I added some examples of virtual double categories that do not have composites described in Crutwell-Shulman.
]]>[Reason for new thread: to all appearances, tricategory did not have one of its own, despite tetracategory having one]
(Updated reference to a representability theorem in arXiv:0711.1761v2 on tricategory; what was Theorem 21 in arXiv:0711.1761v1 has become Theorem 24 in arXiv:0711.1761v2 and its journal version)
]]>What are examples of rig categories which
I cannot seem to come up with an example myself except very artificial ones (such as taking a distributive monoidal category and removing some arrows). I’m sure there’s quite some though, can anyone help?
]]>I started comma double category. Since I care about equipments more than double categories in general, and because it actually is an instance of a comma object, I made the article mostly about virtual double categories. I wrote down a couple of conjectures about when the comma has units and composites, but haven’t verified them yet and not sure when I will.
]]>Is the category Hom of bicategories with homomorphisms as the morphisms, in the sense of
Ross Street, Fibrations in bicategories, Cahiers de topologie et géométrie différentielle catégoriques, tome 21, no 2 (1980), p. 111-160
already (recognizably) documented on the nLab ? (I did a sem-cursory search in this respect, but did not find it documented (in its own right, I mean, the article of Street appears.)
Should it be?
Should it have an article of its own?
To me it seems it should (my motivation is that I am using and documenting bicategories currently, and are studying Street’s 1980 paper as a sort of background reading to Garner–Shulman, Adv. Math. 289), but its traditional name Hom seems unfortunate, creating yet another meaning of Hom.
My suggestion would be to call it (and its article)
$BiCat$
]]>Like suggested by someone else in this forum, here I propose creating an article, but will wait for agreement or disagreement before creating it.
The article would be called
category of simple graphs with embeddings
and would
be partly modelled on, and aiming for consistency with, the article category of simple graphs
treat and compare at least three wide subcategories of category of simple graphs, namely
(weak.emb) countable simple graphs with weak graph-embeddings
(strong.emb) countable simple graphs with all strong graph-embeddings
(isom.emb) countable simple graphs with all isometric graph-embeddings
Part of the motivation for this:
Apart from the personal motivation of giving structure to my n-th attempt to get the manuscript into a satisfying form, this comparison would perhaps also be mildy interesting from a pure categorical point of view, since
Part of my motivation for creating left cancellative categories is our interest in category (isom.emb).
Do you agree that such an article could fit into the nlab?
Incidentally, I know that wide subcategories are a concept to which sometimes a certain four-letter-word is applied. Nevertheless, it seems to me that
^{ 1 } Actually, it seems that we can prove something considerably stronger, namely that this class of graphs is not closed under elementary equivalence. (synonyms: the class is not elementarily closed$=$ the class is not $\Sigma\Delta$-elementary) Moreover, what we are mostly interested in is the non-elementarily-closedness (and hence non-elementarity) of various subclasses of the class of all vertex-reconstructible graphs. (Proving that the latter class is not elementarily closed does not need https://arxiv.org/abs/1606.02926) For example, it seems we can prove that the class of all vertex-reconstructible locally-finite forests is not elementarily closed. For proving that, the methods of https://arxiv.org/abs/1606.02926 appear essential.
]]>I just worked through the definition of double profunctor and was really surprised that when you view a double profunctor as a lax functor $P : C^{\text{op}} \times D \to \text{Span}$, that $\text{Span}$ really means the transpose of what you’d expect if you think of vertical arrows as being function-like and horizontal arrows as being relation-like, and that $\text{op}$ meant horizontal reversal.
I think I’ll change the page so that $\text{Span}$ has functions as vertical arrows and write the lax functor as $P : (C^{\text{co}}\times D)^{T} \to \text{Span}$, so that $\text{co}$ means horizontal reversal. The presence of the transpose then looks very presheaf-like, and I can’t tell if that’s a misleading intuition or not. Also, I think this would obviate the need to define a “vertically lax” functor as mentioned later in the article.
]]>There’s a table on several nlab articles, for example at the bottom of accessible categories that relates “rich categories” with “rich preorders/posets”.
Most of it makes sense to me (Topos - Locale, Powerset - Presheaf), but I don’t understand the “accessible” column. Specifically, the definition of accessible categories are categories as having directed colimits and a generating set of compact objects, is almost word for word the definition of algebraic domain used in for example Abramsky and Jung’s notes on domain theory, which is that it is a poset with directed suprema and a basis of compact objects.
