text: 2 T. Hewlett et al. secularly through, for example, disk instabilities (Bournaud et al. occurred (Ueda et al. 2003; Aird et al. 2010). It has been argued 2011), bars (Knapen et al. 2000; Oh et al. 2012), or otherwise by that only the most luminous AGN, with the highest accretion rates (cid:199) minor mergers (Kaviraj 2013). It remains unclear whether alterna- and therefore the most rapid loss of angular momentum, need to tives to major merger triggering can drive several M year-1 of gas be fuelled by mergers (Sanders et al. 1988; Di Matteo et al. 2005; to the central SMBHs, as is necessary to power the most luminous Hopkins et al. 2008). This makes z∼2 a natural place to search AGN. for an AGN-merger connection, since z∼2 also coincides with the The tightness of correlations between the masses of SMBHs predicted peak of galaxy mergers in hierarchical galaxy evolution and host galaxy masses, velocity dispersions and luminosities may (Kauﬀmann & Haehnelt 2000; Volonteri et al. 2002; Conselice et al. imply coevolution between them (Ferrarese & Merritt 2000; Mar- 2003). Because of the diﬃculties inherent in detecting mergers at coni & Hunt 2003; Haring & Rix 2004). Major mergers are a com- such high redshift many studies thus far have focussed on either monly proposed mechanism for growing black holes and galaxies low luminosity sources (e.g. Schawinski et al. 2011; Kocevski et al. synchronously. Detection of connections between AGN activity and 2012; Villforth et al. 2014, 2017) or low redshift AGN (Schawinski major mergers may therefore shed some light on the origins of cor- et al. 2010). In this study we will analyse COSMOS HST F814W relations between the SMBH mass and galaxy properties, such as banddata,comparingasampleofAGNhostmorphologiestocontrol the M- relation (the correlation between the mass of the SMBH galaxies between z=0.5-2.2. It is not enough to show that AGN do σ and the velocity dispersion of the spheroid). Alternatively, these or do not live in merging galaxies; the crucial comparison is to an relations could be caused by stochastic processes, since black holes inactive sample of galaxies matched in stellar mass, redshift and star and star formation are both fed by a supply of cold gas. formingpropertiestodeterminewhetherthemergerrateisenhanced The possibility of a connection between nuclear activity and in active galaxies relative to "normal" galaxies. One aim of this galaxy mergers can be investigated observationally by comparing study is to see if triggering mechanisms changed over cosmic time, galaxy morphologies and AGN properties. Canalizo & Stockton both during and outside the peak of mergers/AGN activity. Another (2001), Treister et al. (2012), Ellison et al. (2013) and Kocevski principal aim was to probe the triggering mechanisms of the highest et al. (2015) ﬁnd evidence of enhancement of AGN associated with luminosity AGN to see if there was some threshold accretion rate mergers, or else that mergers are associated in particular with ob- which required more catastrophic events such as galaxy mergers. scured AGN (Kocevski et al. 2015). However, Allevato et al. (2011) The sample of AGN is introduced in section ; the methodology and Karouzos et al. (2014) suggest that biases can account for ap- in section ; the results are presented in section and discussed in parent correlations between AGN and galaxy interactions, while section before conclusions are presented in section . AB magnitudes Grogin et al. (2005), Gabor et al. (2009), Cisternas et al. (2011), are used throughout and, where needed, a CDM cosmology with Λ Schawinski et al. (2011), Kocevski et al. (2012), Villforth et al. h = 0.7, = 0.3, = 0.7 is assumed. It is worth noting that (2014) and Villforth et al. (2017) ﬁnd no diﬀerence between the w0ith this ΩcomsmologyΩthΛe angular scale is approximately constant morphologies of AGN hosts and control galaxies. Though this is at ∼8.5 kpc/" between z=1-2.2, dropping oﬀ sharply to ∼6.2 kpc/" far from a complete summary of the literature, thus far observa- at z=0.5. The fraction of galaxies in this work with z<1 is small tional evidence is inconclusive with regards to the question of a enough to state here that working with pixel scales throughout is merger-AGN connection. The work presented here seeks to extend approximately equivalent to working with physical scales and will these analyses to higher luminosities and larger redshift ranges than not aﬀect the conclusions. previously attempted in a single coherent study (as opposed to a 2 SAMPLE AND DATA compilation of various surveys and analyses). In the local universe it is generally fairly easy to identify recent The COSMOS survey contains multi-wavelength coverage mergers by eye: low surface brightness features such as tidal tails over a region of sky ∼2 deg2 (Scoville et al. 2007). This includes are relatively easily picked out and mergers can be inferred from this X-ray data from XMM-Newton at a depth of ∼50Ks (Hasinger et al. highly non-equilibrium state (Vorontsov-Velyaminov 1959; Cister- 2007; Cappelluti et al. 2009), allowing for identiﬁcation of a sample nas et al. 2011; Pawlik et al. 2016). In the high redshift universe of AGN using X-rays, and HST F814W data for optical followup. it is far more challenging. Faint features may be of too low sur- X-rays are used for identiﬁcation of AGN because it has been ar- face brightness to reliably detect, even soon after the merger, and gued that they provide the cleanest AGN sample compared to other galaxies are often poorly resolved. In addition, when looking for selection methods (e.g. optical, IR, radio), with very few non-AGN connections between AGN and galaxies, light from the bright point sources with LX > 1042 erg s-1 (Ranalli et al. 2003). Only Compton source is inevitably spread across the plane of the galaxy with some thick sources (NH > 1024 cm-2; Risaliti et al. 1999) may obscure Point Spread Function (PSF) which can drown out galaxy features, X-ray emission from AGN. especially those of low surface brightness. High resolution is thus HST imaging in the F814W band has a point source limiting required for morphological study in the high redshift universe, mak- AB magnitude of 27.2 at 10 and a surface brightness limiting σ ing the Advanced Camera for Surveys on the HST ideal. The COS- magnitude of 26.1 mags/arcsecond2 at 3 (calculated from image MOS ﬁeld is a ∼2 deg2 region of sky utilising many telescopes for noise), otherwise stated as a 5 depth σin a 3" aperture of 25.3 σ multi-wavelength coverage (Scoville et al. 2007), including HST (Capak et al. 2007). Point source depths drop of with (1+z)2, while imaging with a point source limiting magnitude of 27.2. This is extended sources drop oﬀ with (1+z)4. This high sensitivity makes deep enough to detect even faint objects. The HST has a complex the F814W HST ﬁlter ideal for identifying low surface brightness and hard to model PSF. Studies in the past have dealt with this by features such as tidal tails, hence was used in this investigation use of modelling programs such as TinyTim (Krist et al. 2011) or in the search for evidence of mergers. HST images in COSMOS empirically determining the PSF by averaging the distribution of have been corrected for sky background and instrumental noise, light from stars (Villforth et al. 2014). The latter method was used and drizzled to a resolution of 0.03 "/pixel in post-processing (see in this work. Koekemoer et al. 2007; Massey et al. 2010, for details). From the Looking to earlier epochs means looking toward the peak of 545 luminous X-ray detected type 1 AGN in the COSMOS ﬁeld AGN activity (z∼2-3), where the most rapid growth of SMBHs in Lusso et al. (2009), 106 were selected for analysis here over a MNRAS 000, 1–15 (2017)

text: T.

font: JXTPLL+TeXGyreTermes-Italic

text: Hewlett

text: et

text: al.

text: secularly

text: through,

text: for

text: example,

text: disk

text: instabilities

text: (Bournaud

text: et

text: al.

text: occurred

text: (Ueda

text: et

text: al.

text: Aird

text: et

text: al.

text: It

text: has

text: been

text: argued

text: bars

text: (Knapen

text: et

text: al.

text: Oh

text: et

text: al.

text: or

text: otherwise

text: by

text: that

text: only

text: the

text: most

text: luminous

text: AGN,

text: with

text: the

text: highest

text: accretion

text: rates

text: minor

text: mergers

text: (Kaviraj

text: It

text: remains

text: unclear

text: whether

text: alterna-

text: and

text: therefore

text: the

text: most

text: rapid

text: loss

text: of

text: angular

text: momentum,

text: need

text: to

text: tives

text: to

text: major

text: merger

text: triggering

text: can

text: drive

text: several

text: M

text: of

text: gas

text: be

text: fuelled

text: by

text: mergers

text: (Sanders

text: et

text: al.

text: Di

text: Matteo

text: et

text: al.

text: to

text: the

text: central

text: SMBHs,

text: as

text: is

text: necessary

text: to

text: power

text: the

text: most

text: luminous

text: Hopkins

text: et

text: al.

text: This

text: makes

text: a

text: natural

text: place

text: to

text: search

text: AGN.

text: for

text: an

text: AGN-merger

text: connection,

text: since

text: also

text: coincides

text: with

text: the

text: The

text: tightness

text: of

text: correlations

text: between

text: the

text: masses

text: of

text: SMBHs

text: predicted

text: peak

text: of

text: galaxy

text: mergers

text: in

text: hierarchical

text: galaxy

text: evolution

text: and

text: host

text: galaxy

text: masses,

text: velocity

text: dispersions

text: and

text: luminosities

text: may

text: (Kauﬀmann

text: Haehnelt

text: Volonteri

text: et

text: al.

text: Conselice

text: et

text: al.

text: imply

text: coevolution

text: between

text: them

text: (Ferrarese

text: Merritt

text: Mar-

text: Because

text: of

text: the

text: diﬃculties

text: inherent

text: in

text: detecting

text: mergers

text: at

text: coni

text: Hunt

text: Haring

text: Rix

text: Major

text: mergers

text: are

text: a

text: com-

text: such

text: high

text: redshift

text: many

text: studies

text: thus

text: far

text: have

text: focussed

text: on

text: either

text: monly

text: proposed

text: mechanism

text: for

text: growing

text: black

text: holes

text: and

text: galaxies

text: low

text: luminosity

text: sources

text: (e.g.

text: Schawinski

text: et

text: al.

text: Kocevski

text: et

text: al.

text: synchronously.

text: Detection

text: of

text: connections

text: between

text: AGN

text: activity

text: and

text: Villforth

text: et

text: al.

text: or

text: low

text: redshift

text: AGN

text: (Schawinski

text: major

text: mergers

text: may

text: therefore

text: shed

text: some

text: light

text: on

text: the

text: origins

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text: cor-

text: et

text: al.

