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Stream: deprecated: topos theory

Topic: formal (co)limits

James Deikun (Apr 05 2022 at 14:44):

Tracing through the definition of formal (co)limits in $(-)^* : \mathrm{Topos}^{\mathrm{co}} \to \mathrm{TopLex}^{\mathrm{coop}}$:

• First figure out what $\triangleleft$ and $\triangleright$ are:
  $\mathscr{H}(\mathbb{T}\mathrm{opos}^{\mathrm{co}})(K \odot H, L) \cong \mathscr{H}(\mathbb{T}\mathrm{opos}^{\mathrm{co}})(H, K \triangleright L) \cong \mathscr{H}(\mathbb{T}\mathrm{opos}^{\mathrm{co}})(K, L \triangleleft H)$
  $\mathrm{TopLex}^{\mathrm{coop}}(HK,L) \cong \mathrm{TopLex}^{\mathrm{coop}}(H, K \triangleright L) \cong \mathrm{TopLex}^{\mathrm{coop}}(K, L \triangleleft H)$
  $\mathrm{TopLex}(L,KH) \cong \mathrm{TopLex}(K \triangleright L, H) \cong \mathrm{TopLex}(L \triangleleft H, K)$
  $K \triangleright L = \mathrm{Lan}_{K} L, L \triangleleft H = \mathrm{Lift}_{H} L$.

• Now what are limits?
  $C(1,\mathrm{lim}_{W} F) \cong C(1,F) \triangleleft W$ (from [[proarrow equipment]])
  $(\mathrm{lim}_{W} F)^{*} \cong \mathrm{Lift}_{W} F^{*}$
  $\mathrm{Lift}_{(\mathrm{lim}_{W} F)_{*}} 1_{C} \cong \mathrm{Lift}_{W} \mathrm{Lift}_{F_{*}} 1_{C}$
  $(\mathrm{lim}_{W} F)_{*} \cong F_{*} W$.

• And colimits?
  $C(\mathrm{colim}_{W} F,1) \cong W \triangleright C(F,1)$ (is this correct?)
  $(\mathrm{colim}_{W} F)_{*} \cong \mathrm{Lan}_{W} F_{*}$
  $\mathrm{Lan}_{(\mathrm{colim}_{W} F)^{*}} 1_{C} \cong \mathrm{Lan}_{W} \mathrm{Lan}_{F^{*}} 1_{C}$
  $(\mathrm{colim}_{W} F)^{*} \cong W F^{*}$.

You can embed diagrammatic limits and colimits in here using $\mathbb{P}\mathrm{sh}$. Looking at conical limits of diagrams, which we take as geometric morphisms from $\mathbb{P}\mathrm{sh}(D)$ to $\mathcal{E}$, weighted by the locally constant sheaf functor on $\mathbb{P}\mathrm{sh}(D)$:

$(\mathrm{lim} f)_{*} = f_{*} \mathrm {LConst}$ (needs a lex left adjoint to be representable)
$\mathrm{Set}((\mathrm{lim} f)^{*} e, X) \cong \mathcal{E}(e, f_{*}(\mathrm{LConst}(X))) \cong \mathrm{Set}^{D^{\mathrm{op}}}(f^{*}(e), \mathrm{LConst}(X)) \cong \mathrm{Set}(\mathrm{colim} f^{*}(e), X)$.

The left adjoint is $\mathrm{colim} f^{*}(-)$ but since $f^{*}$ is lex it is hard (impossible?) for this to be lex without the colimit commuting with finite limits in general in Set, which is to say being filtered. Because of variance this is the same as the original diagram being cofiltered, so in general it is only cofiltered limits among the small, diagrammatic, conical limits that exist in $\mathbb{T}\mathrm{opos}^{\mathrm{co}}$.

The situation with diagrammatic colimits is worse: you want a limit functor to have a right adjoint and this essentially never happens (only in the trivial case of the limit of the one-point diagram).

However, there are some other interesting limits and colimits that can exist. The conical colimit of the identity morphism exists when $\mathcal{E}$ is a local topos and is equal to $\Gamma \dashv \mathrm{Codisc}$. The conical limit of the identity morphism exists when $\mathcal{E}$ is a totally connected topos and is equal to $\Pi_{0} \dashv \mathrm{Disc}$.

Why all this? I'm hoping someone will check my calculations, or know whether the things that seem impossible really are impossible, or come up with interesting nontrivial examples of nondiagrammatic or nonconical (co)limits that exist...

James Deikun (Apr 05 2022 at 19:19):

Also, these formal limits have "universal cones": one can construct a geometric morphism from $\mathbf{Gl}(W)$ to $\mathcal{E}$ that extends $f$. Call it $d$:

• $d_{*}(x,y,u) = f_{*}(y)$
• $d^{*}(e) = ((\mathrm{Lift}_{W} f^{*})(e), f^{*}(e), u)$ where $u : f^{*}(e) \to (W \circ \mathrm{Lift}_{W} f^{*})(e)$ is the component of the unit(?) of the Lift ...

James Deikun (Apr 05 2022 at 19:31):

This $d$ has a universal property that all domain extensions of $f$ to $\mathbf{Gl}(W)$ through the canonical closed inclusion factor through $d$ by a unique geometric transformation.

James Deikun (Apr 06 2022 at 18:46):

The limit of a connected geometric morphism weighted by its own inverse image functor is the identity. Trying to remember anything this might remind me of ...

James Deikun (Apr 06 2022 at 18:53):

You could call this the "pointwise right Kan extension" of the geometric morphism along itself. Ah, so this is saying that connected geometric morphisms are formally codense!

James Deikun (Apr 06 2022 at 18:56):

... and they seem to also be dense.

James Deikun (Apr 06 2022 at 19:18):

In general it looks like here codensity monads and density comonads are adjoints if both exist. Is that weird? It makes me wonder if I got the definition of colimit right.

James Deikun (Apr 06 2022 at 23:26):

It looks like in this setting, being formally dense, formally codense, and connected, are all equivalent properties. I wonder if nerve and realization works ...

James Deikun (Apr 06 2022 at 23:33):

...but I'm not sure how to define the nerve functor here, or even its codomain.