ind object

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

Recall that the Yoneda embedding $h:C\hookrightarrow \hat{C}$ is right exact. It canonical splits as $C\hookrightarrow ind(C)\hookrightarrow \hat{C}$ and the first part $C\hookrightarrow ind(C)$ is right exact and right small. From that one obtains that $ind(C)$ is

the category of colimits of filtered diagrams of representables such that the hom is given by the formula

$Nat(lim_\lambda h_{X_\lambda}, lim_\mu h_{Y_\mu}) = lim_\mu colim_\lambda Hom(X_\lambda, Y_\mu)$

Regarding the well known descriptions of filtered limits and colimits in $Set$ 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 $C$ admits finite colimits, then a functor $A:C^{op}\to Set$ is **left exact** (flat) iff the *category of elements* $h\downarrow A = \int_C A$ (which consists of pairs $(c,u)$, $c\in Ob(C)$, $u\in A(c)$) is filtered. For a general $C$, $A:C^{op}\to Set$ is in $Ind(C)$ iff $\int_C A$ is filtered and cofinally small. Hence, in the particular case of the category $C$ with finite colimits, an element $A\in \hat{C}$ is in $ind(C)$ iff $A$ is left exact and the comma category $h\downarrow A$ 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.