text: In

text: this

text: study

text: we

text: will

text: analyse

text: COSMOS

text: HST

text: relations

text: between

text: the

text: SMBH

text: mass

text: and

text: galaxy

text: properties,

text: such

text: as

text: the

text: M-

text: relation

text: (the

text: correlation

text: between

text: the

text: mass

text: of

text: the

text: SMBH

text: galaxies

text: between

text: It

text: is

text: not

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text: to

text: show

text: that

text: AGN

text: do

text: σ

text: and

text: the

text: velocity

text: dispersion

text: of

text: the

text: spheroid).

text: Alternatively,

text: these

text: or

text: do

text: not

text: live

text: in

text: merging

text: galaxies;

text: the

text: crucial

text: comparison

text: is

text: to

text: an

text: relations

text: could

text: be

text: caused

text: by

text: stochastic

text: processes,

text: since

text: black

text: holes

text: inactive

text: sample

text: of

text: galaxies

text: matched

text: in

text: stellar

text: mass,

text: redshift

text: and

text: star

text: and

text: star

text: formation

text: are

text: both

text: fed

text: by

text: a

text: supply

text: of

text: cold

text: gas.

text: The

text: possibility

text: of

text: a

text: connection

text: between

text: nuclear

text: activity

text: and

text: in

text: active

text: galaxies

text: relative

text: to

text: "normal"

text: galaxies.

text: One

text: aim

text: of

text: this

text: galaxy

text: mergers

text: can

text: be

text: investigated

text: observationally

text: by

text: comparing

text: study

text: is

text: to

text: see

text: if

text: triggering

text: mechanisms

text: changed

text: over

text: cosmic

text: time,

text: galaxy

text: morphologies

text: and

text: AGN

text: properties.

text: Canalizo

text: Stockton

text: both

text: during

text: and

text: outside

text: the

text: peak

text: of

text: mergers/AGN

text: activity.

text: Another

text: Treister

text: et

text: al.

text: Ellison

text: et

text: al.

text: and

text: Kocevski

text: principal

text: aim

text: was

text: to

text: probe

text: the

text: triggering

text: mechanisms

text: of

text: the

text: highest

text: et

text: al.

text: ﬁnd

text: evidence

text: of

text: enhancement

text: of

text: AGN

text: associated

text: with

text: luminosity

text: AGN

text: to

text: see

text: if

text: there

text: was

text: some

text: threshold

text: accretion

text: rate

text: mergers,

text: or

text: else

text: that

text: mergers

text: are

text: associated

text: in

text: particular

text: with

text: ob-

text: which

text: required

text: more

text: catastrophic

text: events

text: such

text: as

text: galaxy

text: mergers.

text: scured

text: AGN

text: (Kocevski

text: et

text: al.

text: However,

text: Allevato

text: et

text: al.

text: The

text: sample

text: of

text: AGN

text: is

text: introduced

text: in

text: section

text: the

text: methodology

text: and

text: Karouzos

text: et

text: al.

text: suggest

text: that

text: biases

text: can

text: account

text: for

text: ap-

text: in

text: section

text: the

text: results

text: are

text: presented

text: in

text: section

text: and

text: discussed

text: in

text: parent

text: correlations

text: between

text: AGN

text: and

text: galaxy

text: interactions,

text: while

text: section

text: before

text: conclusions

text: are

text: presented

text: in

text: section

text: AB

text: magnitudes

text: Grogin

text: et

text: al.

text: Gabor

text: et

text: al.

text: Cisternas

text: et

text: al.

text: are

text: used

text: throughout

text: and,

text: where

text: needed,

text: a

text: CDM

text: cosmology

text: with

text: Λ

text: Schawinski

text: et

text: al.

text: Kocevski

text: et

text: al.

text: Villforth

text: et

text: al.

text: h

text: is

text: assumed.

text: It

text: is

text: worth

text: noting

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text: and

text: Villforth

text: et

text: al.

text: ﬁnd

text: no

text: diﬀerence

text: between

text: the

text: this

text: ΩcomsmologyΩthΛe

text: angular

text: scale

text: is

text: approximately

text: constant

text: morphologies

text: of

text: AGN

text: hosts

text: and

text: control

text: galaxies.

text: Though

text: this

text: is

text: at

text: kpc/"

text: between

text: dropping

text: oﬀ

text: sharply

text: to

text: kpc/"

text: far

text: from

text: a

text: complete

text: summary

text: of

text: the

text: literature,

text: thus

text: far

text: observa-

text: at

text: The

text: fraction

text: of

text: galaxies

text: in

text: this

text: work

text: with

text: is

text: small

text: tional

text: evidence

text: is

text: inconclusive

text: with

text: regards

text: to

text: the

text: question

text: of

text: a

text: enough

text: to

text: state

text: here

text: that

text: working

text: with

text: pixel

text: scales

text: throughout

text: is

text: merger-AGN

text: connection.

text: The

text: work

text: presented

text: here

text: seeks

text: to

text: extend

text: approximately

text: equivalent

text: to

text: working

text: with

text: physical

text: scales

text: and

text: will

text: these

text: analyses

text: to

text: higher

text: luminosities

text: and

text: larger

text: redshift

text: ranges

text: than

text: not

text: aﬀect

text: the

text: conclusions.

text: previously

text: attempted

text: in

text: a

text: single

text: coherent

text: study

text: (as

text: opposed

text: to

text: a

text: SAMPLE

text: AND

text: DATA

text: compilation

text: of

text: various

text: surveys

text: and

text: analyses).

text: In

text: the

text: local

text: universe

text: it

text: is

text: generally

text: fairly

text: easy

text: to

text: identify

text: recent

text: The

text: COSMOS

text: survey

text: contains

text: multi-wavelength

text: coverage

text: mergers

text: by

text: eye:

text: low

text: surface

text: brightness

text: features

text: such

text: as

text: tidal

text: tails

text: over

text: a

text: region

text: of

text: sky

text: ∼2

text: (Scoville

text: et

text: al.

text: This

text: includes

text: are

text: relatively

text: easily

text: picked

text: out

text: and

text: mergers

text: can

text: be

text: inferred

text: from

text: this

text: X-ray

text: data

text: from

text: XMM-Newton

text: at

text: a

text: depth

text: of

text: (Hasinger

text: et

text: al.

text: highly

text: non-equilibrium

text: state

text: (Vorontsov-Velyaminov

text: Cister-

text: Cappelluti

text: et

text: al.

text: allowing

text: for

text: identiﬁcation

text: of

text: a

text: sample

text: nas

text: et

text: al.

text: Pawlik

text: et

text: al.

text: In

text: the

text: high

text: redshift

text: universe

text: of

text: AGN

text: using

text: X-rays,

text: and

text: HST

text: data

text: for

text: optical

text: followup.

text: it

text: is

text: far

text: more

text: challenging.

text: Faint

text: features

text: may

text: be

text: of

text: too

text: low

text: sur-

text: X-rays

text: are

text: used

text: for

text: identiﬁcation

text: of

text: AGN

text: because

text: it

text: has

text: been

text: ar-

text: face

text: brightness

text: to

text: reliably

text: detect,

text: even

text: soon

text: after

text: the

text: merger,

text: and

text: gued

text: that

text: they

text: provide

text: the

text: cleanest

text: AGN

text: sample

text: compared

text: to

text: other

text: galaxies

text: are

text: often

text: poorly

text: resolved.

text: In

text: addition,

text: when

text: looking

text: for

text: selection

text: methods

text: (e.g.

text: optical,

text: IR,

text: radio),

text: with

text: very

text: few

text: non-AGN

text: connections

text: between

text: AGN

text: and

text: galaxies,

text: light

text: from

text: the

text: bright

text: point

text: sources

text: with

text: LX

text: erg

text: (Ranalli

text: et

text: al.

text: Only

text: Compton

text: source

text: is

text: inevitably

text: spread

text: across

text: the

text: plane

text: of

text: the

text: galaxy

text: with

text: some

text: thick

text: sources

text: (NH

text: Risaliti

text: et

text: al.

text: may

text: obscure

text: Point

text: Spread

text: Function

text: (PSF)

text: which

text: can

text: drown

text: out

text: galaxy

text: features,

text: X-ray

text: emission

text: from

text: AGN.

text: especially

text: those

text: of

text: low

text: surface

text: brightness.

text: High

text: resolution

text: is

text: thus

text: HST

text: imaging

text: in

text: the

text: band

text: has

text: a

text: point

text: source

text: limiting

text: required

text: for

text: morphological

text: study

text: in

text: the

text: high

text: redshift

text: universe,

text: mak-

text: AB

text: magnitude

text: of

text: at

text: and

text: a

text: surface

text: brightness

text: limiting

text: σ

text: ing

text: the

text: Advanced

text: Camera

text: for

text: Surveys

text: on

text: the

text: HST

text: ideal.

text: The

text: COS-

text: magnitude

text: of

text: at

text: (calculated

text: from

text: image

text: MOS

text: ﬁeld

text: is

text: a

text: ∼2

text: region

text: of

text: sky

text: utilising

text: many

text: telescopes

text: for

text: noise),

text: otherwise

text: stated

text: as

text: a

text: depth

text: σin

text: a

text: aperture

text: of

text: σ

text: multi-wavelength

text: coverage

text: (Scoville

text: et

text: al.

text: including

text: HST

text: (Capak

text: et

text: al.

text: Point

text: source

text: depths

text: drop

text: of

text: with

text: while

text: imaging

text: with

text: a

text: point

text: source

text: limiting

text: magnitude

text: of

text: This

text: is

text: extended

text: sources

text: drop

text: oﬀ

text: with

text: This

text: high

text: sensitivity

text: makes

text: deep

text: enough

text: to

text: detect

text: even

text: faint

text: objects.

text: The

text: HST

text: has

text: a

text: complex

text: the

text: HST

text: ﬁlter

text: ideal

text: for

text: identifying

text: low

text: surface

text: brightness

text: and

text: hard

text: to

text: model

text: PSF.

text: Studies

text: in

text: the

text: past

text: have

text: dealt

text: with

text: this

text: by

text: features

text: such

text: as

text: tidal

text: tails,

text: hence

text: was

text: used

text: in

text: this

text: investigation

text: use

text: of

text: modelling

text: programs

text: such

text: as

text: TinyTim

text: (Krist

text: et

text: al.

