 International Trap Range is across from the fish pond. If the gate is closed you can park your car near the gate and walk to the patterning range.
There is a roll of butcher paper on the target stand and posts to show distances. If the paper runs out please call Roger Wanamaker at 484-788-2935We hope if you download Team Cpt-For Bamoza just for the review purpose only. and then if you like the song Team Cpt-For Bamoza don't forget to buy the orginal one to support the singers. you also can download Team Cpt-For Bamoza for legal on Apple Music or iTunes officiall music artist.15-01-2019: Clone.nl mailing week 2
Your weekly updater!
Andrea Zarza Canova & Matthew Kent
Secrets of Mana
Mana can be understood as a powerful, often magical, force contained within objects. As these objects get passed from one to another, in a cycle of gi... more...
DA BOOK
Vixxen
Recommended by: Alessandro Adriani
A real Techno / House / Electrosynth Stomper!! "Patrik Book has being in our knowledge since we were kids and listening to his early project AUSGANG... more...
Rolando Simmons
Summer Diary One EP
Rolando Simmons finally gets his first full EP at Analogical Force, and there will be more!. SD series pulses with a beauty and squelchy character all......
Eva Geist
Urban Monogamy
Eva Geist's music has the power to open doors. Her enchanting sounds are always an invitation to explore the most concealed nooks of our own perceptio... more...
Steve Reich
Different Trains / Electric Counterpoint (featuring Kronos Quartet / Pat Metheny)
Nonesuch Records' 1989 recording of Steve Reich's Different Trains, the Grammy Award winner for Best Contemporary Composition that year, and Electric... more...
Brian Eno
Ambient 1 (Music For Airports)
Recommended by: Gustav Goodstuff
Though not the earliest entry in the genre (which Eno makes no claim to have invented), ‘Ambient 1 (Music For Airports)’ was the first album ever to b... more...
Unknown Artist
WLSLTD08
8th edition of the Wilson Records limited series dedicated to physical record stores. 350 Hand-Numbered / Hand-Stamped copies.
Merch
Boris Tellegen (Delta)
8697 (Canvas Cover)
''Between 1986 and 1997, I kept a black book. It was an A4-sized book with a hard cover in which I made sketches and pasted photos of my own graffiti... more...
Boris Tellegen (Delta)
OH
This book is a collection of pen drawings on tracing paper made between 2013 and 2018. They were aides for large scale paintings on brown painted meta... more...
Back in stock
Lost Trax
Life Out Of Balance
Recommended by: Sync 24 | Serge | Vin Sol | Ovatow | 214 | Ovatow | MinimalRome | Pervert
Blah blah blah... long story short... just listen!!
214
North Cascades
Recommended by: Martyn | Duplex | Ovatow
...and then the Log Lady said: ''Beauty is in the eye of the beholder. Yet there are those who open many eyes. Eyes are the mirror of the soul, someon......
Reedale Rise
Eternal Return
Recommended by: MinimalRome | Ovatow
Another season awakens with the return of Reedale Rise....
François De Roubaix
Daughters Of Darkness (Les Lèvres Rouges)
The French film composer Francois De Roubaix recorded compositions for a lot of movies. One of them is the Belgian cult horror classic Daughters of Da... more...The diagonal of cellular spaces and effective algebro-homotopical constructions

In this survey article we discuss certain homotopy coherent enhancements of the coalgebra structure on cellular chains defined by an approximation to the diagonal. Over the rational numbers, $C_\infty$-coalgebra structures control the $\mathbb Q$-complete homotopy theory of spaces, and over the integers, $E_\infty$-coalgebras provide an appropriate setting to model the full homotopy category. Effective constructions of these structures, the focus of this work, carry geometric and combinatorial information which has found applications in various fields including deformation theory, higher category theory, and condensed matter physics.


