internal (co-)limit




Speaking in the internal language of a given category H\mathbf{H}, one may try to formulate universal constructions such as of limits/colimits internally.

Given the system of slice categories H /X\mathbf{H}_{/X} of H\mathbf{H} (or more generally any indexed category XH XX \mapsto \mathbf{H}_X), then one may regard XX as the shape of a discrete internal diagram in H\mathbf{H}, and regard the objects in the slice H /X\mathbf{H}_{/X} as discrete actual internal diagrams of this shape.

If H\mathbf{H} happens to have a (small) object classifier h\mathbf{h}, then this is particularly evocative, as then (small) such slice objects F^\hat F over XX equivalent to morphisms F:XhF \colon X \to \mathbf{h}, hence are directly analogous to external diagrams in the form of functors 𝒳H\mathcal{X}\to \mathbf{H}.

Now if H\mathbf{H} has good enough properties as a (self-)indexed category in that it is a hyperdoctrine with dependent sum XfY:H /XH /Y\underset{X \stackrel{f}{\to} Y}{\sum} \colon \mathbf{H}_{/X} \to \mathbf{H}_{/Y} and dependent product XfY:H /XH /Y\underset{X \stackrel{f}{\to} Y}{\prod} \colon \mathbf{H}_{/X} \to \mathbf{H}_{/Y} operations, then it makes sense to define the internal colimit of a discrete diagram as above as

limFXF^ \underset{\longrightarrow}{\lim} F \coloneqq \underset{X}{\sum} \hat F

and the internal limit as

limFXF^. \underset{\longleftarrow}{\lim} F \coloneqq \underset{X}{\prod} \hat F \,.

Of course with XX being, internally, a discrete diagram shape these are, so far, just internal coproducts and internal products, respectively (which is of course the source of the terminology “dependent sum” and “dependent product”).

It is more or less straightforward to extend the above from H\mathbf{H} to Cat(H)Cat(\mathbf{H}), the collection of internal categories in H\mathbf{H}, then consider for a given internal category (hence internal diagram shape) CCat(H)C \in Cat(\mathbf{H}) the objects in Cat(H) /CCat(\mathbf{H})_{/C} as not-necessarily discrete diagrams, and proceed as before. For various specific choices of context H\mathbf{H}, this is considered in (Johnstone, below prop. B2.3.20, Pisani 09, p.18, p. 23, HoTTBook, section 6.12).

To see internally the expected universal property of the above definition, consider the internal XX-diagram constant on any AHA \in \mathbf{H}, namely X *AH /XX^\ast A \in \mathbf{H}_{/X}. If there is an object classifier h\mathbf{h} then this corresponds indeed to the constant map X*AhX \to \ast \stackrel{\vdash A}{\to} \mathbf{h}. Accordingly an internal cone with tip AA over the diagram FF as above is a morphism of XX-diagrams (hence in H /X\mathbf{H}_{/X}) of the fomr

X *AF^. X^\ast A \longrightarrow \hat F \,.

Now by the defining adjunction (X *X)(X^\ast \dashv \underset{X}{\prod}) of the dependent product, such morphisms are equivalent to morphisms

AXF^ A \longrightarrow \underset{X}{\prod} \hat F

in H\mathbf{H}, hence by the above definition to morphisms

AlimF A \longrightarrow \underset{\longleftarrow}{\lim} F

from AA into the internal colimit.

This is clearly the internal version of the statement that is extrernally true Set, that the limit of sets over a diagram is the set of all its cones.

Dually, an internal cocone is a morphism

F^X *A \hat F \longrightarrow X^\ast A

in H /X\mathbf{H}_{/X}, and by the defining adjunction (XX *)(\underset{X}{\sum} \dashv X^\ast) of the dependent sum this is equivalently a morphism

limFA \underset{\longrightarrow}{\lim} F \longrightarrow A

exhibiting internally the statement familiar externally from Set that maps out of colimits of a diagram are equivalently cocones under that diagram.


\infty-Groupoidal homotopy limits and colimits of \infty-groupoids

The small object classifier of the (∞,1)-topos ∞Grpd is Core(Grpd small)Core(\infty Grpd_{small}) itself. Hence an \infty-groupoidal-shaped diagram of ∞-groupoids is internally in ∞Grpd a discrete diagram F:XCore(Grpd small) F \colon X\to Core(\infty Grpd_{small}). The slice object classified by this is the pullback of the universal right fibration, which is equivalently its (∞,1)-Grothendieck construction F^= XF\hat F = \int_X F.

Accordingly the above gives the internal limits and colimits

limF=XF^ \underset{\longleftarrow}{\lim} F = \underset{X}{\sum} \hat F


limF=XF^. \underset{\longrightarrow}{\lim} F = \underset{X}{\prod} \hat F \,.

There are indeed also the correct external (∞,1)-colimits and (∞,1)-limits, by the discussion here.

Borel construction and homotopy-quotients, -invariants, -coinvariants

For H\mathbf{H} an (∞,1)-topos and GG an ∞-group object, consider X=BGX = \mathbf{B}G its delooping. Externally this is in general highly non-discrete, but internally this, being \infty-groupoidal, is a disrete diagram shape.

An internal diagram of this shape

ρ:BGh \rho \;\colon\; \mathbf{B}G \longrightarrow \mathbf{h}

is equivalently an ∞-action of GG on the object VV named by

*BGh. \ast \to\mathbf{B}G \longrightarrow \mathbf{h} \,.

Now the internal colimit

limρ=BGρ^V//G \underset{\longrightarrow}{\lim} \rho = \underset{\mathbf{B}G}{\sum} \hat \rho \simeq V/\!/G

is the homotopy quotient of VV by GG, equivalently the object of homotopy coinvariants. In H=\mathbf{H} = ∞Grpd this is given by the Borel construction.

Similarly, the internal limit

limρ=BGρ^Ext G(*,V) \underset{\longleftarrow}{\lim} \rho = \underset{\mathbf{B}G}{\prod} \hat\rho \simeq Ext_G(\ast, V)

is the homotopy invariants of the action.

More generally, the internal left and right Kan extension give the induced representation and coinduced representation constructions. See at ∞-action for more on this.


Last revised on March 23, 2021 at 01:54:27. See the history of this page for a list of all contributions to it.