HOMEWORK FOR MATH 676
RENZO
(1) What does a morphism from BG to BH correspond to?
(2) What are all possible morphisms from BZ2 to itself?
(3) Compute all possible fiber products:
BZ2f ×g BZ2
as f and g vary among all possible morphisms from question (2).
(4) Formulate the natural notion of diagonal morphism
∆ : BG → BG × BG
(5) Compute:
BZ2∆ ×∆ BZ2
BS3∆ ×∆ BS3
Can you give a description of the above fiber product for the arbitrary symmetric group Sn ?
(6) Given a groupoid G, with x, y ∈ G0 , prove that if M or(x, y) 6= ∅, then
there are natural (but not canonical) bijections between M or(x, x), M or(x, y)
and M or(y, y).
(7) Given a groupoid G, define the quotient groupoid by contracting all arrows
between different objects (first make sense of what I mean by this vague
statement!) Show that the corresponding “contraction morphism” is an
equivalence.
(8) Give an example of two groupoids that are Morita equivalent but not equivalent.
Date: September 17, 2011.
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