text: or

text: in

text: the

text: search

text: for

text: evidence

text: of

text: mergers.

text: HST

text: images

text: in

text: COSMOS

text: empirically

text: determining

text: the

text: PSF

text: by

text: averaging

text: the

text: distribution

text: of

text: have

text: been

text: corrected

text: for

text: sky

text: background

text: and

text: instrumental

text: noise,

text: light

text: from

text: stars

text: (Villforth

text: et

text: al.

text: The

text: latter

text: method

text: was

text: used

text: and

text: drizzled

text: to

text: a

text: resolution

text: of

text: "/pixel

text: in

text: post-processing

text: (see

text: in

text: this

text: work.

text: Koekemoer

text: et

text: al.

text: Massey

text: et

text: al.

text: for

text: details).

text: From

text: the

text: Looking

text: to

text: earlier

text: epochs

text: means

text: looking

text: toward

text: the

text: peak

text: of

text: luminous

text: X-ray

text: detected

text: type

text: AGN

text: in

text: the

text: COSMOS

text: ﬁeld

text: AGN

text: activity

text: where

text: the

text: most

text: rapid

text: growth

text: of

text: SMBHs

text: in

text: Lusso

text: et

text: al.

text: were

text: selected

text: for

text: analysis

text: here

text: over

text: a

text: MNRAS


text: ∼ AGN merger enhancement at z 2 11 supportstheﬁndingsofotherobservationalstudies,thoughdiﬀering AGN selection methods may obfuscate the issue. The ﬁnal piece of evidence is the correlation between residual ﬂux or visual classiﬁcation and redshift in ﬁgures 9 and 10. First we will discuss the residual ﬂux result. Figure 9 shows similar- ity between AGN and control galaxies in residual ﬂux across the redshift range. One may be concerned that merging galaxies will have similar enhancement of residual ﬂux compared to galaxies with non-axisymmetric features, such as spiral arms and clumpy disks. Over this redshift range the F814W band measures rest frame UV through to visual; since star formation is intrinsically clumpy (Bournaud et al. 2011; Trump et al. 2014) this may manifest itself as excess residuals at high redshift. We did not attempt to ﬁt non- axisymmetric structures (section ), instead choosing to compare residual ﬂux to visual classiﬁcation. Visual class and residual ﬂux correlate well (see ﬁgure A2).Visual classiﬁers had the opportunity to highlight non-axisymmetric structures for each galaxy: their ex- clusion did not aﬀect the result, and further inspection of images with high residual ﬂux reveals tidal tails and structures that do not Figure 11. Probability density functions, calculated in accordance with Cameron (2011), for the fraction of AGN (wider red line) and controls look like clumpy disks or spiral arms, ruling this out as the probable (narrow blue line) classiﬁed as either mergers or disturbed. cause of similarity (See ﬁgure A4). Considering visual classiﬁcation, ﬁgure 10 shows the fraction of AGN hosts and control galaxies classiﬁed as mergers in bins of sample than presented here. Randomly drawing 19 AGN and 84 redshift. AGN hosts are found to have a higher incidence of mergers control galaxies from the sample (matching to the number of AGN at high redshift, with a factor of ∼4 enhancement of AGN in merg- and controls in Mechtley et al. 2016), ﬁnding the statistical diﬀer- ers at z∼2 with 99% conﬁdence. Considering disturbed and merger ence in merger fraction between the two populations, and repeating classiﬁcations together as one class gives a factor of ∼2 enhance- multiple times yields a wide (Gaussian) distribution of statistical ment with 98.5% conﬁdence at z∼2. We consider whether a mass signiﬁcances. The probability that Mechtley et al. (2016) would mismatch between the AGN hosts and control galaxies could lead to ﬁnd a signiﬁcance of <1 is ∼24%. Work to date has not found a σ a diﬀerence in merger incidence between AGN hosts and controls. statistically signiﬁcant redshift evolution of AGN in mergers (Gabor If, for instance, the M-M relationship evolves with redshift et al. 2009; Allevato et al. 2011; Cisternas et al. 2011; Schawinski bulge this could lead to a mass mismatch, and galaxies of diﬀerent masses et al. 2011; Treister et al. 2012; Bruce et al. 2015; Mechtley et al. may undergo major mergers with diﬀerent frequencies. Stewart et al. 2016); it will be interesting to see if this elevation persists in future (2009) and Rodriguez-Gomez et al. (2015) predict the merger fre- large high redshift samples, since we are now possibly approaching quency to be a function of mass, with more massive galaxies having large enough samples to detect such an enhancement. (cid:199) slightly higher merger rates. At z∼2 there are approximately twice Overall AGN are not found more commonly in galaxies ex- as many major m(cid:199)ergers for galaxies with M ∼1011M as there are hibiting signs of recent major mergers, but an excess of mergers * for M ∼1010M (Rodriguez-Gomez et al. 2015). This suggests is found in the highest redshift bin (z∼2). No evidence is found in * any mass mismatch here would have to be implausibly severe to support of conjecture that the most luminous AGN must be fuelled entirely explain the observed factor of ∼4 enhancement. Were the predominantly by mergers, in contradiction to Treister et al. (2012) mass mismatch so severe, the model magnitude distributions for but in accordance with other studies (Cisternas et al. 2011; Allevato AGN hosts and control galaxies would be extremely diﬀerent to et al. 2011; Karouzos et al. 2014; Mechtley et al. 2016; Villforth each other: in reality they are comparable. The M-M relation et al. 2017). An enhancement of merger fraction in AGN hosts is would be expected to evolve slowly, if at all, to have avbouildgeed robust found in the highest redshift bin, z∼2, despite the results of other detection in previous studies (Robertson et al. 2006a; Lauer et al. studies (Gabor et al. 2009; Allevato et al. 2011; Cisternas et al. 2011; 2007; Shankar et al. 2009; Shen et al. 2015), so it is likely that Schawinski et al. 2011; Bruce et al. 2015; Mechtley et al. 2016). control and AGN host galaxies are similarly massive. The majority of galaxies here are unlikely to have recently under- gone a merger, given their disk-dominated light proﬁles (Toomre & Next, we consider the possibility that this result has remained Toomre 1972; Barnes 1988; Di Matteo et al. 2005; Cox et al. 2008; undetected in previous studies due to poor number statistics. Work Gabor et al. 2009; Cisternas et al. 2011; Schawinski et al. 2011; by Mechtley et al. (2016) constitutes the best comparison to the Kocevski et al. 2012; Villforth et al. 2014, 2017). study presented here, since they consider a sample of 19 high lu- minosity AGN at z∼2, compared to the 32 high luminosity AGN at 6 CONCLUSIONS z∼2 here. Mechtley et al. (2016) found a statistically insigniﬁcant The aim of this study was to determine if a link exists between enhancement in the probability of the fraction of AGN in mergers galaxy mergers and AGN activity, and whether this link evolves with being greater than the fraction of controls of 0.78 . Similar to ﬁg- redshift or AGN luminosity. High resolution HST images of 106 X- σ ure 3 in Mechtley et al. (2016), we show the probability density ray selected AGN and 486 host galaxy mass and redshift matched functions of the 32 z∼2 AGN and their respective control galaxies control galaxies from the COSMOS survey are analysed. Artiﬁ- in ﬁgure 11. Assuming the AGN in Mechtley et al. (2016) are in- cial AGN (stars), matched in apparent magnitude to the relevant trinsically drawn from the same underlying population of AGN as AGN, are placed into control galaxies to make images equivalent. A those in this work, we can explore the likelihood of ﬁnding a statis- new method for detecting disturbance in galaxies is developed: sub- tically signiﬁcant enhancement of the merger fraction in a smaller tracting 2D photometric decomposition models from images (using MNRAS 000, 1–15 (2017) ##LTFigure##

label: paragraph

font: IFMCGE+txsys

text: AGN

text: merger

text: enhancement

text: at

text: z

font: ITVAOD+TeXGyreTermes-Regular

text: AGN

text: selection

text: methods

text: may

text: obfuscate

text: the

text: issue.

text: The

text: ﬁnal

text: piece

text: of

text: evidence

text: is

text: the

text: correlation

text: between

text: residual

text: ﬂux

text: or

text: visual

text: classiﬁcation

text: and

text: redshift

text: in

text: ﬁgures

text: and

text: First

text: we

text: will

text: discuss

text: the

text: residual

text: ﬂux

text: result.

text: Figure

text: shows

text: similar-

text: ity

text: between

text: AGN

text: and

text: control

text: galaxies

text: in

text: residual

text: ﬂux

text: across

text: the

text: redshift

text: range.

text: One

text: may

text: be

text: concerned

text: that

text: merging

text: galaxies

text: will

text: have

text: similar

text: enhancement

text: of

text: residual

text: ﬂux

text: compared

text: to

text: galaxies

text: with

text: non-axisymmetric

text: features,

text: such

text: as

text: spiral

text: arms

text: and

text: clumpy

text: disks.

text: Over

text: this

text: redshift

text: range

text: the

text: band

text: measures

text: rest

text: frame

text: UV

text: through

text: to

text: visual;

text: since

text: star

text: formation

text: is

text: intrinsically

text: clumpy

text: (Bournaud

text: et

text: al.

text: Trump

text: et

text: al.

text: this

text: may

text: manifest

text: itself

text: as

text: excess

text: residuals

text: at

text: high

text: redshift.

text: We

text: did

text: not

text: attempt

text: to

text: ﬁt

text: non-

text: axisymmetric

text: structures

text: (section

text: instead

text: choosing

text: to

text: compare

text: residual

text: ﬂux

text: to

text: visual

text: classiﬁcation.

text: Visual

text: class

text: and

text: residual

text: ﬂux

text: correlate

text: well

text: (see

text: ﬁgure

text: classiﬁers

text: had

text: the

text: opportunity

text: to

text: highlight

text: non-axisymmetric

text: structures

text: for

text: each

text: galaxy:

text: their

text: ex-

text: clusion

text: did

text: not

text: aﬀect

text: the

text: result,

text: and

text: further

text: inspection

text: of

text: images

text: with

text: high

text: residual

text: ﬂux

text: reveals

text: tidal

text: tails

text: and

text: structures

text: that

text: do

text: not

text: Figure

text: Probability

text: density

text: functions,

text: calculated

text: in

text: accordance

text: with

text: Cameron

text: for

text: the

text: fraction

text: of

text: AGN

text: (wider

text: red

text: line)

text: and

text: controls

text: look

text: like

text: clumpy

text: disks

text: or

text: spiral

text: arms,

text: ruling

text: this

text: out

text: as

text: the

text: probable

text: (narrow

text: blue

text: line)

text: classiﬁed

text: as

text: either

text: mergers

text: or

text: disturbed.

