gaunt category

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.

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.

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

- Clark Barwick, Chris Schommer-Pries,
*On the Unicity of the Homotopy Theory of Higher Categories*(arXiv:1112.0040, slides)

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).