Subject: Re: two postings
From: "Vigleik Angeltveit"
Date: Fri, 15 Sep 2006 08:58:07 -0500
In reply to Tom Goodwillie's question. The concept of a V-model
category has been studied. A good place to start is with Mark Hovey's
book on model categories.
Vigleik
> From: Tom Goodwillie
> Date: Thu, 14 Sep 2006 21:54:29 -0400
>
> Is there a literature on enriched model categories?
>
> A bit more precisely: If V is a symmetric monoidal category then one can
> speak of a V-enriched category C. If V is also a model category, and if
> its model structure is suitably compatible with its symmetric monoidal
> structure, then one can imagine that there is then a serviceable notion
> of V-enriched model category C. In the case when V is simplicial sets,
> this should be the usual notion of simplicial model category. Has some
> version of this been worked out?
>
> Thanks,
> Tom Goodwillie
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Subject:Re: two postings
From:Georg Biedermann
Date:Fri, 15 Sep 2006 13:16:14 -0400
here is a short response to Tom Goodwillie's lates question:
Phil Hirschhorn used to have a chapter about enriched model categories in
his book, but very unfortunately it has vanished from the final version.
Georg Biedermann
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Subject: Re: two postings
From: Mark Hovey
Date: 15 Sep 2006 14:47:32 -0400
Tom,
Michael Shulman, a student of Peter May's, has written a long paper that
addresses this question. It is currently under revision, but if you
write him you might be able to get a copy. His web page is
http://www.math.uchicago.edu/~shulman/
Mark
_________________________________________________________________________
Subject: Re: two postings
From: "V. Schmitt"
Date: Sat, 16 Sep 2006 10:19:01 +0100
Regarding enriched category theory, there
is Max Kelly's book "basic concepts of enriched category
theory". This is mainly about the theory of enriched categories
based on a symmetric monoidal closed category.
That i know a bit about.
Regarding simplicial model category.
I had a look and asked myself exactly the same question:
Are they just simple simple categories enriched over
V=SSet with tensors and cotensors defined
as Kelly & Co did ? (and with extra axioms
model categories + SM7).
Actually I found many different definitions
of simplicial model categories and never got
clear answers from anyone (!?).
So i really like to know!
Anyway this is what i worked out. For what it is worth.
For instance there are definitions of simplicial
model cat C where the Hom is a bifunctor
C^{op}xC -> SSet and the tensor is defined
by means of *mere natural isomorphism*
in SSet: C(k@a,b) ~ SSet(k, C(a,b))
(if i remember well!) where C(a,b) is in SSet.
I am thinking for instance of the definition in chapter 2
of Goerss-Jardines'book - where they use
many different homs C(-,-) defined for a simplicial model cat C.
I doubt that this definition of simplicial (model)
category corresponds to tensored and cotensored
SSet-categories in the sense of Kelly & al.
For instance, the definition of tensors for V-categories
is as follows: a V-cat C is tensored iff and only
the V-presheaf C -> V: b |--> [v,C(a,b)]
is representable (and I should say *V-representable*),
that one can write:
C(v@a,b) iso_{in b} [v,C(a,b)]
the latter iso being a *V-isomorphism*
(i.e. a V-natural tranformation that is iso) and
[-,-] above denotes the internal in V.
The difference with the previous notion
of tensors is the use of V-naturality.
Now one can ask whether SSet-naturality amounts
to the same thing as the mere naturality for
SSet-presheaves A->SSet when A is an SSet-category.
The answer is no.
Actually these SSet-natural transformations
between SSet-presheaves A->V=SSet are those called
"maps" in Goerss_Jardines' book in chapter 9.
Now if one considers the SSet-category [A,S]
where A is a SSet-category and S is the SSet-category
SSet enriched over itself then it happens that
[A,S] is an SSet-category with all the SSet-indexed limits
and colimits and thus it has in particular the cotensors
and tensors as defined by Kelly and al.
I hope that i did not write too many typos
but i am in a rush this morning.
By the way i really would like to know a clear
answer on Tom Goodwillie's question:
Are "simplicial model categories" exactly
SSet-Complete & SSet-cocomplete SSet-categories
with model categories axioms and SM7.
Best regards,
Vincent Schmitt.