Zoran Skoda
ind object

The category of ind objects ind(C)ind(C) in the category CC is the full subcategory of the category of presheaves C^=C Set op\hat{C} = C^{Set^{op}} whose objects are small filtered colimits of representable presheaves. The category of pro objects pro(C)pro(C) is the opposite of the full subcategory of the category of functors CSetC\to Set whose objects small filtered limits of corepresentables on CC, i.e. of representables on C opC^{op}. In other words, pro(C)=ind(C op) oppro(C) = ind(C^{op})^{op}. In other words, the morphisms between pro objects are the opposites of the natural transformations.

Recall that the Yoneda embedding h:CC^h:C\hookrightarrow \hat{C} is right exact. It canonical splits as Cind(C)C^C\hookrightarrow ind(C)\hookrightarrow \hat{C} and the first part Cind(C)C\hookrightarrow ind(C) is right exact and right small. From that one obtains that ind(C)ind(C) is
the category of colimits of filtered diagrams of representables such that the hom is given by the formula

Nat(lim λh X λ,lim μh Y μ)=lim μcolim λHom(X λ,Y μ) Nat(lim_\lambda h_{X_\lambda}, lim_\mu h_{Y_\mu}) = lim_\mu colim_\lambda Hom(X_\lambda, Y_\mu)

Regarding the well known descriptions of filtered limits and colimits in SetSet this formula can be written in terms of threads and germs as it is usual in the theory of direct and inverse systems.

If the category CC admits finite colimits, then a functor A:C opSetA:C^{op}\to Set is left exact (flat) iff the category of elements hA= CAh\downarrow A = \int_C A (which consists of pairs (c,u)(c,u), cOb(C)c\in Ob(C), uA(c)u\in A(c)) is filtered. For a general CC, A:C opSetA:C^{op}\to Set is in Ind(C)Ind(C) iff CA\int_C A is filtered and cofinally small. Hence, in the particular case of the category CC with finite colimits, an element AC^A\in \hat{C} is in ind(C)ind(C) iff AA is left exact and the comma category hAh\downarrow A is cofinally small.

Last revised on February 2, 2011 at 22:27:02. See the history of this page for a list of all contributions to it.