Introduction
There is a tense trade-off in algebraic topology having roots reaching back to the beginning of its modern form. This tension can be illustrated with the concept of cohomology. The first approaches, dating back to Poincaré, are based on the subdivision of a space into simple contractible pieces. These elementary shapes are made to generate a free graded module and their spatial relations define the differential used to compute it. This definition makes certain geometric properties of cohomology, for example excision, fairly clear. Yet, it is not easy to show that a continuous map of spaces induces a map between their associated cohomologies. The functoriality just alluded to is trivial when defining cohomology in terms of homotopy classes of maps to Eilenberg-MacLane spaces, but the passage to the homotopy category erases geometric and combinatorial information and the resulting definition is not well suited for concretely presented spaces.
Cohomology as a graded group is a fairly computable invariant but it has noticeable limitations, for example, CP 2 and S 2 ∨ S 4 are not distinguished by it. This invariant can be refined to a graded ring by endowing cohomology with the cup product. In the spectral context, this finer structure is defined through the wedge product of Eilenberg-MacLane spaces, whereas in the cellular setting it is obtained from a choice of cellular approximation to the diagonal. Such cellular map induces a chain map (1) ∆ : C(X) → C(X) ⊗2 making the cellular chains of X into a (differential graded) coalgebra. The fact that the cup product on cohomology, induced by the linear dual of (1), is associative and (graded) commutative hints at the presence of additional structure extending the coalgebra structure on C(X).
In this survey article we will present, from a viewpoint that emphasizes their constructive nature, C ∞ and E ∞ extensions of ∆ over the rationals and integers respectively. The resulting algebraic structures control much of the homotopy theory of spaces. For example, over the rationals, the quasi-isomorphism type of a C ∞coalgebra extension of the symmetrization of ∆ determines the Q-completion of X under certain assumptions [Qui69;Bui+20]. Similarly, over the algebraic closure of F p, the quasi-isomorphism type of the dual of an E ∞ -coalgebra extension of ∆ determines the p-completion of X provided certain finiteness assumptions are met [Man01]. Both of these statements follow from the stronger fact that the relevant model categories are Quillen equivalent under certain assumptions.
Rational coefficients. In Section 2 we will study extensions of chain approximations to the diagonal with rational coefficients. Over this field, a chain approximation to the diagonal can be symmetrized, giving rise to a cocommutative coalgebra. This coalgebra cannot be made simultaneously coassociative, but this relation can be imposed in a derive sense through a family of coherent chain homotopies -which also respect certain symmetry constrains -and give rise to a so called C ∞ -coalgebra structure. One can think of C ∞ -coalgebras in terms of the somewhat more familiar notion of A ∞ -coalgebra where cocommutativity is satisfied strictly. As a manifestation of Koszul duality, a C ∞ -coalgebra structure on cellular chains is equivalent to a differential on the completion of the free graded Lie algebra generated by the cells shifted downwards in degree by one. This relates C ∞ -coalgebras to deformation theory, but we do not explore this deep connection here. For cell complexes whose closed cells have the Q-homology of a point, Dennis provided in [Sul07] a local inductive construction defining a C ∞ -coalgebra structure on their cellular chains. We reprint a challenge he posted regarding the resulting structure.
Problem. Study this free differential Lie algebra attached to a cell complex, determine its topological and geometric meaning as an intrinsic object. Give closed form formulae for the differential and for the induced maps associated to subdivisions. [LS14,p.2] As proven by Quillen, the quasi-isomorphism type of this C ∞ -coalgebra is a complete invariant of the rational homotopy type of simply-connected spaces. For the C ∞ -coalgebra structure on the interval, Dennis and Ruth Lawrence addressed the challenge reprinted above introducing a formula for it which can be interpreted in terms of parallel transport of flat connections [LS14], and for which the subdivision map is described by the Baker-Campbell-Hausdorff formula.
To generalize Quillen's equivalence of homotopy categories to one between (not necessarily 1-connected) simplicial sets, Buijs, Félix, Murillo, and Tanré [Bui+20] extended to n-simplices the Lawrence-Sullivan structure building, constructively for n ∈ {2, 3} and inductively otherwise, C ∞ -coalgebra structures on their chains. Their construction agrees after linear dualization with the one obtained by Cheng and Getzler in [CG08], where they showed that the Kontsevich-Soibelman sumover-trees formula defining the transfer of A