text: cause

text: of

text: similarity

text: (See

text: ﬁgure

text: Considering

text: visual

text: classiﬁcation,

text: ﬁgure

text: shows

text: the

text: fraction

text: of

text: AGN

text: hosts

text: and

text: control

text: galaxies

text: classiﬁed

text: as

text: mergers

text: in

text: bins

text: of

text: sample

text: than

text: presented

text: here.

text: Randomly

text: drawing

text: AGN

text: and

text: redshift.

text: AGN

text: hosts

text: are

text: found

text: to

text: have

text: a

text: higher

text: incidence

text: of

text: mergers

text: control

text: galaxies

text: from

text: the

text: sample

text: (matching

text: to

text: the

text: number

text: of

text: AGN

text: at

text: high

text: redshift,

text: with

text: a

text: factor

text: of

text: ∼4

text: enhancement

text: of

text: AGN

text: in

text: merg-

text: and

text: controls

text: in

text: Mechtley

text: et

text: al.

text: ﬁnding

text: the

text: statistical

text: diﬀer-

text: ers

text: at

text: with

text: conﬁdence.

text: Considering

text: disturbed

text: and

text: merger

text: ence

text: in

text: merger

text: fraction

text: between

text: the

text: two

text: populations,

text: and

text: repeating

text: classiﬁcations

text: together

text: as

text: one

text: class

text: gives

text: a

text: factor

text: of

text: ∼2

text: enhance-

text: multiple

text: times

text: yields

text: a

text: wide

text: (Gaussian)

text: distribution

text: of

text: statistical

text: ment

text: with

text: conﬁdence

text: at

text: We

text: consider

text: whether

text: a

text: mass

text: signiﬁcances.

text: The

text: probability

text: that

text: Mechtley

text: et

text: al.

text: would

text: mismatch

text: between

text: the

text: AGN

text: hosts

text: and

text: control

text: galaxies

text: could

text: lead

text: to

text: ﬁnd

text: a

text: signiﬁcance

text: of

text: is

text: Work

text: to

text: date

text: has

text: not

text: found

text: a

text: σ

text: a

text: diﬀerence

text: in

text: merger

text: incidence

text: between

text: AGN

text: hosts

text: and

text: controls.

text: statistically

text: signiﬁcant

text: redshift

text: evolution

text: of

text: AGN

text: in

text: mergers

text: (Gabor

text: If,

text: for

text: instance,

text: the

text: M-M

text: relationship

text: evolves

text: with

text: redshift

text: et

text: al.

text: Allevato

text: et

text: al.

text: Cisternas

text: et

text: al.

text: Schawinski

text: bulge

text: this

text: could

text: lead

text: to

text: a

text: mass

text: mismatch,

text: and

text: galaxies

text: of

text: diﬀerent

text: masses

text: et

text: al.

text: Treister

text: et

text: al.

text: Bruce

text: et

text: al.

text: Mechtley

text: et

text: al.

text: may

text: undergo

text: major

text: mergers

text: with

text: diﬀerent

text: frequencies.

text: Stewart

text: et

text: al.

text: it

text: will

text: be

text: interesting

text: to

text: see

text: if

text: this

text: elevation

text: persists

text: in

text: future

text: and

text: Rodriguez-Gomez

text: et

text: al.

text: predict

text: the

text: merger

text: fre-

text: large

text: high

text: redshift

text: samples,

text: since

text: we

text: are

text: now

text: possibly

text: approaching

text: quency

text: to

text: be

text: a

text: function

text: of

text: mass,

text: with

text: more

text: massive

text: galaxies

text: having

text: large

text: enough

text: samples

text: to

text: detect

text: such

text: an

text: enhancement.

text: slightly

text: higher

text: merger

text: rates.

text: At

text: there

text: are

text: approximately

text: twice

text: Overall

text: AGN

text: are

text: not

text: found

text: more

text: commonly

text: in

text: galaxies

text: ex-

text: as

text: many

text: major

text: for

text: galaxies

text: with

text: M

text: as

text: there

text: are

text: hibiting

text: signs

text: of

text: recent

text: major

text: mergers,

text: but

text: an

text: excess

text: of

text: mergers

text: for

text: M

text: (Rodriguez-Gomez

text: et

text: al.

text: This

text: suggests

text: is

text: found

text: in

text: the

text: highest

text: redshift

text: bin

text: No

text: evidence

text: is

text: found

text: in

text: any

text: mass

text: mismatch

text: here

text: would

text: have

text: to

text: be

text: implausibly

text: severe

text: to

text: support

text: of

text: conjecture

text: that

text: the

text: most

text: luminous

text: AGN

text: must

text: be

text: fuelled

text: entirely

text: explain

text: the

text: observed

text: factor

text: of

text: ∼4

text: enhancement.

text: Were

text: the

text: predominantly

text: by

text: mergers,

text: in

text: contradiction

text: to

text: Treister

text: et

text: al.

text: mass

text: mismatch

text: so

text: severe,

text: the

text: model

text: magnitude

text: distributions

text: for

text: but

text: in

text: accordance

text: with

text: other

text: studies

text: (Cisternas

text: et

text: al.

text: Allevato

text: AGN

text: hosts

text: and

text: control

text: galaxies

text: would

text: be

text: extremely

text: diﬀerent

text: to

text: et

text: al.

text: Karouzos

text: et

text: al.

text: Mechtley

text: et

text: al.

text: Villforth

text: each

text: other:

text: in

text: reality

text: they

text: are

text: comparable.

text: The

text: M-M

text: relation

text: et

text: al.

text: An

text: enhancement

text: of

text: merger

text: fraction

text: in

text: AGN

text: hosts

text: is

text: would

text: be

text: expected

text: to

text: evolve

text: slowly,

text: if

text: at

text: all,

text: to

text: have

text: avbouildgeed

text: robust

text: found

text: in

text: the

text: highest

text: redshift

text: bin,

text: despite

text: the

text: results

text: of

text: other

text: detection

text: in

text: previous

text: studies

text: (Robertson

text: et

text: al.

text: Lauer

text: et

text: al.

text: studies

text: (Gabor

text: et

text: al.

text: Allevato

text: et

text: al.

text: Cisternas

text: et

text: al.

text: Shankar

text: et

text: al.

text: Shen

text: et

text: al.

text: so

text: it

text: is

text: likely

text: that

text: Schawinski

text: et

text: al.

text: Bruce

text: et

text: al.

text: Mechtley

text: et

text: al.

text: control

text: and

text: AGN

text: host

text: galaxies

text: are

text: similarly

text: massive.

text: The

text: majority

text: of

text: galaxies

text: here

text: are

text: unlikely

text: to

text: have

text: recently

text: under-

text: gone

text: a

text: merger,

text: given

text: their

text: disk-dominated

text: light

text: proﬁles

text: (Toomre

text: Next,

text: we

text: consider

text: the

text: possibility

text: that

text: this

text: result

text: has

text: remained

text: Toomre

text: Barnes

text: Di

text: Matteo

text: et

text: al.

text: Cox

text: et

text: al.

text: undetected

text: in

text: previous

text: studies

text: due

text: to

text: poor

text: number

text: statistics.

text: Work

text: Gabor

text: et

text: al.

text: Cisternas

text: et

text: al.

text: Schawinski

text: et

text: al.

text: by

text: Mechtley

text: et

text: al.

text: constitutes

text: the

text: best

text: comparison

text: to

text: the

text: Kocevski

text: et

text: al.

text: Villforth

text: et

text: al.

text: study

text: presented

text: here,

text: since

text: they

text: consider

text: a

text: sample

text: of

text: high

text: lu-

text: minosity

text: AGN

text: at

text: compared

text: to

text: the

text: high

text: luminosity

text: AGN

text: at

text: CONCLUSIONS

text: here.

text: Mechtley

text: et

text: al.

text: found

text: a

text: statistically

text: insigniﬁcant

text: The

text: aim

text: of

text: this

text: study

text: was

text: to

text: determine

text: if

text: a

text: link

text: exists

text: between

text: enhancement

text: in

text: the

text: probability

text: of

text: the

text: fraction

text: of

text: AGN

text: in

text: mergers

text: galaxy

text: mergers

text: and

text: AGN

text: activity,

text: and

text: whether

text: this

text: link

text: evolves

text: with

text: being

text: greater

text: than

text: the

text: fraction

text: of

text: controls

text: of

text: Similar

text: to

text: ﬁg-

text: redshift

text: or

text: AGN

text: luminosity.

text: High

text: resolution

text: HST

text: images

text: of

text: X-

text: σ

text: ure

text: in

text: Mechtley

text: et

text: al.

text: we

text: show

text: the

text: probability

text: density

text: ray

text: selected

text: AGN

text: and

text: host

text: galaxy

text: mass

text: and

text: redshift

text: matched

text: functions

text: of

text: the

text: AGN

text: and

text: their

text: respective

text: control

text: galaxies

text: control

text: galaxies

text: from

text: the

text: COSMOS

text: survey

text: are

text: analysed.

text: Artiﬁ-

text: in

text: ﬁgure

text: Assuming

text: the

text: AGN

text: in

text: Mechtley

text: et

text: al.

text: are

text: in-

text: cial

text: AGN

text: (stars),

text: matched

text: in

text: apparent

text: magnitude

text: to

text: the

text: relevant

text: trinsically

text: drawn

text: from

text: the

text: same

text: underlying

text: population

text: of

text: AGN

text: as

text: AGN,

text: are

text: placed

text: into

text: control

text: galaxies

text: to

text: make

text: images

text: equivalent.

