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Stream: learning: questions

Topic: Semi-double category?

Kevin Carlson (Mar 20 2026 at 16:24):

Well, virtual double categories are a common and even more general concept, where not only $U$ but even $\circ$ need not exist. One has to add more structure, namely, cells with a natural number of input horizontal arrows, rather than the usual squares. Then you can talk about the existence of certain composites in terms of representability.

Vincent R.B. Blazy (Mar 20 2026 at 16:41):

@Adittya Chaudhuri That’s not a double semicategory in the sense of $D_0, D_1$ being semi, right? Rather of semi-(double category), perhaps semidouble category… (Don’t have the time to check out)

Adittya Chaudhuri (Mar 20 2026 at 17:02):

Kevin Carlson said:

Well, virtual double categories are a common and even more general concept, where not only $U$ but even $\circ$ need not exist. One has to add more structure, namely, cells with a natural number of input horizontal arrows, rather than the usual squares. Then you can talk about the existence of certain composites in terms of representability.

Thanks! Thats interesting! In my case, everything else exists and behaves exactly like a usual strict double category but only the functor $U$ is missing. So, I think according to what you said, my case will be a very special case of the virtual double category. Since I do not need such generality at the moment in my setup, I wanted to have a notion of a double categorical analogue of a semicategory. However, I am not sure whether such a special virtual double category has a name in the exisiting literature.

Adittya Chaudhuri (Mar 20 2026 at 17:05):

Vincent R.B. Blazy said:

Adittya Chaudhuri That’s not a double semicategory in the sense of $D_0, D_1$ being semi, right? Rather of semi-(double category), perhaps semidouble category… (Don’t have the time to check out)

Thanks for asking about this clarification. In my case both $D_0$ and $D_1$ are categories(not semicategories) and only the functor $U$ is missing in my structure, and hence, from the analogy to a semicategory, I was calling it a double semicategory.

Kevin Carlson (Mar 20 2026 at 17:49):

We've used such virtual double categories in recent work and just said they "have non-nullary composites", as an identity loose arrow can be seen as the composite of an empty list. I'm not aware of another name.

Adittya Chaudhuri (Mar 20 2026 at 17:56):

Kevin Carlson said:

We've used such virtual double categories in recent work and just said they "have non-nullary composites", as an identity loose arrow can be seen as the composite of an empty list. I'm not aware of another name.

I see. Thanks.

Adittya Chaudhuri (Mar 20 2026 at 18:00):

@Vincent R.B. Blazy I think you are right that the name double semicategory is a bit misleading and probably saying more like $D_1$ and $D_0$ are themselves semicategories. I think the name semi-double category would be more appropriate for the case I am interested in. Thanks!

Just now, I changed the subject from Double semicategory to Semi-double category.

Adittya Chaudhuri (Mar 22 2026 at 16:59):

Is it reasonable to call the notion defined below a semi-double category ? (Although I fully agree that it is a very special case of a virtual double category)

A semi-double category $\mathsf{D}$ consists of the following:

• A category of objects $\mathsf{D}_0$ and a category of arrows $\mathsf{D}_{1}$.
• Source and target functors $\mathsf{S, T} \colon \mathsf{D}_1 \to \mathsf{D}_{0},$
• A composition functor $\circ_{H} \colon \mathsf{D}_{1} \times_{\mathsf{S} , \mathsf{D}_{0}, \mathsf{T}} \mathsf{D}_{1} \to \mathsf{D}_{1}$ such that for any compatible triple of objects $d_1,d_2,d_3$ in $\mathsf{D}_{1}$ and any compatible triple of morphisms $\phi_1, \phi_2, \phi_{3}$ in $\mathsf{D}_{1}$, we have the following: $d_3 \circ_{H}(d_{2} \circ_{H}d_{1})= (d_3 \circ_{H}d_{2}) \circ_{H}d_{1}$ and $\phi_3 \circ_{H}(\phi_{2} \circ_{H}\phi_{1})= (\phi_3 \circ_{H}\phi_{2}) \circ_{H}\phi_{1}.$

To be more precise, I would like to call it a strict semi-double category , but it sounds very long to me!

John Baez (Mar 22 2026 at 20:12):

A strict double category is a category internal to the category Cat, so there seem to be two ways to "semify" this:

• categories internal to SemiCat
• semicategories internal to Cat

By commutativity of internalization these should be equivalent. We can also do both and consider

• semicategories internal to SemiCat.

We might call such a thing a strict double semicategory.

(I believe SemiCat is a category, not just a semicategory, but someone should check me on this!)

