Relative Endomorphisms

April 30, 2016

Posted by Qiaochu Yuan

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Let $(M, \otimes)$ be a monoidal category and let $C$ be a left module category over $M$ , with action map also denoted by $\otimes$ . If $m \in M$ is a monoid and $c \in C$ is an object, then we can talk about an action of $m$ on $c$ : it’s just a map

$\alpha : m \otimes c \to c$

satisfying the usual associativity and unit axioms. (The fact that all we need is an action of $M$ on $C$ to define an action of $m$ on $c$ is a cute instance of the microcosm principle.)

This is a very general definition of monoid acting on an object which includes, as special cases (at least if enough colimits exist),

actions of monoids in $\text{Set}$ on objects in ordinary categories,
actions of monoids in $\text{Vect}$ (that is, algebras) on objects in $\text{Vect}$ -enriched categories,
actions of monads (letting $M = \text{End}(C)$ ), and
actions of operads (letting $C$ be a symmetric monoidal category and $M$ be the monoidal category of symmetric sequences under the composition product)

This definition can be used, among other things, to straightforwardly motivate the definition of a monad (as I did here): actions of a monoidal category $M$ on a category $C$ correspond to monoidal functors $M \to \text{End}(C)$ , so every action in the above sense is equivalent to an action of a monad, namely the image of the monoid $m$ under such a monoidal functor. In other words, monads on $C$ are the universal monoids which act on objects $c \in C$ in the above sense.

Corresponding to this notion of action is a notion of endomorphism object. Say that the relative endomorphism object $\text{End}_M(c)$ , if it exists, is the universal monoid in $M$ acting on $c$ : that is, it’s a monoid acting on $c$ , and the action of any other monoid on $c$ uniquely factors through it.

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This is again a very general definition which includes, as special cases (again if enough colimits exist),

the endomorphism monoid in $\text{Set}$ of an object in an ordinary category,
the endomorphism algebra of an object in a $\text{Vect}$ -enriched category,
the endomorphism monad of an object in an ordinary category, and
the endomorphism operad of an object in a symmetric monoidal category.

If the action of $M$ on $C$ has a compatible enrichment $[-, -] : C^{op} \times C \to M$ in the sense that we have natural isomorphisms

$\text{Hom}_C(m \otimes c_1, c_2) \cong \text{Hom}_M(m, [c_1, c_2])$

then $\text{End}_M(c)$ is just the endomorphism monoid $[c, c]$ , and in fact the above discussion could have been done in the context of enrichments only, but in the examples I have in mind the actions are easier to notice than the enrichments. (Has anyone ever told you that symmetric monoidal categories are canonically enriched over symmetric sequences? Nobody told me, anyway.)

Here’s another example where the action is easier to notice than the enrichment. If $D, C$ are two categories, then the monoidal category $\text{End}(C) = [C, C]$ has a natural left action on the category $[D, C]$ of functors $D \to C$ . If $G : D \to C$ is a functor, then the relative endomorphism object $\text{End}_{\text{End}(C)}(G)$ , if it exists, turns out to be the codensity monad of $G$ !

This actually follows from the construction of an enrichment: the category $[D, C]$ of functors $D \to C$ is (if enough limits exist) enriched over $\text{End}(C)$ in a way compatible with the natural left action. This enrichment takes the following form (by a straightforward verification of universal properties): if $G_1, G_2 \in [D, C]$ are two functors $D \to C$ , then their hom object

$[G_1, G_2] = \text{Ran}_{G_1}(G_2) \in \text{End}(C)$

is, if it exists, the right Kan extension of $G_2$ along $G_1$ . When $G_1 = G_2$ this recovers the definition of the codensity monad of a functor $G : D \to C$ as the right Kan extension of $G$ along itself, and neatly explains why it’s a monad: it’s an endomorphism object.

Question: Has anyone seen this definition of relative endomorphisms before?

It seems pretty natural, but I tried guessing what it would be called on the nLab and failed. It also seems that “relative endomorphisms” is used to mean something else in operad theory.