text: A

text: those

text: in

text: this

text: work,

text: we

text: can

text: explore

text: the

text: likelihood

text: of

text: ﬁnding

text: a

text: statis-

text: new

text: method

text: for

text: detecting

text: disturbance

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text: galaxies

text: is

text: developed:

text: sub-

text: tically

text: signiﬁcant

text: enhancement

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text: the

text: merger

text: fraction

text: in

text: a

text: smaller

text: tracting

text: photometric

text: decomposition

text: models

text: from

text: images

text: (using

text: MNRAS

text: ##LTFigure##


text: 16 T. Hewlett et al. Figure A4. All galaxies with residual ﬂux > 5 counts; high degrees of disturbance are seen in almost all cases. The ﬁrst 11 galaxies (counting left to right, top to bottom and starting in the top left) are controls and the last 5 are AGN hosts. 5 Galaxies (1 AGN and 4 control) were removed due to image defects; see text for details. MNRAS 000, 1–15 (2017) ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure##

text: T.

text: Hewlett

text: et

text: al.

text: Figure

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text: All

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text: galaxies

text: with

text: residual

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text: high

text: degrees

text: of

text: disturbance

text: are

text: seen

text: in

text: almost

text: all

text: cases.

text: The

text: ﬁrst

text: galaxies

text: (counting

text: left

text: to

text: right,

text: top

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text: bottom

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text: starting

text: in

text: the

text: top

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text: are

text: controls

text: and

text: the

text: last

text: are

text: AGN

text: hosts.

text: Galaxies

text: AGN

text: and

text: control)

text: were

text: removed

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text: image

text: defects;

text: see

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text: details.

text: MNRAS

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text: ∼ AGN merger enhancement at z 2 9 Figure 9. Relation between redshift and residual ﬂux. As in other plots blue points are control galaxies; small red triangles are AGN; and the large blue triangles and large red stars represent the median values of controls and AGN respectively, with associated bootstrap errors for diﬀerent bins of redshift. As with ﬁgure 7 all residual ﬂux values have had +2 added for clarity; this is merely because ∼15% of points have negative values once the erroneous point source contribution has been subtracted, and these points do not appear on a logarithmic scale. The signiﬁcant comparison is between the relative median values, which is unaﬀected by a uniform addition. Errors are as in ﬁgure A2. disturbance in the galaxies. Galaxies are visually classiﬁed by three experts, identifying likely mergers. Comparison to a group of mass and redshift matched control galaxies allows for analysis of the role of galaxy mergers in triggering AGN. Most galaxies are found to be best ﬁt by exponentially declining light proﬁles, indicative of disk dominated morphologies (Lin & Pringle 1987), in contradiction to Figure 8. The fraction of galaxies classiﬁed by humans as mergers (upper panel) or disturbed (lower panel) in bins of point source absolute magnitude. post-merger morphologies as predicted by some models (Toomre 1 sigma conﬁdence levels are calculated according to the prescription of & Toomre 1972; Barnes 1988; Di Matteo et al. 2005; Cox et al. Cameron (2011). 2008). It is found that the fractions of AGN and control galaxies are equivalent for all residual ﬂux values. Similarly, the fractions of AGN and control galaxies in diﬀerent visual classes are equivalent. In addition, no correlation is found between AGN luminosity and slightly reducing the robustness of the classiﬁcation, gives a factor residual ﬂux, and visual inspection of morphology yields no indi- of two enhancement of AGN in morphologically disrupted galaxies cation that more luminous AGN are hosted by merging galaxies. at 98.5% conﬁdence level. This provides evidence that AGN are Residual ﬂux values are approximately constant across the whole more common in disturbed galaxies at high redshift, where mergers range of redshifts, but there is a 2.4 enhancement of the visually are most common (Cedric & Shaun 1994; Fakhouri et al. 2010), but σ classiﬁed merger fraction of AGN hosts relative to control galaxies at lower redshifts other processes are dominant. This enhancement of merger fraction at z∼2 has not been found in other studies (Gabor at z∼2. This may imply a possible evolution in the dominance of merger-triggering through time. These results will now be discussed et al. 2009; Allevato et al. 2011; Cisternas et al. 2011; Schawin- in the context of the wider literature. ski et al. 2011; Kocevski et al. 2012; Treister et al. 2012; Villforth Firstweconsidertheimpactofselectionbiasontheresults.The et al. 2014; Bruce et al. 2015; Villforth et al. 2017). This is the AGN studied here are X-ray selected. Are these representative of the largest study to date of the morphologies of high redshift luminous underlying AGN population? It has been argued that merger induced AGN. The better number statistics may allow for detection of this AGN are more likely to be Compton-thick (Cattaneo et al. 2005; mild enhancement; this possibility is explored in some detail in the Kocevski et al. 2015), therefore selecting AGN in the X-ray biases discussion below. against observing merger signatures contemporaneously with the 5 DISCUSSION AGNemission,whichisobscured.Itisimportanttobearthisinmind In this study 106 high luminosity AGN in COSMOS are host when interpreting these results, but also to recall biases associated galaxy mass and redshift matched to 486 control galaxies to com- with other selection methods. Infrared (IR) emission, for instance, pare their morphological properties in order to attempt to answer allows for selection of AGN by relying on the approximately power- the question: are mergers a dominant AGN triggering mechanism? lawspectrumofAGNintheIR(Sternetal.2012).Thiscan,however, galfit models of galaxy light proﬁles are made. Models are sub- biasinfavourofﬁndingmergers,sincemergerscaninducestarbursts tracted from images, and the residual ﬂux serves as a measure of (Mihos & Hernquist 1994, 1996; Hopkins et al. 2006) which emit MNRAS 000, 1–15 (2017) ##LTFigure## ##LTFigure## ##LTFigure##

text: AGN

text: merger

text: enhancement

text: at

text: z

text: Figure

text: Relation

text: between

text: redshift

text: and

text: residual

text: ﬂux.

text: As

text: in

text: other

text: plots

text: blue

text: points

text: are

text: control

text: galaxies;

text: small

text: red

text: triangles

text: are

text: AGN;

text: and

text: the

text: large

text: blue

text: triangles

text: and

text: large

text: red

text: stars

text: represent

text: the

text: median

text: values

text: of

text: controls

text: and

text: AGN

text: respectively,

text: with

text: associated

text: bootstrap

text: errors

text: for

text: diﬀerent

text: bins

text: of

text: redshift.

text: As

text: with

text: ﬁgure

text: all

text: residual

text: ﬂux

text: values

text: have

text: had

text: added

text: for

text: clarity;

text: this

text: is

text: merely

text: because

text: of

text: points

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text: negative

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text: once

text: the

text: erroneous

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text: been

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text: these

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text: The

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text: a

text: uniform

text: addition.

text: Errors

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text: the

text: galaxies.

text: Galaxies

text: are

text: visually

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text: by

text: three

text: experts,

text: identifying

text: likely

text: mergers.

text: Comparison

text: to

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text: group

text: of

text: mass

text: and

text: redshift

text: matched

text: control

text: galaxies

text: allows

text: for

text: analysis

text: of

text: the

text: role

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text: mergers

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text: triggering

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text: Most

text: galaxies

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text: best

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text: exponentially

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text: (Lin

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text: contradiction

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text: Figure

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text: fraction

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text: galaxies

text: classiﬁed

text: by

text: humans

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text: mergers

text: (upper

text: panel)

text: or

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text: (lower

text: panel)

text: in

text: bins

text: of

text: point

text: source

text: absolute

text: magnitude.

text: post-merger

text: morphologies

text: as

text: predicted

text: by

text: some

text: models

text: (Toomre

text: sigma

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text: fractions

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text: addition,

text: no

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text: AGN

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text: robustness

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text: classiﬁcation,

text: gives

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text: enhancement

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text: redshift,

text: where

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text: redshifts,

text: but

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text: the

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text: merger-triggering

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text: time.

text: These

text: results

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text: discussed

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text: literature.

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text: therefore

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text: discussion

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text: against

text: observing

text: merger

text: signatures

text: contemporaneously

text: with

text: the

text: DISCUSSION

text: In

text: this

text: study

text: high

text: luminosity

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text: COSMOS

text: are

text: host

text: when

text: interpreting

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text: but

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text: recall

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text: with

text: other

text: selection

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text: emission,

text: for

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text: their

text: morphological

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text: SEMI-SIMPLICIAL SPACES 13 the maps h and h − are weak homotopy equivalences, and it follows that the right square is p p 1 homotopy cartesian. The lower square in (cid:47)(cid:47) (cid:107) (cid:107) X X• 0 (cid:107) (cid:107) (cid:15)(cid:15) h0 (cid:15)(cid:15) h• (cid:47)(cid:47) (cid:107) (cid:107) Z Z• 0 (cid:107) (cid:107) g0 g• (cid:15)(cid:15) (cid:15)(cid:15) (cid:47)(cid:47) (cid:107) (cid:107) Y Y• 0 is homotopy cartesian by the ﬁrst part of the proof, and the upper square is homotopy cartesian (cid:107) (cid:107) (cid:3) as h and h• are both weak equivalences, by Theorem 2.2. 0 → → Lemma 2.13. Let (cid:15) : X• X− and (cid:15) : Y• Y− be augmented semi-simplicial spaces and let → 1 1 ≥ (f•, f) : (X•, X− ) (Y•, Y− ) be a map of augmented semi-simplicial spaces. If for each p 0 1 1 the square X fp (cid:47)(cid:47) Y p p (cid:15)p (cid:15)p (cid:15)(cid:15) (cid:15)(cid:15) f (cid:47)(cid:47) X− Y− 1 1 is homotopy cartesian, then so is the square (cid:107) (cid:107) (cid:107)f•(cid:107) (cid:47)(cid:47) (cid:107) (cid:107) X• Y• (cid:107) (cid:107) (cid:107) (cid:107) (cid:15)• (cid:15)• (cid:15)(cid:15) (cid:15)(cid:15) f (cid:47)(cid:47) X− Y− . 1 1 Proof. The diagram X di (cid:47)(cid:47) X − (cid:15)p−1 (cid:47)(cid:47) X− p p 1 1 fp fp−1 f (cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15) Y di (cid:47)(cid:47) Y − (cid:15)p−1 (cid:47)(cid:47) Y− p p 1 1 has right-hand and outer squares homotopy cartesian by hypothesis, so the left-hand square is also homotopy cartesian. Thus the map f• is homotopy cartesian and we can apply Theorem 2.12, which shows that the left-hand square in X ι (cid:47)(cid:47) (cid:107)X•(cid:107) (cid:107)(cid:15)•(cid:107) (cid:47)(cid:47) X− 0 1 (cid:107) (cid:107) Y(cid:15)(cid:15) f0 ι (cid:47)(cid:47) (cid:107)Y(cid:15)(cid:15)•(cid:107)f• (cid:107)(cid:15)•(cid:107) (cid:47)(cid:47) Y−(cid:15)(cid:15) f 0 1 (cid:107) (cid:107) ◦ is homotopy cartesian. As (cid:15)• ι = (cid:15) the outer square is homotopy cartesian by hypothesis, and → (cid:107) (cid:107) 0 (cid:3) ι : Y Y• is 0-connected, so the right-hand square is also homotopy cartesian as required. 0 → → Lemma 2.14. Let (cid:15)• : X• X− be an augmented semi-simplicial space such that each (cid:15) : X 1 ∈ p p X− is a quasiﬁbration. Then for each x X− , the natural map 1 1 (cid:107) − (cid:107) −→ (cid:107) (cid:107) (cid:15)• 1(x) hoﬁb (cid:15)• x is a weak homotopy equivalence. ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: SEMI-SIMPLICIAL

