$ PK_{n}:=\operatornam …

Let $\mathcal{C}$ be a site. Consider $sPsh(\mathcal{C})$ be the equipped with the local projective model structure. Let $C_{\bullet}$ be a cofibrant object in $\mathcal{C}$ and let $y(C_\bullet)$ be …

Let $\mathcal{C}$ be a site and let $sPh(\mathcal{C})_{proj}$ be the category of simplicial presheaves equipped with the projective model structure. This category is a closed monoidal model category ( …

Let $\boldsymbol{\Delta}$ be the simplex category. Let ${\Box}$ be the cube category. I denote with $\Delta[n]$ the standard $n$-simplex and with $\Delta_{geo}[n]$ its geometric realization. On the ot …