category theory

Contents

Definition

A category is called gaunt if all its isomorphisms are in fact identities. This is a property of strict categories; that is, it is not invariant under equivalence of categories. See below for some related concepts that are invariant.

Properties

Relation to skeletal categories, thin categories, poset categories

Gaunt categories are necessarily skeletal; a skeletal category is gaunt iff every automorphism is an identity morphism. Consequently a thin gaunt category is skeletal, and since a thin skeletal category is a poset category a thin gaunt category is also a poset category.

Note that a gaunt category need not be thin, since we may have parallel non-isomorphisms which are not equal. Similarly, a thin category need not be gaunt since we may have isomorphisms that aren’t the identity.

Relation to complete Segal spaces

The nerve simplicial set of a category, regarded as a simplicial object in homotopy types under the inclusion $Set \hookrightarrow \infty Grpd$, is a complete Segal space precisely if the category is gaunt. More discussion of this is at Segal space – Examples – In Set.

To make sense of the definition of a gaunt category, we need to use equality of objects: For every isomorphism $f : a \simeq b$, there is an equality $p : a = b$, relative to which $f$ equals the identity at $a$. Replacing the equality $p$ by an isomorphism $g : a \simeq b$, the resulting condition holds for all categories. This echoes how one might understand the definition in univalent foundations: the univalence condition for a univalent category is another way of saying that every isomorphism is an identity (and uniquely so).

Alternatively, we could avoid the equality on objects by requiring only that every endoisomorphism $f : a \simeq a$ be equal to the identity at $a$. This amounts to requiring that the core be a thin category, i.e., that parallel isomorphisms are equal.

Incidentally, we may view both strict categories and categories up to equivalence as embedded in the type of flagged categories?. Recall that a flagged category consists of a category $C$, a groupoid $X$, and a surjection $p:X\to C$ of groupoids from $X$ to the underlying groupoid of objects of $C$. In this way, we can view categories as those flagged categories where $p$ is an equivalence, and strict categories as those flagged categories where $X$ is a set (up to homotopy). The intersection of the categories and the strict categories within the type of flagged categories is then exactly this type of core-thin categories.

The term “gaunt category” was apparently introduced in

in the context of a discussion of (infinity,n)-categories.

In the Elephant, gaunt categories are briefly mentioned under the name “stiff categories”, in the paragraph preceding B1.3.11 (about splitting of Grothendieck fibrations).