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text: 16 JOHANNES EBERT AND OSCAR RANDAL-WILLIAMS (cid:32) (cid:33)   the maps g give a chain contract(cid:88)ion of (E1 ∗, d1), as we ha(cid:88)ve q+1 p,   q q+1 − − (g )∗d1 + d1(g )∗ = ( 1)i(g )∗(d )∗ + ( 1)j(d )∗(g )∗ q q+1 q (cid:88)i j q+1 i=0 j=0 q − − = (d )∗(g )∗ + ( 1)i ((g )∗(d )∗ (d )∗(g )∗) 0 q+1 q i i+1 q+1 i=0 = Id (cid:107) (cid:107) Z Thus E∗2 ∗ = 0 and hence H∗(PX− , PX• ; ) = 0 as claimed. , (cid:107) (cid:107) 1 (cid:48) To show that PX• is simply-connected, let PX• be obtained by collapsing down the 0- simplices of PX• to a point. Consider the map of homotopy coﬁbre sequences (cid:47)(cid:47) (cid:107) (cid:107) (cid:47)(cid:47) (cid:107) (cid:48) (cid:107) X PX• PX• 1 (cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15) (cid:47)(cid:47) (cid:107) (cid:107) (cid:47)(cid:47) (cid:107) (cid:48) (cid:107) π X π PX• π PX• 0 1 0 0 → (cid:48) → (cid:48) − The map X π X is 1-connected. The map PX π (PX ) is (2 p)-connected, so (cid:107) (cid:48) (cid:107) → (cid:107)1 (cid:48) (cid:107)0 1 p 0 p PX• π PX• is 2-connected by Lemma 2.4. The semi-simplicial set π PX• is in bijec- 0 0 tion with π (X )p+1 in degree p, and can be identiﬁed with E•π (X ) for the group π (X ), so (cid:107) (cid:107) (cid:39)0 ∗ 1 → (cid:107) (cid:107) 0 1 (cid:55)→ 0 1 ∈ π PX• . Now the map X PX• is nullhomotopic (it is homotopic to x µ(x, e) 0⊂ (cid:107) (cid:107) 1 (cid:55)→ ∈ ⊂ (cid:107) (cid:107) X PX• which in turn is homotopic to x π (x, e) = e X PX• ), so the middle map 1 2 1 (cid:107) (cid:107) is a retract of the right-hand map, so is also an isomorphism on fundamental groups. Thus PX• (cid:3) is simply-connected. Remark 2.17. If X• is a simplicial space satisfying (i) and (ii) of Theorem 2.16, then instead of the hypothesis of (iv) it is enough to just ask for the square X d1 (cid:47)(cid:47) X 2 1 (cid:15)(cid:15) d2 (cid:15)(cid:15) d1 X d1 (cid:47)(cid:47) X 1 0 to be homotopy cartesian, for the same conclusion to hold. This is because the maps h = s : → p+1 p+1 PX = X PX = X form a system of extra degeneracies, so Lemma 1.12 shows that p p+1 p+1(cid:107) p(cid:107)+→2 the augmentation map PX• X is a weak homotopy equivalence, and we have assumed that (cid:39) ∗ 0 X . 0 3. (Non-unital) topological categories C C C Deﬁnition 3.1. A non-unital topological category consists of an object space = Ob( ), a C C 0 morphism space = Mor( ) and three maps 1 C −→ C C × C { ∈ C × C | −→ C s, t : and m : C := (f, g) t(f) = s(g) , 1 0 1 0 1 1 1 1 such that m(m(f, g), h) = m(f, m(g, h)); t(m(f, g)) = t(g); s(m(f, g)) = s(f) for all f, g, h for which these expressions are deﬁned. One thinks of s as the map associating to a morphism its source, t as the map associating to a ◦ morphism its target, and m as the composition of morphisms, whence we write g f := m(f, g). C − C → D We write (b , b ) := (s, t) 1(b , b ) for the space of morphisms from b to b . A functor F : 0 1 0 1 0 1C → D between non-unital topological categories is a pair of continuous maps F : , i = 0, 1 such i i i ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: JOHANNES

text: EBERT

text: AND

text: OSCAR

text: RANDAL-WILLIAMS

text: the

text: maps

text: g

text: give

text: a

text: chain

text: of

text: as

text: we

text: p,

text: q

text: (g

text: )∗(d

text: )∗(g

text: q

text: q

text: j

text: q

text: (d

text: )∗(g

text: ((g

text: )∗(d

text: (d

text: )∗(g

text: q

text: i

text: Id

text: Z

text: Thus

text: and

text: hence

text: H∗(PX−

text: PX•

text: as

text: claimed.

text: To

text: show

text: that

text: PX•

text: is

text: simply-connected,

text: let

text: PX•

text: be

text: obtained

text: by

text: collapsing

text: down

text: the

text: simplices

text: of

text: PX•

text: to

text: a

text: point.

text: Consider

text: the

text: map

text: of

text: homotopy

text: coﬁbre

text: sequences

text: X

text: PX•

text: PX•

text: π

text: X

text: π

text: PX•

text: π

text: PX•

text: The

text: map

text: X

text: π

text: X

text: is

text: The

text: map

text: PX

text: π

text: (PX

text: is

text: p)-connected,

text: so

text: p

text: p

text: PX•

text: π

text: PX•

text: is

text: by

text: Lemma

text: The

text: semi-simplicial

text: set

text: π

text: PX•

text: is

text: in

text: bijec-

text: tion

text: with

text: π

text: (X

text: in

text: degree

text: p,

text: and

text: can

text: be

text: identiﬁed

text: with

text: E•π

text: (X

text: for

text: the

text: group

text: π

text: (X

text: so

text: π

text: PX•

text: Now

text: the

text: map

text: X

text: PX•

text: is

text: nullhomotopic

text: (it

text: is

text: homotopic

text: to

text: x

text: µ(x,

text: e)

text: 0⊂

text: X

text: PX•

text: which

text: in

text: turn

text: is

text: homotopic

text: to

text: x

text: π

text: (x,

text: e)

text: e

text: X

text: PX•

text: so

text: the

text: middle

text: map

text: is

text: a

text: retract

text: of

text: the

text: right-hand

text: map,

text: so

text: is

text: also

text: an

text: isomorphism

text: on

text: fundamental

text: groups.

text: Thus

text: PX•

text: is

text: simply-connected.

text: Remark

text: If

text: X•

text: is

text: a

text: simplicial

text: space

text: satisfying

text: (i)

text: and

text: (ii)

text: of

text: Theorem

text: then

text: instead

text: of

text: the

text: hypothesis

text: of

text: (iv)

text: it

text: is

text: enough

text: to

text: just

text: ask

text: for

text: the

text: square

text: X

text: X

text: X

text: X

text: to

text: be

text: homotopy

text: cartesian,

text: for

text: the

text: same

text: conclusion

text: to

text: hold.

text: This

text: is

text: because

text: the

text: maps

text: h

text: s

text: PX

text: X

text: PX

text: X

text: form

text: a

text: system

text: of

text: extra

text: degeneracies,

text: so

text: Lemma

text: shows

text: that

text: p

text: the

text: augmentation

text: map

text: PX•

text: X

text: is

text: a

text: weak

text: homotopy

text: equivalence,

text: and

text: we

text: have

text: assumed

text: that

text: X

text: (Non-unital)

text: topological

text: categories

text: C

text: C

text: C

text: Deﬁnition

text: A

text: non-unital

text: topological

text: category

text: consists

text: of

text: an

text: object

text: space

text: Ob(

text: a

text: C

text: C

text: morphism

text: space

text: Mor(

text: and

text: three

text: maps

text: C

text: C

text: C

text: C

text: C

text: C

text: C

text: s,

text: t

text: and

text: m

text: C

text: (f,

text: g)

text: t(f)

text: s(g)

text: such

text: that

text: m(m(f,

text: g),

text: h)

text: m(f,

text: m(g,

text: h));

text: t(m(f,

text: g))

text: t(g);

text: s(m(f,

text: g))

text: s(f)

text: for

text: all

text: f,

text: g,

text: h

text: for

text: which

text: these

text: expressions

text: are

text: deﬁned.

text: One

text: thinks

text: of

text: s

text: as

text: the

text: map

text: associating

text: to

text: a

text: morphism

text: its

text: source,

text: t

text: as

text: the

text: map

text: associating

text: to

text: a

text: morphism

text: its

text: target,

text: and

text: m

text: as

text: the

text: composition

text: of

text: morphisms,

text: whence

text: we

text: write

text: g

text: f

text: m(f,

text: g).

