By Chris Hillman
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Additional info for A Categorical Primer
Y be two arrows in a category C. Then ' = i for all Z in C and all x 2Z X , ' x] = x]. Exercise: let C be a category. 1. Show that F is nal in C i for all objects Z of C, there exists a unique generalized element z 2Z F. 2. Show that A ! X is monic i for all Z and all z; z 0 2Z A, z] = z 0] implies z = z0. 3. Show that initial objects and epic arrows have no such characterizations. x X of X is in fact an element of the subobject ], a X such that x = exactly when there is an arrow 1 ! a. Examining the UMP for the classi er, we conclude this happens i >= A x This gives an internal criterion in T for when an element x of X is in fact an element of A.
Let T be a topos. The collection of sheaf maps between j -sheaves is a subtopos of T, written Tj , which is a re ective subcategory of T; that is, the inclusion functor Ij from Tj to T has a left adjoint Lj a Ij . Moreover, Lj preserves nite limits and all colimits and Ij preserves all limits. An adjoint pair L a I satisfying the stated conditions is called a geometric morphism. Exercise: show that the collection of geometric morphisms between topoi forms a category. 7. If T is a topos, then for any category C, the category TC is a topos.
Z ???? Y ?? y ?? y ?? y A ????! C ???? B Verify that here ; ; are the components of a natural transformation. Verify ' that the map taking an arrow X ! Y to the commuting diagram 1X 1X X X ???? X ????! ? '? y '? y '? y ?? y ?? y ?? y 1Y 1Y Y Y ???? Y ????! de nes a functor, called the diagonal functor, from C to C&. 3. De ne a category with three objects U; V; W and ve arrows 1U ; 1V ; 1W , and U W ! V . It is denoted Push or -%. Verify that the objects of C-% are diagrams in C with \shape" X ' Z !
A Categorical Primer by Chris Hillman