A weak double category is a category weakly internal to the 2-category $\mathbf{Cat}$. One could try to systematically semify this idea too, and consider

• categories weakly internal to $\mathbf{SemiCat}$
• semicategories weakly internal to $\mathbf{Cat}$
• semicategories weakly internal to $\mathbf{SemiCat}$

But be careful: it seems that $\mathbf{SemiCat}$ is not a 2-category, because while it has identity 1-morphisms, it doesn't have identity 2-morphisms. So, it seems to be some sort of '2-semicategory', and this needs to be understood before doing internalization.

Maybe someone has done all this stuff already.

Next time I rent a hotel room I'll ask for a semi-double bed and see if they give me a single bed.

Adittya Chaudhuri (Mar 23 2026 at 05:54):

Thanks very much! I am now trying to understand your ideas.

Adittya Chaudhuri (Mar 23 2026 at 06:03):

John Baez said:

(I believe SemiCat is a category, not just a semicategory, but someone should check me on this!)

Yes, SemiCat seems to be a category whose objects are semicategories and morphisms are semifunctors as stated after the Definition 2.4 in the nLab page semicategories.

Adittya Chaudhuri (Mar 23 2026 at 06:11):

John Baez said:

A strict double category is a category internal to the category Cat, so there seem to be two ways to "semify" this:

• categories internal to SemiCat
• semicategories internal to Cat
  

Yes, I agree, and I think my description here #learning: questions > Semi-double category? @ 💬 is same as the semicategories internal to Cat.

Adittya Chaudhuri (Mar 23 2026 at 07:48):

John Baez said:

By commutativity of internalization these should be equivalent. We can also do both and consider

• semicategories internal to SemiCat.

We might call such a thing a strict double semicategory.

I am not completely sure that I fully understand the equivalence you said. May be I am missing something! I am thinking.

Adittya Chaudhuri (Mar 23 2026 at 07:51):

John Baez said:

A weak double category is a category weakly internal to the 2-category $\mathbf{Cat}$. One could try to systematically semify this idea too, and consider

• categories weakly internal to $\mathbf{SemiCat}$
• semicategories weakly internal to $\mathbf{Cat}$
• semicategories weakly internal to $\mathbf{SemiCat}$
  

Thanks! Yes, I agree.

Adittya Chaudhuri (Mar 23 2026 at 07:55):

John Baez said:

But be careful: it seems that $\mathbf{SemiCat}$ is not a 2-category, because while it has identity 1-morphisms, it doesn't have identity 2-morphisms. So, it seems to be some sort of '2-semicategory', and this needs to be understood before doing internalization.
.

Thanks! Yes, I agree that it does not have identity 2-morphisms and it is some sort of '2-semicategory' which needs to be understood in the first place.

Adittya Chaudhuri (Mar 23 2026 at 08:02):

John Baez said:

Next time I rent a hotel room I'll ask for a semi-double bed and see if they give me a single bed.

They may also give you two single beds with some less features (like without bed sheets) in each, but provide a single large bedsheet which can cover both the beds when they are placed next to each other side by side.

Adittya Chaudhuri (Mar 23 2026 at 12:46):

Below, I am writing some thoughts on the correct definition of a semified strict double category:

There is an evident forgetful functor $U \colon { {\rm{Cat}} \to \rm{SemCat}}$, which satisfies the Proposition 3.3 in semicategories.

Now, since the notion of a strict double catgory is standard, we may expect that a correct notion of a semified strict double category should also be related to a strict double category via the image of forgetful 2-functor,

But how?

The collection of strict double categories, strict double functors and double transformations forms a 2-category Dbl. Now, I think if we take the definition of a semified strict double category as a semicatetgory internal to Cat, then the collection of semified strict double categories, semified double functors and transformations is expected to form a 2-category, which I am calling as SemDbl. However, I think if we take the definition of a semified strict double category as a category internal to SemiCat, then, I think we can not expect to get a 2-category because of the absence of identity 2-morphisms. Thus, I am a bit confused whether the notion of a semicategory internal to Cat is equivalent to the notion of a category internal to SemiCat.

However, I think it is possible to define a 2-functor $\mathbb{U} \colon$ Dbl $\to$ SemDbl , which takes a caetgory internal to Cat to a semicaetgory internal to Cat by forgetting the identity morphism assigning functor. In this case, $\mathbb{U}$ may also be expected to satisfy a double category theoretic analogue of the Proposition 3.3 in semicategories.

John Baez (Mar 23 2026 at 19:11):

I am not completely sure that I fully understand the equivalence you said. May be I am missing something! I am thinking.

The equivalence should switch vertical and horizontal composition. For ordinary strict double categories this is familiar: any strict double category gives another one where the roles of horizontal and vertical are switched. More generally, I'm claiming here that if you have a kind of strict double semicategory that is 'semi' only the vertical direction, you'll get one that's semi in the horizontal direction - and vice versa.

Adittya Chaudhuri (Mar 23 2026 at 19:57):

Thanks very much! Yes, I got your point.