Given categories $X$ and $Y$ and a strong functor

$$D:X^{op}\times Y\to Cat$$

we can of course build the oplax colimit

$$\mathrm{colim}^{oplax}_{X^{op}\times Y}D$$

via the usual (covariant) grothendieck construction:

Objects are triples $(x,d,y)$ with $d\in D(x,y)$.

Morphisms $(x,d,y)\to (x',d',y')$ are triples $(f,\phi, g)$ with $f:x'\to x$, $g:y\to y'$ and $$\varphi: f^*x_*g\to x'.$$

There is however another possible construction that corresponds better to the slogan "presheaves are distributors into the one-point category and copresheaves are distributors out of the one-point category":

Objects are the same triples as above.

Morphisms however are now triples $(f,\varphi, g)$ with $f:x\to x'$ (the direction changed!) and $g:y\to y'$ and

$$\varphi:x_*g \to f^*x'.$$

Why does it correspond better to the slogan stated above? Taking distributors having at one side the one-point category then specialises to the usual grothendieck constructions yielding fibered and opfibered categories respectively.

Question: What can be said about the second construction? Is it some kind of colimit as well?

Related question

  • $\begingroup$ Consider one step lower $X^{op}\times Y\to Set$. Then the diagonal part of the second construction will give you the coend. $\endgroup$
    – Ma Ming
    Apr 5 '14 at 21:56

[I guess that by $x_*g$ you mean $D(\mathit{id}, g)(x)$ and by $f^*y$ you mean $D(f, \mathit{id})(y)$.]

Actually, your second construction is the usual Grothendieck construction for "$\mathbf{Cat}$-valued distributors" (BTW, this term may be misleading a bit, because in a $\mathbf{Cat}$-valued distributor $\mathbb{X}^{op} \times \mathbb{Y} \rightarrow \mathbf{Cat}$ 2-categories $\mathbb{X}$ and $\mathbb{Y}$ are not necessary degenerated). It may be universally characterised as a $(- \downarrow \mathbb{Y}) \times (\mathbb{X} \downarrow =)$-weighted colimit of $D \colon \mathbb{X}^{op}\times \mathbb{Y} \rightarrow \mathbf{Cat}$.

See: "Cosmoi of Internal Categories" by Ross Street, Transactions American Math. Soc. 258 (1980) 271-318; MR82a:18007.

  • $\begingroup$ Thanks for the refrerence. This seems to be the characterisation i was looking for; let's see if this helps with my other question. :) $\endgroup$ Apr 22 '13 at 21:29

Let $f^*:= D(f, 1) $, $g_*:= D(1, g)$ then $f^* g_*= g_*f^*$, the lax.colimit is the Grothendieck construction i.e. the cofibrated category (with base $\textbf{X}^{op}\times \textbf{Y}$) associated to $D$, its morphisms are $(f, g, \phi : (x, d, y) \to (x', d', y')$ with $f \in \textbf{X}^{op}( x, x')$ i.e. $f \in \textbf{X}( x', x)$, $g\in \textbf{Y}(y , y')$, and $\phi: g_*f^*(d) \to d' $. Now there is a second point of view: the functor $D$ describe a "span fibration" from $\textbf{X}$ to $\textbf{Y}$

(see R.Street article cited above, or Categorical Logic and Type Theory (B. JAcobs) p.517) or "Bifibration Induced Adjoint Pairs" (M. Bunge) Reports of the Midwest Category Seminar V - Lnm 195)

and (the associated category of) this span-fibration has morphisms $(f, g, \psi : (x, d, y) \to (x', d', y')$ with $f \in \textbf{X}( x', x)$, $g\in \textbf{Y}(y , y')$ and $\psi: g_*(d) \to f^*(d')$.

PS. excuse my poor English


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.