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• CommentRowNumber1.
• CommentAuthorMike Shulman
• CommentTimeNov 4th 2009

Created Ho(Cat), mainly as a place to put a counterexample showing that it doesn't have pullbacks. If anyone has a simpler one, please contribute it.

• CommentRowNumber2.
• CommentAuthorUrs
• CommentTimeNov 4th 2009
• (edited Nov 4th 2009)
Thanks.

I would just ask that we specify the first sentence a bit further. You are thinking of the (how is it called?) "trivial homotopical structure" (weak equivalences are precisely the isos) as opposed to some other model structure. We should say that explicitly, because there are good reasons to imagine something different when seeing the symbols Ho(Cat).
• CommentRowNumber3.
• CommentAuthorTodd_Trimble
• CommentTimeNov 4th 2009

What are your desiderata for such counterexamples? For example, the category of nonempty finite sets doesn't admit all pullbacks. Nor does the category of smooth manifolds.

• CommentRowNumber4.
• CommentAuthorMike Shulman
• CommentTimeNov 4th 2009

@Urs: No, the weak equivalences are the equivalences of categories.

@Todd: I meant a counterexample to the existence of pullbacks in Ho(Cat) specifically. (I had a reason to want to show someone a counterexample like that, so I thought I might as well record it in case anyone else wanted it.) I know there are lots of other categories that don't have pullbacks!

• CommentRowNumber5.
• CommentAuthorUrs
• CommentTimeNov 4th 2009
• (edited Nov 4th 2009)
oh, sorry, I didn't read correctly. I was in a haste
• CommentRowNumber6.
• CommentAuthorZhen Lin
• CommentTimeAug 29th 2013

The article states without citation that

Any $Ho(Cat)$-category which is equivalent, as a $Ho(Cat)$-category, to a bicategory, is itself in fact a bicategory.

but is this really so trivial?

More precisely, let $\mathcal{C}$ be a $Ho(Cat)$-enriched category, let $\mathfrak{D}$ be a bicategory, let $\mathcal{D}$ be the $Ho(Cat)$-enriched category underlying $\mathfrak{D}$, and suppose $F : \mathcal{C} \to \mathcal{D}$ is a $Ho(Cat)$-enriched equivalence. Then, is there a bicategory $\mathfrak{C}$ whose underlying $Ho(Cat)$-enriched category is not just isomorphic to $\mathcal{C}$ but equal? The former is quite easy, since we can just take the hom-categories and coherence data for $\mathfrak{C}$ from those of $\mathfrak{D}$; but the nature of $Ho(Cat)$ as a localisation of $Cat$ means that there is already a canonical choice of hom-categories for $\mathfrak{C}$. Is it true that coherence data exists for that choice of hom-categories? If so, is this in the literature somewhere?

• CommentRowNumber7.
• CommentAuthorDavidRoberts
• CommentTimeAug 29th 2013

Sesquicategories rather than bicategories?

• CommentRowNumber8.
• CommentAuthorMike Shulman
• CommentTimeAug 29th 2013

I don’t know that it’s in the literature in exactly that form. However, it’s a general fact that any sufficiently flexible 2-categorical structure can be transported across equivalences of categories. More precisely, if $T$ is a 2-monad or pseudomonad, $A$ is a pseudo $T$-algebra, and $B\simeq A$, then $B$ inherits a pseudo $T$-algebra structure. Apply that to the 2-monad on Cat-graphs whose algebras are 2-categories and whose pseudoalgebras are (unbiased) bicategories, and you get a bicategory whose hom-categories are exactly those of $\mathcal{C}$. Then the fact that $F$ is a Ho(Cat)-equivalence, rather than just a Cat-graph-equivalence, implies that the underlying Ho(Cat)-category of this bicategory is $\mathcal{C}$.

• CommentRowNumber9.
• CommentAuthorZhen Lin
• CommentTimeAug 29th 2013

Wonderful, thanks! It will take some time for me to understand the details but an affirmative answer is a good start.

• CommentRowNumber10.
• CommentAuthorUrs
• CommentTimeJul 6th 2018

touched the formatting of this ancient entry, and cross-linked with Ho(CombModCat). There is some old query-box-discussion sitting here, which would deserve to be dealt with

• CommentRowNumber11.
• CommentAuthorMike Shulman
• CommentTimeNov 22nd 2019

Added link to MO answer by Kevin Carlson giving a $Ho(Cat)$-category that does not arise from any bicategory.

• CommentRowNumber12.
• CommentAuthorUrs
• CommentTimeOct 3rd 2021

moving the following old query box discussion out of the entry to here:

+–{: .query} David Roberts: I would think that $\tau_1(C)$ for a strict 2-category is the underlying 1-category. What is described here could be called the Poincaré category (I think that Benabou’s monograph on bicategories has this term). Maybe terminology as developed in the meantime, though.

Mike Shulman: Well, the uses of “truncation” I’ve seen always involves quotienting by equivalences, rather than discarding them. Discarding them only even makes sense in the strict situation (a bicategory has no underlying 1-category) and is an evil (and not often very useful) thing to do, so it doesn’t seem to me worth giving an important name to. “Poincare category” may also be a name for the same thing, but I prefer “truncation” as more evocative.

Beppe Metere: If I remember well, Benabou introduces two different constructions related to this discussion: the Poincarè category of a bicategory, where the arrows are connected components of 1-cells, and the classifying category, where the arrows are iso classes of 1-cells. Of course, these two categories coincide when the bicategory is locally groupoidal. =–