• 検索結果がありません。

Show that the map that to a sieveS onX assigns the subfunctorh(S) ofh(X) defined by h(S)(Y) ={f:Y →X |f ∈ob(S)} ⊂h(X)(Y) is a bijection from the set of sieves onX to the set of subfunctors of h(X)

N/A
N/A
Protected

Academic year: 2021

シェア "Show that the map that to a sieveS onX assigns the subfunctorh(S) ofh(X) defined by h(S)(Y) ={f:Y →X |f ∈ob(S)} ⊂h(X)(Y) is a bijection from the set of sieves onX to the set of subfunctors of h(X)"

Copied!
1
0
0

読み込み中.... (全文を見る)

全文

(1)

Problems for Recitation 1

1. LetCbe a category and leth:C →Cˆ be the Yoneda embedding. Show that the map that to a sieveS onX assigns the subfunctorh(S) ofh(X) defined by

h(S)(Y) ={f:Y →X |f ∈ob(S)} ⊂h(X)(Y)

is a bijection from the set of sieves onX to the set of subfunctors of h(X). Show that ifg: X0→X is a morphism inCandS a sieve onX, then the diagram

h(g(S)) //

h(S)

h(X0) h(g) //h(X)

is a fiber product in Cˆ. Here the vertical maps are the canonical inclusions and the top horizontal map is the natural transformation whose value atY0 is the map that to f0: Y0 →X0 to g◦f0: Y0 → X. Restate the definitions of a topologyJ onCand of a sheaf onCwith respect to the topologyJ in terms of subfunctors of representable presheaves.

2. LetCbe a category which admits fiber products. If K is a pretopology on C, then we define the topology JK onCgenerated by K as follows. For every object X of C, the set JK(X) consists of all sieves S onX with the property that there exists a family (fi: Xi → X)i∈I in K(X) all of whose members are objects of S.

Show that JK is indeed a topology. Show that a presheaf F on Cis a sheaf with respect toK if and only if it is a sheaf with respect toJK.

1

参照

関連したドキュメント

As concrete applications of the monotonicities and properties of the generalized weighted mean values M p,f (r, s; x, y), some monotonicity re- sults and inequalities of the gamma

2 A Hamiltonian tree of faces in the spherical Cayley map of the Cayley graph of S 4 giving rise to a Hamiltonian cycle, the associated modified hexagon graph Mod H (X) shown in

In this section, we establish a purity theorem for Zariski and etale weight-two motivic cohomology, generalizing results of [23]... In the general case, we dene the

Massoudi and Phuoc 44 proposed that for granular materials the slip velocity is proportional to the stress vector at the wall, that is, u s gT s n x , T s n y , where T s is the

We provide an accurate upper bound of the maximum number of limit cycles that this class of systems can have bifurcating from the periodic orbits of the linear center ˙ x = y, y ˙ =

In the second section, we study the continuity of the functions f p (for the definition of this function see the abstract) when (X, f ) is a dynamical system in which X is a

We study a Neumann boundary-value problem on the half line for a second order equation, in which the nonlinearity depends on the (unknown) Dirichlet boundary data of the solution..

Lang, The generalized Hardy operators with kernel and variable integral limits in Banach function spaces, J.. Sinnamon, Mapping properties of integral averaging operators,