It seems to me like these are the appropriate analogous concepts, for example the Ind-completion of a small category is the analogue of the Ideal completion of a poset.
So why is it that in the table posets are the analogue of accessible categories? I’m guessing there is a different analogy than I am thinking about?
]]>Small A_∞-dg-categories and small dg-categories admit a model structure and the forgetful functor from dg-categories to A_∞-dg-categories is a right Quillen functor so that the resulting Quillen adjunction is actually a Quillen equivalence, which can be seen as a many-objects version of the Quillen equivalence between A_∞-dg-algebras and dg-algebras, as described, for example, by Berger and Moerdijk, “Axiomatic homotopy theory for operads”.
Is there a written source for this Quillen equivalence?
]]>Hey all,
So I’ve been kind of bugging out trying to find some kind of coherent theory of comonoids in $\infty$-categories. This, for instance, would apply to comonads (as co-associative comonoids in endomorphism categories) among other things. When I try to use Lurie’s stuff, I end up having to trace further and further back to try to prove anything, and end up feeling like I need a theory of cooperads. Somehow comonoid structures seem fundamentally different than monoid structures. Does anyone know how to do this, or if it’s written down clearly anywhere? For instance, Lurie has this nice theorem in one of the DAGs where he shows that monoids are essentially simplicial objects, and this seems to generalize pieces of Emily Riehl’s work with Dominic Verity, except for the fact that there’s no analog for comonoidal objects. It’d be nice to have the analogous statement saying that comonoids are cosimplicial objects in some essential way.
Thanks for any ideas!
-Jon
]]>In reduced homology#relation to relative homology, at the bottom is a computation where one step is that $\ker H_0(\epsilon) \cong \operatorname{coker}{H_0(x)}.$ Can you explain that step to me? If you think of kernels and cokernels as morphisms, then there’s no way such an isomorphism can hold, since $\ker H_0(\epsilon)$ is a morphism into $H_0(X)$, while $\operatorname{coker}{H_0(x)}$ is a morphism out of $H_0(X).$ But if you think of kernels and cokernels as objects, then I guess it’s possible for them to be isomorphic. My guess is it must follow from the fact that $H_0(\epsilon)\circ H_0(x)$ is an iso, so the statement is something like “the kernel of a split epimorphism is the cokernel of its right-inverse,” but I can’t figure it out.
]]>Dear all,
I recently got involved with enriched category theory and I want to apply the machinery in a computer science environment. I am interested in non-complete enrichments and here particular in functor categories.
I am aware that if $\mathcal{V}$ is a complete symmetric monoidal closed category and $\mathcal{A}$ and $\mathcal{B}$ are $\mathcal{V}$-categories, then $[\mathcal{A},\mathcal{B}]$ can be enriched over $\mathcal{V}$. In Kelly it is then shown that under this assumption the enriched Yoneda lemma and enriched Yoneda embedding hold. There is also a short explanation of what happens when $\mathcal{V}$ is not necessarily small. I would now be interested what happens when $\mathcal{V}$ is not necessarily complete.
In Borceux (Handbook II Chapter 6) this is made a bit more precise. Here, it is made clear that we do not actually need completeness for the enriched Yoneda lemma. Still, for the enriched Yoneda embedding, we appartently need completeness. But it is not made precise why we actually need it. I assume that this is related to the functor category “problem”. Nevertheless, there is a difference between giving a recipe to get an enrichement when $\mathcal{V}$ is complete, but this does not mean that we cannot find an enrichment when $\mathcal{V}$ is not complete. Is anyone aware of results in this direction?
To reduce the problem, it would be enough to consider functor categories $[\mathcal{A},\mathcal{V}]$, where $\mathcal{V}$ is our enrichment. Here, the enriched Yoneda lemma indicates that $[\mathcal{A},\mathcal{V}](H^{A}, F)$ has a hom-object. In Borceux we can find a definition for an object of $\mathcal{V}$-natural transformations between two functors $F,G:\mathcal{A} \rightarrow \mathcal{V}$ when $\mathcal{V}$ is not complete. This means in the special case we do not have problem to find the right hom-object, but what about a general $\mathcal{V}$-functor $F$. I have not found nice examples that points out a problem. Maybe for some enrichements we can still get an enrichable $[\mathcal{A},\mathcal{V}]$ functor category.
I would appreciate any comment or reference to the literature that might answer some of these questions.
Kind regards,
franeb
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