Posted at April 30, 2016 1:40 AM UTC

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Re: Relative Endomorphisms

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I would say that the general construction is that of the right (Kan) extension in a bicategory $\mathcal{A}$ of a morphism $f : A \rightarrow B$ along itself, yielding a monad $[f,f]$ on $B$ acting on $f$ , as in the following diagram. This is the codensity monad construction generalised from Cat to an arbitrary bicategory. $\begin{matrix} A & \overset{f}{\longrightarrow} & B \\ {}_{f} \searrow & \overset{\alpha}{\Leftarrow} & \swarrow_{[f,f]} \\ & B \\ \end{matrix}$

This incorporates your setting because a monoidal category $M$ acting on a category $C$ gives a bicategory $\mathcal{A}$ with two objects $0$ and $1$ , with $\mathcal{A}(0,0) = 1$ , $\mathcal{A}(0,1) = C$ , $\mathcal{A}(1,1) = M$ , $\mathcal{A}(1,0) = 0$ , and composition given by the action. The relative endomorphism monad of $c$ is then the right extension of $c$ along itself, $c$ being seen as a morphism $c : 0\rightarrow 1$ .

Posted by: Alexander Campbell on April 30, 2016 7:09 AM | Permalink | Reply to this

Re: Relative Endomorphisms

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Your “codensity monad” example has appeared as the “endomorphism monad” in various places. For instance, here: http://www.math.uchicago.edu/~may/IMA/Incoming/Lack/lack.pdf.

I also talked about these in my thesis (a rediscovery: I think they were first discussed long before I ever thought about them; I can’t find the reference now.) I haven’t seen your general formulation of “relative endomorphism object” before.

Posted by: Charles Rezk on April 30, 2016 2:02 PM | Permalink | Reply to this

Re: Relative Endomorphisms

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Hi Charles,

We had a similar conversation back here (comments on the post that Qiaochu linked to). I still haven’t tracked down who first wrote about endomorphism monads, but if you want to find out, asking Anders Kock might be a good start.

Posted by: Tom Leinster on May 1, 2016 4:57 PM | Permalink | Reply to this

Re: Relative Endomorphisms

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Thanks for the reference! This whole post only exists because Lurie uses exactly this universal property to describe the monad induced by an adjunction in section 4.7 of Higher Algebra, although there he just calls it the endomorphism object. It reminded me of Tom’s post on the codensity monad and one thing led to another.

Posted by: Qiaochu Yuan on May 1, 2016 6:35 PM | Permalink | Reply to this

Re: Relative Endomorphisms

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In the finite abelian rigid setting this is called “internal endomorphisms.” It appears in Ostrik’s first paper on module categories. It was generalized somewhat in his later paper with Etingof “Finite Tensor Categories.” They comment in their new book (specifically the endnotes to Chapter 7) that existence follows from Barr-Beck, but leave that as an exercise to the reader. We did that exercise in Section 2.2 of our paper, mostly because we wanted to be more precise in understanding the role of rigidity.

Posted by: Noah Snyder on May 6, 2016 2:26 PM | Permalink | Reply to this

Re: Relative Endomorphisms

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Sorry, that last comment was not quite right. Existence of internal Homs in the finite abelian setting follows quickly from an appropriate adjoint functor theorem, and only uses right exactness of the action. The harder thing that uses Barr-Beck and rigidity is that if you pick your object carefully then the module category is equivalent to the category of modules over the internal end algebra.

Posted by: Noah Snyder on May 6, 2016 2:40 PM | Permalink | Reply to this

Re: Relative Endomorphisms

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Thanks, this is helpful. I would definitely say “internal endomorphisms” if I were only concerned with fixed enrichments, but I suspect there are interesting examples even when the enrichments don’t exist in full generality (although I don’t know any off the top of my head), and also I might want to do things like fix $C$ and let $M$ vary, in which case “internal” feels off to me somehow, since the construction really depends on this “external” choice of $M$ .

Posted by: Qiaochu Yuan on May 6, 2016 4:49 PM | Permalink | Reply to this

Re: Relative Endomorphisms

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Bit late to the party here, but you can find your general definition in Sec 2 of Janelidze-Kelly “A note on actions of a monoidal category”, where they call it “oft-discovered folklore”. The earlier, more general reference cited there is Gordon-Power “Enrichment through variation”.

Kelly uses this construction a lot in the 2-dimensional context (maybe even as early as SLNM420).

Posted by: Richard Garner on August 24, 2016 12:47 PM | Permalink | Reply to this

Re: Relative Endomorphisms

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Kelly uses this construction a lot in the 2-dimensional context (maybe even as early as SLNM420).

Posted by: Richard Garner on August 24, 2016 1:08 PM | Permalink | Reply to this

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