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Stream: learning: questions

Topic: simplicial categories

ADITTYA CHAUDHURI (Dec 22 2020 at 17:24):

By a simplicial object in Cat, the category of all small categories, we mean a functor $\Delta^{op} \rightarrow Cat$ where $\Delta$ is the simplex category. Now since $\Delta$ can be seen as a full-subcategory of Cat, hence it has a strict 2-categorical structure induced from $Cat$. Now, if instead of considering the functor $\Delta^{op} \rightarrow Cat$ we consider a 2-functor $\Delta^{op} \rightarrow Cat$ between the two 2-categories, then can I expect that it will capture more structures than usual simplicial sets?. Is there a standard name for such objects? Where do these objects appear generally? Are they not interesting?

Reid Barton (Dec 22 2020 at 23:28):

I thought about this question a long time ago and I think such objects are interesting, though I never saw them written down anywhere.

Reid Barton (Dec 22 2020 at 23:31):

Oddly enough I was just thinking about this again the other night. A nice place to start is to think about the guy represented by the object $[1] \in \Delta$ ($[0]$ is a bit too trivial).

ADITTYA CHAUDHURI (Dec 23 2020 at 06:04):

@Reid Barton Thank you.

Todd Trimble (Dec 25 2020 at 04:05):

The following might be helpful. Please allow me to use $(n)$ to denote the ordinal with $n$ elements. If we include the empty ordinal $(0)$, then we obtain the augmented simplicial category $\Delta_a$ whereby functors $\Delta_a^{op} \to Cat$ are called augmented simplicial categories. It's well known that $\Delta_a$ is "the walking monoid": there is a monoidal product + (called ordinal sum) on $\Delta_a$, and the category of strong monoidal functors $\Delta_a \to M$ to a monoidal category, and monoidal transformations between them, is equivalent to the category of monoids in $M$.

Let $u: (0) \to (1)$ and $m: (2) \to (1)$ be the unique maps. We have maps $i_0 = (1) + u: (1) \to (2)$ and $i_1 = (u) + (1): (1) \to (2)$, and considered as a 2-category, we have a 2-cell $i_0 \leq i_1$. More is true: we clearly have $m \circ i_0 = 1_{(1)} = m \circ i_1$, and also we have that $i_0 \circ m \leq 1_{(2)} \leq i_1 \circ m$. In short, we have an adjunction $i_0 \dashv m \dashv i_1$. In other words, we have a 2-monad for which the multiplication $m$ is left adjoint to the unit $i_1$.

There is a name for this in the categorical literature: a Kock-Zöberlein or KZ-monad, or alternatively a lax idempotent monad. Notice that 2-functors $\Delta_a \to Cat$ preserve adjunctions.

Thus, a fairly reasonable source of 2-functors $\Delta_a \to Cat$ (and 2-functors $\Delta \to Cat$, by restricting the former) is given by KZ-monads. In greater detail (and putting set-theoretic matters aside) we have a monoidal bicategory $[Cat, Cat]$. Given a KZ-monad $M$ on $Cat$, there is a monoidal 2-functor $\Delta_a \to [Cat, Cat]$ that takes $(1)$ to $M$ -- in fact KZ-monads are tantamount to such monoidal 2-functors. Then, we can do further things, like compose with a 2-functor $[Cat, Cat] \to Cat$, for example one given by evaluation at a particular category $A$.

Morally speaking, KZ-monads should be thought of as cocompletion monads of various sorts. See the nLab article above for more.

Todd Trimble (Dec 26 2020 at 16:15):

No response? Alrighty, then.

Amar Hadzihasanovic (Dec 26 2020 at 18:28):

For what it's worth, I found your answer very interesting. I had seen the names "lax idempotent monad" and "KZ-monad" before and I had no idea why they were 'natural' or interesting. Whereas your perspective certainly got me interested.

Todd Trimble (Dec 26 2020 at 19:17):

Thanks! As you might guess, they were introduced by Anders Kock (early 1970's). I think it was Street who observed that the augmented simplex category as 2-category is the "walking KZ-monoid".

John Baez (Dec 26 2020 at 19:20):

That's a pleasant way of understanding them.

ADITTYA CHAUDHURI (Dec 26 2020 at 21:34):

Todd Trimble said:

The following might be helpful. Please allow me to use $(n)$ to denote the ordinal with $n$ elements. If we include the empty ordinal $(0)$, then we obtain the augmented simplicial category $\Delta_a$ whereby functors $\Delta_a^{op} \to Cat$ are called augmented simplicial categories. It's well known that $\Delta_a$ is "the walking monoid": there is a monoidal product + (called ordinal sum) on $\Delta_a$, and the category of strong monoidal functors $\Delta_a \to M$ to a monoidal category, and monoidal transformations between them, is equivalent to the category of monoids in $M$.

Let $u: (0) \to (1)$ and $m: (2) \to (1)$ be the unique maps. We have maps $i_0 = (1) + u: (1) \to (2)$ and $i_1 = (u) + (1): (1) \to (2)$, and considered as a 2-category, we have a 2-cell $i_0 \leq i_1$. More is true: we clearly have $m \circ i_0 = 1_{(1)} = m \circ i_1$, and also we have that $i_0 \circ m \leq 1_{(2)} \leq i_1 \circ m$. In short, we have an adjunction $i_0 \dashv m \dashv i_1$. In other words, we have a 2-monad for which the multiplication $m$ is left adjoint to the unit $i_1$.

There is a name for this in the categorical literature: a Kock-Zöberlein or KZ-monad, or alternatively a lax idempotent monad. Notice that 2-functors $\Delta_a \to Cat$ preserve adjunctions.

Thus, a fairly reasonable source of 2-functors $\Delta_a \to Cat$ (and 2-functors $\Delta \to Cat$, by restricting the former) is given by KZ-monads. In greater detail (and putting set-theoretic matters aside) we have a monoidal bicategory $[Cat, Cat]$. Given a KZ-monad $M$ on $Cat$, there is a monoidal 2-functor $\Delta_a \to [Cat, Cat]$ that takes $(1)$ to $M$ -- in fact KZ-monads are tantamount to such monoidal 2-functors. Then, we can do further things, like compose with a 2-functor $[Cat, Cat] \to Cat$, for example one given by evaluation at a particular category $A$.

Morally speaking, KZ-monads should be thought of as cocompletion monads of various sorts. See the nLab article above for more.

Thank you very much Sir for the detailed description and for introducing the notion of KZ-monads and lax idempotent monads. I had not seen these notions before.

ADITTYA CHAUDHURI (Dec 26 2020 at 21:35):

Todd Trimble said:

No response? Alrighty, then.

I apologise for my late response.