total

A locally small category

\C

is called total if it satisfies one of the following equivalent conditions:

The covariant Yoneda embedding $y : \C \to [\C^{\op}, \Set]$ has a left adjoint. For a concrete example of how such a left adjoint could look, see here.
Every discrete fibration $D : \I \to \C$ (with $\I$ not necessarily essentially small) with the property that each fiber $D^*(\{X\}) = \{i \in \Ob(\I) : D_i = X\}$ is essentially small (bijective to a set), has a colimit in $\C$ .
Let $D : \I \to \C$ be a diagram (with $\I$ not necessarily essentially small) such that for each object $X$ of $\C$ , the collection $\pi_0(X \downarrow D)$ of connected components of the comma category $X \downarrow D$ is essentially small. Then $D$ has a colimit in $\C$ .

The equivalence is proven in Kelly, Thm. 5.2 and 5.5. In that paper, Thm. 5.6 also proves that every total category is hypercomplete, meaning that every diagram

D : \I \to \C

with the property that for every object

X

\C

the collection of cones

\Hom(\Delta(X),D)

is essentially small, has a limit.
A general category is called total if it is equivalent to a locally small category which is total.

Dual cototal Related cocomplete, complete, locally presentable External nLab link

Relevant implications

cocomplete andgenerating collection andlocally essentially small andwell-copowered implies total
cototal andself-dual implies total
self-dual andtotal implies cototal
total implies cocomplete
total implies complete
total implies locally essentially small

Examples

There are 60 categories with this property.

Counterexamples

There are 74 categories without this property.

Unknown

There are 0 categories for which the database has no information on whether they satisfy this property.

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