text: C

text: C

text: D

text: We

text: write

text: (b

text: b

text: (s,

text: t)

text: b

text: for

text: the

text: space

text: of

text: morphisms

text: from

text: b

text: to

text: b

text: A

text: functor

text: F

text: D

text: between

text: non-unital

text: topological

text: categories

text: is

text: a

text: pair

text: of

text: continuous

text: maps

text: F

text: i

text: such

text: i

text: i

text: i

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: SEMI-SIMPLICIAL SPACES 25 C → C The left-hand maps deﬁne s, t : X X, and the universal property of the pullback provides C × C → C 1 0 a map c : X C X X; this deﬁnes a non-unital topological category, and the F deﬁne a 1 0X 1 C →1 C C C i continuous functor F : X . (If had units, then X does too.) C Theorem 5.2. If is ﬁbrant and has weak right (or left) units, and f is 0-connected, then C → C BF : B X B is a weak equivalence. C C Proof. We consider the resolution (F/ )• • of the functor F. As is left ﬁbrant and has weak right , (say) units, it has soft right units by Lemma 3.13, and so Corollary 4.5 applies and shows that (cid:107) (cid:107) (cid:107) C (cid:107) → C (cid:107) (cid:107) (cid:107) C (cid:107) → (cid:15)• • : (F/ )• • B X is a weak equivalence. It remains to show that η• • : (F/ )• • C, , , , B is a weak equivalence. C The space (F/ ) ﬁts in to a cartesian square 0,0 C (cid:47)(cid:47) C (F/ ) 0,0 1 × × (cid:15)(cid:15) (cid:15)0,0 η0,0 (cid:15)(cid:15) s t CX × C F0×id(cid:47)(cid:47) C × C , 0 0 0 0 C and as is ﬁbrant the right-hand vertical map is a ﬁbration, and so η is a ﬁbration too. Fur- C C 0,0 thermore, as is ﬁbrant, (5.1) shows that X is too. Hence by applying Lemma 4.6 (ii) then (i), → · · · → ∈ C each η is a ﬁbration. Hence, by Lemma 2.14, for each b = (b b ) the map p,q 0 q q C (cid:107) − (cid:107) −→ (cid:107) (cid:107) B( X/b ) = η• 1(b) hoﬁb η• 0 ,q b ,q C is a weak equivalence, so it is enough that the over-categories B( X/b ) be contractible for some ∈ C → C 0 object b in each path-component. As f : X is 0-connected, we may suppose that 0 0 C C 1 C b = F(x ), but in this case X/F(x ) = X/x , by (5.1), so it is enough to show that X has soft 0 0 C 0 0C right units. As has weak right units so does X (by Remark 3.12 and because both categories C (cid:3) are ﬁbrant), so by Lemma 3.13 X has soft right units as required. C C A typical application of this result is to take X = δ to be the set of objects of with the C → C 0 discrete topology, and f : δ to be the identity function (which is 0-connected). This yields C 0 0 a category δ with discrete space of objects but the same space of maps between any two objects, which has a homotopy equivalent classifying spaces under the conditions given above. 6. The Group-Completion Theorem We shall take care to formulate and prove the group-completion theorem, and the main tech- nical result underlying it, for homology with local coeﬃcients. We therefore make the following deﬁnitions. L Deﬁnition 6.1. Let be a local coeﬃcient system of R-modules on a space X. L ∈ → L (i) The monodromy of at x X if the homomorphism µ : π (X, x) Aut ( (x)) L x 1 R-Mod induced from . L (ii) is called constant if all monodromy homomorphisms are trivial. L (iii) is called abelian if the images of all monodromy homomorphisms are abelian groups. A Assumptions 6.2. In the sequel, let be either (i) the class of constant local coeﬃcient systems of R-modules, or (ii) the class of abelian local coeﬃcient systems of R-modules, or (iii) the class of all local coeﬃcient systems of R-modules. ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: SEMI-SIMPLICIAL

text: SPACES

text: C

text: C

text: The

text: left-hand

text: maps

text: deﬁne

text: s,

text: t

text: X

text: X,

text: and

text: the

text: universal

text: property

text: of

text: the

text: pullback

text: provides

text: C

text: C

text: C

text: a

text: map

text: c

text: X

text: C

text: X

text: X;

text: this

text: deﬁnes

text: a

text: non-unital

text: topological

text: category,

text: and

text: the

text: F

text: deﬁne

text: a

text: C

text: →1

text: C

text: C

text: C

text: i

text: continuous

text: functor

text: F

text: X

text: (If

text: had

text: units,

text: then

text: X

text: does

text: too.)

text: C

text: Theorem

text: If

text: is

text: ﬁbrant

text: and

text: has

text: weak

text: right

text: (or

text: left)

text: units,

text: and

text: f

text: is

text: then

text: C

text: C

text: BF

text: B

text: X

text: B

text: is

text: a

text: weak

text: equivalence.

text: C

text: C

text: Proof.

text: We

text: consider

text: the

text: resolution

text: (F/

text: of

text: the

text: functor

text: F.

text: As

text: is

text: left

text: ﬁbrant

text: and

text: has

text: weak

text: right

text: (say)

text: units,

text: it

text: has

text: soft

text: right

text: units

text: by

text: Lemma

text: and

text: so

text: Corollary

text: applies

text: and

text: shows

text: that

text: C

text: C

text: C

text: (F/

text: B

text: X

text: is

text: a

text: weak

text: equivalence.

text: It

text: remains

text: to

text: show

text: that

text: η•

text: (F/

text: C,

text: B

text: is

text: a

text: weak

text: equivalence.

text: C

text: The

text: space

text: (F/

text: ﬁts

text: in

text: to

text: a

text: cartesian

text: square

text: C

text: C

text: (F/

text: s

text: t

text: CX

text: C

text: C

text: C

text: C

text: and

text: as

text: is

text: ﬁbrant

text: the

text: right-hand

text: vertical

text: map

text: is

text: a

text: ﬁbration,

text: and

text: so

text: η

text: is

text: a

text: ﬁbration

text: too.

text: Fur-

text: C

text: C

text: thermore,

text: as

text: is

text: ﬁbrant,

text: shows

text: that

text: X

text: is

text: too.

text: Hence

text: by

text: applying

text: Lemma

text: (ii)

text: then

text: (i),

text: C

text: each

text: η

text: is

text: a

text: ﬁbration.

text: Hence,

text: by

text: Lemma

text: for

text: each

text: b

text: (b

text: b

text: the

text: map

text: p,q

text: q

text: q

text: C

text: B(

text: X/b

text: η•

text: hoﬁb

text: η•

text: ,q

text: b

text: ,q

text: C

text: is

text: a

text: weak

text: equivalence,

text: so

text: it

text: is

text: enough

text: that

text: the

text: over-categories

text: B(

text: X/b

text: be

text: contractible

text: for

text: some

text: C

text: C

text: object

text: b

text: in

text: each

text: path-component.

text: As

text: f

text: X

text: is

text: we

text: may

text: suppose

text: that

text: C

text: C

text: C

text: b

text: F(x

text: but

text: in

text: this

text: case

text: X/F(x

text: X/x

text: by

text: so

text: it

text: is

text: enough

text: to

text: show

text: that

text: X

text: has

text: soft

text: C

text: right

text: units.

text: As

text: has

text: weak

text: right

text: units

text: so

text: does

text: X

text: (by

text: Remark

text: and

text: because

text: both

text: categories

text: C

text: are

text: ﬁbrant),

text: so

text: by

text: Lemma

text: X

text: has

text: soft

text: right

text: units

text: as

text: required.

text: C

text: C

text: A

text: typical

text: application

text: of

text: this

text: result

text: is

text: to

text: take

text: X

text: δ

text: to

text: be

text: the

text: set

text: of

text: objects

text: of

text: with

text: the

text: C

text: C

text: discrete

text: topology,

text: and

text: f

text: δ

text: to

text: be

text: the

text: identity

text: function

text: (which

text: is

text: This

text: yields

text: C

text: a

text: category

text: δ

text: with

text: discrete

text: space

text: of

text: objects

text: but

text: the

text: same

text: space

text: of

text: maps

text: between

text: any

text: two

text: objects,

text: which

text: has

text: a

text: homotopy

text: equivalent

text: classifying

text: spaces

text: under

text: the

text: conditions

text: given

text: above.

text: The

text: Group-Completion

text: Theorem

text: We

text: shall

text: take

text: care

text: to

text: formulate

text: and

text: prove

text: the

text: group-completion

text: theorem,

text: and

text: the

text: main

text: tech-

text: nical

text: result

text: underlying

text: it,

text: for

text: homology

text: with

text: local

text: coeﬃcients.

text: We

text: therefore

text: make

text: the

text: following

text: deﬁnitions.

text: L

text: Deﬁnition

text: Let

text: be

text: a

text: local

text: coeﬃcient

text: system

text: of

text: R-modules

text: on

text: a

text: space

text: X.

text: L

text: L

text: (i)

text: The

text: monodromy

text: of

text: at

text: x

text: X

text: if

text: the

text: homomorphism

text: µ

text: π

text: (X,

text: x)

text: Aut

text: (x))

text: L

text: x

text: R-Mod

text: induced

text: from

text: L

text: (ii)

text: is

text: called

text: constant

text: if

text: all

text: monodromy

text: homomorphisms

text: are

text: trivial.

text: L

text: (iii)

text: is

text: called

text: abelian

text: if

text: the

text: images

text: of

text: all

text: monodromy

text: homomorphisms

text: are

text: abelian

text: groups.

text: A

text: Assumptions

text: In

text: the

text: sequel,

text: let

text: be

text: either

text: (i)

text: the

text: class

text: of

text: constant

text: local

text: coeﬃcient

text: systems

text: of

text: R-modules,

text: or

text: (ii)

text: the

text: class

text: of

text: abelian

text: local

text: coeﬃcient

text: systems

text: of

text: R-modules,

text: or

text: (iii)

text: the

text: class

text: of

text: all

text: local

text: coeﬃcient

text: systems

text: of

text: R-modules.

