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category theory

# Contents

## Definition

In the 2-topos Cat, the pair of classes of morphisms

forms a factorization system in a 2-category. This factorization system can also be restricted to the (2,1)-topos Grpd.

In fact, an analogous factorization system exists in any 2-exact 2-category and any (2,1)-exact (2,1)-category, including any Grothendieck 2-topos or (2,1)-topos; see here.

## Properties

• When restricted to Grpd, this is the special case of the n-connected/n-truncated factorization system in the (∞,1)-topos ∞Grpd for the case that $(n = 0)$ and restricted to 1-truncated objects.

• For $f : X \to Y$ a functor between groupoids, its factorization is through a groupoid $im_2 f$ which is, up to equivalence, given as follows;

• objects are those of $X$;

• a morphism $[\phi] : x_1 \to x_2$ is an equivalence class of morphisms in $X$ where $[\phi] = [\phi']$ if $f(\phi) = f(\phi')$.

More on this is at infinity-image – Of Functors between groupoids.

Last revised on May 27, 2020 at 12:45:18. See the history of this page for a list of all contributions to it.