Sentence examples for Cartesian square from inspiring English sources

cartesian square Open image in new window (3.11)in (mathcal {C}_1) for any (X = langle X_0,l,m,l,m rangle in mathsf{G}(L,M,q)).

Therefore when we extend the diagram (3.3) to a cartesian square, the square is homotopy cartesian, and (3.6) is indeed a weak equivalence as required.

A model category (langle mathcal {C},W,C,F rangle ) is right-proper if for any cartesian square (2.4) in (mathcal {C}) with (f' in F), (g' in W), we have (g in E).

Hence for the canonical group schemes over O L attached to z and σ ( z ), there is a natural cartesian square Open image in new window Put y = j ( x ) ∈ B ( G, L ) Γ.

A fiber square in (mathcal {C}) is a cartesian square (2.4) such that (Y') is fibrant and (f',g' in F) (or equivalently, the square is a cofiber square in (mathcal {C}^o)).

Using these structures in the Quillen Adjunction Theorem, one can prove, for example, that a cartesian square (2.4) with fibrant (Y') is homotopy cartesian as soon as either (f') or (g') is a fibration, and dually for cocartesian squares.

An important particular type of homotopy limits are homotopy cartesian squares.

In the cartesian case, note that (mu ^*) sends weak equivalence to weak equivalences, fibrations to fibrations, and cartesian squares to cartesian squares, so that (2.4) is homotopy cartesian iff it becomes homotopy cartesian after applying the functor (mu ^*).

By Definition 2.12 (i), replacing homotopy cartesian squares in Definition 3.2 with homotopy cocartesian squares gives the same notion.

This is obvious: by Proposition 3.12, a square in the category ({text {Sec}}(M(I),rho ^*mathcal {C})) is homotopy cartesian or cocartesian if and only if this holds pointwise, and the transition functor (f^*) for any map f in M(I) vertical with respect to (rho ) is an equivalence of categories, thus sends homotopy cartesian squares to homotopy cartesian squares.

(R^cdot Phi ) sends ({text {Ho}}(mathcal {C}_0') subset {text {Ho}}(mathcal {C}_0)) into ({text {Ho}}(mathcal {C}_1') subset {text {Ho}}(mathcal {C}_1)) and homotopy cartesian squares of Definition 2.12 (i) in ({text {Ho}}(mathcal {C}_0')) to homotopy cartesian squares in ({text {Ho}}(mathcal {C}_1')).