text: ##LTLine##

text: ##LTLine##

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text: SEMI-SIMPLICIAL SPACES 27 Lemma 6.8. If Y• is a simplicial set with contractible geometric realisation then it is basic. ∈ (cid:107) (cid:107) Proof. The proof only uses the semi-simplicial structure. Let y Y be a basepoint. Since Y• is (cid:107) (cid:107) → (cid:107) (cid:107) 0 contractible, the natural map η : hoﬁb ( f• ) X• is a weak equivalence. Hence any coeﬃcient L(cid:48) (cid:107) (cid:107) y ∗L (cid:107) (cid:107) L(cid:48) system on hoﬁb ( f• ) is of the form η for a coeﬃcient system on X• , and if lies in the A y L ∈ class , then so does . Therefore, we have to prove that for each point y Y , the inclusion map − → (cid:107) (cid:107) − ∗L → (cid:107) (cid:107) L 0 j : f 1(y) X• induces an isomorphism H∗(f 1(y); j ) H∗( X• ; ). 0 0 L The spectral sequence of the semi-simplicial space X• with coeﬃcients in discussed in Section takes the form L ⇒ (cid:107) (cid:107) L E1 = H (X ; ) H ( X• ; ). p,q q p p p+q (cid:77) Since Y is discrete, we can write the E1-term as p L − L | H (X ; ) = H (f 1(s); − ). q p p q p f 1(s) ∈ s Yp − L − L | To simplify notation, we write H (f 1(s); ) := H (f 1(s); − ). Because the map f• is A − qL → p − q L p f 1(s) -cartesian, the map H (f 1(s); ) H (f 1(d s); − ) induced by the face map d is an (cid:55)→q − p L q i p 1 L i isomorphism. Hence s H (f 1(s); ) is a locally constant coeﬃcient system H (f; ) on the q (cid:107) p(cid:107) L (cid:107) (cid:107) q simplicial set Y•. Hence E2 = H ( Y• ; H (f; )). Because Y• is contractible, it follows that ∈ p,q p q → E2 = 0 for p > 0. If y Y is a basepoint, the induced map ∆0• Y• of simplicial sets gives a p,q 0 comparison diagram − (cid:47)(cid:47) f 1(s) X• f• (cid:15)(cid:15) (cid:15)(cid:15) y (cid:47)(cid:47) ∆0• Y•. − → (cid:107) (cid:107) It induces an isomorphism on the E2-term of the spectral sequence, and therefore f 1(y) X• L (cid:3) induces an isomorphism in homology with coeﬃcients in , as claimed. The next step is a discretisation argument. For a simplicial space Y•, we consider the bi-simplicial (cid:55)→ set (p, q) Sing Y and the associated diagonal simplicial set δY := Sing Y . By Theorem 7.1, q p p p p Lemma 1.11 and Theorem 2.2, the maps (cid:107) (cid:107) −→ (cid:107) (cid:107) −→ (cid:107) (cid:107) δY• D Sing•Y• Y• are weak equivalences. Lemma 6.9. If Y• is a simplicial space such that δY• is basic, then Y• is basic. Proof. The proof uses the simplicial structure in an essential way. As in the second proof of (cid:96) → Lemma 2.12 we may assume that f : X Y is a ﬁbration for each p. Let Y := Sing Y , p p p p,q → q p giving a bi-simplicial set Y• •, and deﬁne a bi-simplicial space X• • and a map f• • : X• • Y• • , , , , → , as follows. Let X := ∈ Lift(σ, f ), where Lift(σ, f ) is the space of all maps h : ∆q X ◦ p,q σ Yp,q p p p with f h = σ, equipped with the compact-open topology. The simplicial structure in the p p (cid:55)→ ◦ (cid:55)→ ◦ direction is given by h d h and in the q-direction by h h dj (similarly for the degeneracy i → maps). The evident maps f : X Y are the components of a bi-simplicial map. Because −p,q →p,q − p,q f is a ﬁbration, the map f 1(σ) f 1− (d σ) is a weak equivalence, for each q and i. Hence the p →p,q p,q 1 i simplicial map f • : X • Y • is homotopy cartesian. p, p, p, (cid:107) (cid:107) → (cid:107) (cid:107) → Analogous to the evaluation map Y • Y , let u : X • X be the map which sends ∈ × ∈ p, p p p, p (h, t) X ∆q to h(t) X . These are the components of a map of simplicial spaces, and the p,q p ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: SEMI-SIMPLICIAL

text: SPACES

text: Lemma

text: If

text: Y•

text: is

text: a

text: simplicial

text: set

text: with

text: contractible

text: geometric

text: realisation

text: then

text: it

text: is

text: basic.

text: Proof.

text: The

text: proof

text: only

text: uses

text: the

text: semi-simplicial

text: structure.

text: Let

text: y

text: Y

text: be

text: a

text: basepoint.

text: Since

text: Y•

text: is

text: contractible,

text: the

text: natural

text: map

text: η

text: hoﬁb

text: f•

text: X•

text: is

text: a

text: weak

text: equivalence.

text: Hence

text: any

text: coeﬃcient

text: y

text: ∗L

text: system

text: on

text: hoﬁb

text: f•

text: is

text: of

text: the

text: form

text: η

text: for

text: a

text: coeﬃcient

text: system

text: on

text: X•

text: and

text: if

text: lies

text: in

text: the

text: A

text: y

text: L

text: class

text: then

text: so

text: does

text: Therefore,

text: we

text: have

text: to

text: prove

text: that

text: for

text: each

text: point

text: y

text: Y

text: the

text: inclusion

text: map

text: ∗L

text: L

text: j

text: f

text: X•

text: induces

text: an

text: isomorphism

text: H∗(f

text: j

text: H∗(

text: X•

text: L

text: The

text: spectral

text: sequence

text: of

text: the

text: semi-simplicial

text: space

text: X•

text: with

text: coeﬃcients

text: in

text: discussed

text: in

text: Section

text: takes

text: the

text: form

text: L

text: L

text: H

text: (X

text: H

text: X•

text: p,q

text: q

text: p

text: p

text: p+q

text: Since

text: Y

text: is

text: discrete,

text: we

text: can

text: write

text: the

text: as

text: p

text: L

text: L

text: H

text: (X

text: H

text: (f

text: q

text: p

text: p

text: q

text: p

text: f

text: s

text: Yp

text: L

text: L

text: To

text: simplify

text: notation,

text: we

text: write

text: H

text: (f

text: H

text: (f

text: Because

text: the

text: map

text: f•

text: is

text: A

text: qL

text: p

text: q

text: L

text: p

text: f

text: -cartesian,

text: the

text: map

text: H

text: (f

text: H

text: (f

text: s);

text: induced

text: by

text: the

text: face

text: map

text: d

text: is

text: an

text: p

text: L

text: q

text: i

text: p

text: L

text: i

text: isomorphism.

text: Hence

text: s

text: H

text: (f

text: is

text: a

text: locally

text: constant

text: coeﬃcient

text: system

text: H

text: (f;

text: on

text: the

text: q

text: L

text: q

text: simplicial

text: set

text: Y•.

text: Hence

text: H

text: Y•

text: H

text: (f;

text: Because

text: Y•

text: is

text: contractible,

text: it

text: follows

text: that

text: p,q

text: p

text: q

text: for

text: p

text: If

text: y

text: Y

text: is

text: a

text: basepoint,

text: the

text: induced

text: map

text: ∆0•

text: Y•

text: of

text: simplicial

text: sets

text: gives

text: a

text: p,q

text: comparison

text: diagram

text: f

text: X•

text: f•

text: y

text: ∆0•

text: Y•.

text: It

text: induces

text: an

text: isomorphism

text: on

text: the

text: of

text: the

text: spectral

text: sequence,

text: and

text: therefore

text: f

text: X•

text: L

text: induces

text: an

text: isomorphism

text: in

text: homology

text: with

text: coeﬃcients

text: in

text: as

text: claimed.

text: The

text: next

text: step

text: is

text: a

text: discretisation

text: argument.

text: For

text: a

text: simplicial

text: space

text: Y•,

text: we

text: consider

text: the

text: bi-simplicial

text: set

text: (p,

text: q)

text: Sing

text: Y

text: and

text: the

text: associated

text: diagonal

text: simplicial

text: set

text: δY

text: Sing

text: Y

text: By

text: Theorem

text: q

text: p

text: p

text: p

text: p

text: Lemma

text: and

text: Theorem

text: the

text: maps

text: δY•

text: D

text: Sing•Y•

text: Y•

text: are

text: weak

text: equivalences.

text: Lemma

text: If

text: Y•

text: is

text: a

text: simplicial

text: space

text: such

text: that

text: δY•

text: is

text: basic,

text: then

text: Y•

text: is

text: basic.

text: Proof.

text: The

text: proof

text: uses

text: the

text: simplicial

text: structure

text: in

text: an

text: essential

text: way.

text: As

text: in

text: the

text: second

text: proof

text: of

text: Lemma

text: we

text: may

text: assume

text: that

text: f

text: X

text: Y

text: is

text: a

text: ﬁbration

text: for

text: each

text: p.

text: Let

text: Y

text: Sing

text: Y

text: p

text: p

text: p

text: p,q

text: q

text: p

text: giving

text: a

text: bi-simplicial

text: set

text: Y•

text: and

text: deﬁne

text: a

text: bi-simplicial

text: space

text: X•

text: and

text: a

text: map

text: f•

text: X•

text: Y•

text: as

text: follows.

text: Let

text: X

text: Lift(σ,

text: f

text: where

text: Lift(σ,

text: f

text: is

text: the

text: space

text: of

text: all

text: maps

text: h

text: ∆q

text: X

text: p,q

text: σ

text: Yp,q

text: p

text: p

text: p

text: with

text: f

text: h

text: σ,

text: equipped

text: with

text: the

text: compact-open

text: topology.

text: The

text: simplicial

text: structure

text: in

text: the

text: p

text: p

text: direction

text: is

text: given

text: by

text: h

text: d

text: h

text: and

text: in

text: the

text: q-direction

text: by

text: h

text: h

text: dj

text: (similarly

text: for

text: the

text: degeneracy

text: i

text: maps).

text: The

text: evident

text: maps

text: f

text: X

text: Y

text: are

text: the

text: components

text: of

text: a

text: bi-simplicial

text: map.

text: Because

text: −p,q

text: →p,q

text: p,q

text: f

text: is

text: a

text: ﬁbration,

text: the

text: map

text: f

text: f

text: 1−

text: (d

text: σ)

text: is

text: a

text: weak

text: equivalence,

text: for

text: each

text: q

text: and

text: i.

text: Hence

text: the

text: p

text: →p,q

text: p,q

text: i

text: simplicial

text: map

text: f

text: X

text: Y

text: is

text: homotopy

text: cartesian.

text: p,

text: p,

text: p,

text: Analogous

text: to

text: the

text: evaluation

text: map

text: Y

text: Y

text: let

text: u

text: X

text: X

text: be

text: the

text: map

text: which

text: sends

text: p,

text: p

text: p

text: p,

text: p

text: (h,

text: t)

text: X

text: ∆q

text: to

text: h(t)

text: X

text: These

text: are

text: the

text: components

text: of

text: a

text: map

text: of

text: simplicial

text: spaces,

text: and

text: the

text: p,q

text: p

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